http://wiki.math.uwaterloo.ca/statwiki/api.php?action=feedcontributions&user=Wtjung&feedformat=atomstatwiki - User contributions [US]2023-02-01T13:15:41ZUser contributionsMediaWiki 1.28.3http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Semantic_Relation_Classification%E2%80%94%E2%80%94via_Convolution_Neural_Network&diff=49221Semantic Relation Classification——via Convolution Neural Network2020-12-05T17:30:40Z<p>Wtjung: /* Critiques */</p>
<hr />
<div><br />
<br />
<br />
== Presented by ==<br />
Rui Gong, Xinqi Ling, Di Ma,Xuetong Wang<br />
<br />
== Introduction ==<br />
A Semantic Relation can imply a relation between different words and a relation between different sentences or phrases. For example, the pair of words "white" and "snowy" can be synonyms, while "white" and "black" can be opposites. It can be used for recommendation systems like YouTube, and understanding sentiment analysis. The study of semantic analysis involves determining the exact meaning of a text. For example, the word "date", can have different meanings in different contexts, like a calendar "date", a "date" fruit, or a romantic "date". <br />
<br />
One of the emerging trends of natural language technologies is their use for the humanities and sciences (Gbor et al., 2018). SemEval 2018 Task 7 mainly solves the problem of relation extraction and classification of two entities in the same sentence into 6 potential relations. The 6 relations are USAGE, RESULT, MODEL-FEATURE, PART WHOLE, TOPIC, and COMPARE.<br />
<br />
SemEval 2018 Task 7 extracted data from 350 scientific paper abstracts, which has 1228 and 1248 annotated sentences for two tasks, respectively. For each data, an example sentence was chosen with its right and left sentences, as well as an indicator showing whether the relation is reserved, then a prediction is made. <br />
<br />
Three models were used for the prediction: Linear Classifiers, Long Short-Term Memory(LSTM), and Convolutional Neural Networks (CNN). Linear Classifier achieves the goal of classification by making a classification decision based on the value of a linear combination of the characteristics. LSTM is an artificial recurrent neural network (RNN) architecture well suited to classifying, processing and making predictions based on time series data. In the end, the prediction based on the CNN model was ultimately submitted since it performed the best among all models. By using the learned custom word embedding function, the research team added a variant of negative sampling, thereby improving performance and surpassing ordinary CNN.<br />
<br />
== Previous Work ==<br />
SemEval 2010 Task 8 (Hendrickx et al., 2010) explored the classification of natural language relations and studied the 9 relations between word pairs. However, it is not designed for scientific text analysis, and their challenge differs from the challenge of this paper in its generalizability; this paper’s relations are specific to ACL papers (e.g. MODEL-FEATURE), whereas the 2010 relations are more general, and might necessitate more common-sense knowledge than the 2018 relations. Xu et al. (2015a) and Santos et al. (2015) both applied CNN with negative sampling to finish task7. The 2017 SemEval Task 10 also featured relation extraction within scientific publications.<br />
<br />
== Algorithm ==<br />
<br />
[[File:CNN.png|800px|center]]<br />
<br />
This is the architecture of CNN. We first transform a sentence via Feature embeddings. Word representations are encoded by the column vector in the embedding matrix <math> W^{word} \in \mathbb{R}^{d^w \times |V|}</math>, where <math>V</math> is the vocabulary of the dataset. Each colummn is the word embedding vector for the <math>i^{th}</math> word in the vocabulary. This matrix is trainale during the optimization process and initialized by pre-trained emmbedding vectors. Basically, we transform each sentence into continuous word embeddings:<br />
<br />
$$<br />
(e^{w_i})<br />
$$<br />
<br />
And word position embeddings:<br />
$$<br />
(e^{wp_i}): e_i = [e^{w_i}, e^{wp_i}]<br />
$$<br />
<br />
In the word embeddings, we generated a vocabulary <math> V </math>. We will then generate an embedding word matrix based on the position of the word in the vocabulary. This matrix is trainable and needs to be initialized by pre-trained embedding vectors such as through GloVe or Word2Vec.<br />
<br />
In the word position embeddings, we first need to input some words named ‘entities,’ and they are the key for the machine to determine the sentence’s relation. During this process, if we have two entities, we will use the relative position of them in the sentence to make the<br />
embeddings. We will output two vectors, and one of them keeps track of the first entity relative position in the sentence ( we will make the entity recorded as 0, the former word recorded as -1 and the next one 1, etc. ). And the same procedure for the second entity. Finally, we will get two vectors concatenated as the position embedding. For example, in the sentence "the black '''cat''' jumped", the position embedding of "'''cat'''" is -2,-1,0,1.<br />
<br />
<br />
<br />
After the embeddings, the model will transform the embedded sentence into a fix-sized representation of the whole sentence via the convolution layer. Finally, after the max-pooling to reduce the dimension of the output of the layers, we will get a score for each relation class via a linear transformation.<br />
<br />
<br />
After featurizing all words in the sentence. The sentence of length N can be expressed as a vector of length <math> N </math>, which looks like <br />
$$e=[e_{1},e_{2},\ldots,e_{N}]$$<br />
and each entry represents a token of the word. Also, to apply <br />
convolutional neural network, the subsets of features<br />
$$e_{i:i+j}=[e_{i},e_{i+1},\ldots,e_{i+j}]$$<br />
are given to a weight matrix <math> W\in\mathbb{R}^{(d^{w}+2d^{wp})\times k}</math> to <br />
produce a new feature, defiend as <br />
$$c_{i}=\text{tanh}(W\cdot e_{i:i+k-1}+bias)$$<br />
This process is applied to all subsets of features with length <math> k </math> starting <br />
from the first one. Then a mapped feature factor is produced:<br />
$$c=[c_{1},c_{2},\ldots,c_{N-k+1}]$$<br />
<br />
<br />
The max pooling operation is used, the <math> \hat{c}=max\{c\} </math> was picked.<br />
With different weight filter, different mapped feature vectors can be obtained. Finally, the original <br />
sentence <math> e </math> can be converted into a new representation <math> r_{x} </math> with a fixed length. For example, if there are 5 filters,<br />
then there are 5 features (<math> \hat{c} </math>) picked to create <math> r_{x} </math> for each <math> x </math>.<br />
<br />
Then, the score vector <br />
$$s(x)=W^{classes}r_{x}$$<br />
is obtained which represented the score for each class, given <math> x </math>'s entities' relation will be classified as <br />
the one with the highest score. The <math> W^{classes} </math> here is the model being trained.<br />
<br />
To improve the performance, “Negative Sampling" was used. Given the trained data point <br />
<math> \tilde{x} </math>, and its correct class <math> \tilde{y} </math>. Let <math> I=Y\setminus\{\tilde{y}\} </math> represent the <br />
incorrect labels for <math> x </math>. Basically, the distance between the correct score and the positive margin, and the negative <br />
distance (negative margin plus the second largest score) should be minimized. So the loss function is <br />
$$L=\log(1+e^{\gamma(m^{+}-s(x)_{y})})+\log(1+e^{\gamma(m^{-}-\mathtt{max}_{y'\in I}(s(x)_{y'}))})$$<br />
with margins <math> m_{+} </math>, <math> m_{-} </math>, and penalty scale factor <math> \gamma </math>.<br />
The whole training is based on ACL anthology corpus and there are 25,938 papers with 136,772,370 tokens in total, <br />
and 49,600 of them are unique.<br />
<br />
== Results ==<br />
Unlike traditional hyper-parameter optimization, there are some modifications to the model that are necessary in order to increase performance on the test set. There are 5 modifications that can be applied:<br />
<br />
'''1.''' Merged Training Sets. It combined two training sets to increase the data set<br />
size and it improves the equality between classes to get better predictions.<br />
<br />
'''2.''' Reversal Indicate Features. It added a binary feature.<br />
<br />
'''3.''' Custom ACL Embeddings. It embedded a word vector to an ACL-specific<br />
corps.<br />
<br />
'''4.''' Context words. Within the sentence, it varies in size on a context window<br />
around the entity-enclosed text.<br />
<br />
'''5.''' Ensembling. It used different early stop and random initializations to improve<br />
the predictions.<br />
<br />
These modifications performances well on the training data and they are shown<br />
in table 3.<br />
<br />
[[File:table3.PNG|center]]<br />
<br />
<br />
<br />
As we can see the best choice for this model is ensembling as the random initialization made the data more natural and avoided the overfit.<br />
During the training process, there are some methods such that they can only<br />
increase the score on the cross-validation test sets but hurt the performance on<br />
the overall macro-F1 score. Thus, these methods were eventually ruled out.<br />
<br />
<br />
[[File:table4.PNG|center]]<br />
<br />
There are six submissions in total. Three for each training set and the result<br />
is shown in figure 2.<br />
<br />
The best submission for the training set 1.1 is the third submission which does not<br />
use a separate cross-validation dataset. Instead, a constant number of<br />
training epochs are run with cross-validation based on the training data.<br />
<br />
This compares to the other submissions in which 10% of the training data are<br />
removed to form the validation set (both with and without stratification).<br />
Predictions for these are made when the model has the highest validation accuracy. <br />
<br />
The best submission for the training set 1.2 is the submission which<br />
extracted 10% of the training data to form the validation dataset. Predictions are<br />
made when maximum accuracy is reached on the validation data.<br />
<br />
All in all, early stopping cannot always be based on the accuracy of the validation set<br />
since it cannot guarantee to get better performance on the real test set. Thus,<br />
we have to try new approaches and combine them to see the prediction<br />
results. Also, doing stratification will certainly improve the performance of<br />
the test data.<br />
<br />
== Conclusions ==<br />
Throughout the process, we have experimented with linear classifiers, sequential random forest, LSTM, and CNN models with various variations applied, such as two models of attention, negative sampling, entity embedding or sentence-only embedding, etc. <br />
<br />
Among all variations, vanilla CNN with negative sampling and ACL-embedding, without attention, has significantly better performance than all others. Attention-based pooling, up-sampling, and data augmentation are also tested, but they barely perform positive increment on the behavior.<br />
<br />
== Critiques == <br />
<br />
- Applying this in news apps might be beneficial to improve readability by highlighting specific important sections.<br />
<br />
- The data set come from 350 scientist papers, this could be more explained by the author on how those paper are selected and why those paper are important to discuss.<br />
<br />
- In the section of previous work, the author mentioned 9 natural language relationships between the word pairs. Among them, 6 potential relationships are USAGE, RESULT, MODEL-FEATURE, PART WHOLE, TOPIC, and COMPARE. It would help the readers to better understand if all 9 relationships are listed in the summary.<br />
<br />
-This topic is interesting and this application might be helpful for some educational websites to improve their website to help readers focus on the important points. I think it will be nice to use Latex to type the equation in the sentence rather than center the equation on the next line. I think it will be interesting to discuss applying this way to other languages such as Chinese, Japanese, etc.<br />
<br />
- It would be a good idea if the authors can provide more details regarding ACL Embeddings and Context words modifications. Scores generated using these two modifications are quite close to the highest Ensembling modification generated score, which makes it a valid consideration to examine these two modifications in detail.<br />
<br />
- This paper is dealing with a similar problem as 'Neural Speed Reading Via Skim-RNN', num 19 paper summary. It will be an interesting approach to compare these two models' performance based on the same dataset.<br />
<br />
- I think it would be highly practical to implement this system as a page-rank system for search engines (such as google, bing, or other platforms like Facebook, Instagram, etc.) by finding the most prevalent information available in a search query and then matching the search to the related text which can be found on webpages. This could also be implemented in search bars on specific websites or locations as well.<br />
<br />
- It would be interesting to see in the future how the model would behave if data not already trained was used. This pre-trained data as mentioned in the paper had noise included. Using cleaner data would give better results maybe.<br />
<br />
- The selection of the training dataset, i.e. the abstracts of scientific papers, is an excellent idea since the abstracts usually contain more information than the body. But it may be also a good idea to train the model with the conclusions. Other than that, the result of applying the model to the body part of the scientific papers may show some interesting features of the model.<br />
<br />
- From Table 4 we find that comparing with using a fixed number of training periods, early stopping based on the accuracy of the validation set does not guarantee better test set performance. The label ratio of the validation set is layered according to the training set, which helps to improve the performance of the test set. Whether it is beneficial to add entity embedding as an additional feature could be an interesting point of discussion.<br />
<br />
- The author mentioned the use of CNNs for contextual understanding. NLP models based on CNNs have sometimes been inadequate for the use of understanding the context of words used in a body of text, being very prone to overfitting the context of the training data. It would help if more evidence was shown as to how the paper deals with this problem.<br />
<br />
- Deep neural network with a complex structure and huge parameter set is good at fitting the model. However, overfitting is a problem. Some strategies can be discussed like dropout strategies. Also, the choice of the most appropriate number of hidden layers is related to many factors like the scale of the training corpus. This can also be talked about in the paper.<br />
<br />
- It will be interesting to see: since it is CNN, applying dropout can improve the performance of the model like the one introduced in this paper [https://arxiv.org/pdf/1207.0580.pdf].<br />
<br />
== References ==<br />
[1] Diederik P Kingma and Jimmy Ba. 2014. Adam: A<br />
method for stochastic optimization. arXiv preprint<br />
arXiv:1412.6980.<br />
<br />
[2] DragomirR. Radev, Pradeep Muthukrishnan, Vahed<br />
Qazvinian, and Amjad Abu-Jbara. 2013. The ACL<br />
anthology network corpus. Language Resources<br />
and Evaluation, pages 1–26.<br />
<br />
[3] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey<br />
Dean. 2013a. Efficient estimation of word<br />
representations in vector space. arXiv preprint<br />
arXiv:1301.3781.<br />
<br />
[4] Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado,<br />
and Jeff Dean. 2013b. Distributed representations<br />
of words and phrases and their compositionality.<br />
In Advances in neural information processing<br />
systems, pages 3111–3119.<br />
<br />
[5] Kata Gbor, Davide Buscaldi, Anne-Kathrin Schumann, Behrang QasemiZadeh, Hafa Zargayouna,<br />
and Thierry Charnois. 2018. Semeval-2018 task 7:Semantic relation extraction and classification in scientific papers. <br />
In Proceedings of the 12th International Workshop on Semantic Evaluation (SemEval2018), New Orleans, LA, USA, June 2018.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Adversarial_Attacks_on_Copyright_Detection_Systems&diff=49220Adversarial Attacks on Copyright Detection Systems2020-12-05T17:26:08Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Presented by == <br />
Luwen Chang, Qingyang Yu, Tao Kong, Tianrong Sun<br />
<br />
==Introduction ==<br />
The copyright detection system is one of the most commonly used machine learning systems. Important real-world applications include tools such as Google Jigsaw which can identify and remove videos promoting terrorism, or companies like YouTube who use machine learning systems to flag content that infringes copyrights. Failure to do so will result in legal consequences. However, the adversarial attacks on these systems have not been widely addressed by the public and remain largely unexplored. Adversarial attacks are instances where inputs are intentionally designed by people to cause misclassification in the model. <br />
<br />
Copyright detection systems are vulnerable to attacks for three reasons:<br />
<br />
1. Unlike physical-world attacks where adversarial samples need to survive under different conditions like resolutions and viewing angles, any digital files can be uploaded directly to the web without going through a camera or microphone.<br />
<br />
2. The detection system is an open-set problem, which means that the uploaded files may not correspond to an existing class. In this case, it will prevent people from uploading unprotected audio/video whereas most of the uploaded files nowadays are not protected.<br />
<br />
3. The detection system needs to handle a vast majority of content which have different labels but similar features. For example, in the ImageNet classification task, the system is easily attacked when there are two cats/dogs/birds with high similarities but from different classes.<br />
<br />
The goal of this paper is to raise awareness of the security threats faced by copyright detection systems. In this paper, different types of copyright detection systems will be introduced. A widely used detection model from Shazam, a popular app used for recognizing music, will be discussed. As a proof-of-concept, the paper generates audio fingerprints using convolutional neural networks and formulates the adversarial loss function using standard gradient methods. An example of remixing music is given to show how adversarial examples can be created. Then, the adversarial attacks are applied to the industrial systems like AudioTag and YouTube Content ID to evaluate the effectiveness of the systems.<br />
<br />
== Types of copyright detection systems ==<br />
Fingerprinting algorithms work by extracting the features of a source file as a hash and then utilizing a matching algorithm to compare that to the materials protected by copyright in the database. If enough matches are found between the source and existing data, then the two samples are considered identical. Thus, the copyright detection system is able to reject the copyright declaration of the source. Most audio, image, and video fingerprinting algorithms work by training a neural network to output features or extracting hand-crafted features.<br />
<br />
In terms of video fingerprinting, a useful algorithm is to detect the entering/leaving time of the objects in the video (Saviaga & Toxtli, 2018). The final hash consists of the entering/leaving of different objects and a unique relationship of the objects. However, most of these video fingerprinting algorithms only train their neural networks by using simple distortions such as adding noise or flipping the video rather than adversarial perturbations. This leads to algorithms that are strong against pre-defined distortions, but not adversarial attacks.<br />
<br />
Moreover, some plagiarism detection systems also depend on neural networks to generate a fingerprint of the input document. Though using deep feature representations as a fingerprint is efficient in detecting plagiarism, it still might be weak to adversarial attacks.<br />
<br />
Audio fingerprinting may perform better than the algorithms above. This is because the hash is usually generated by extracting hand-crafted features rather than training a neural network. That being said, it still is easy to attack.<br />
<br />
== Case study: evading audio fingerprinting ==<br />
<br />
=== Audio Fingerprinting Model===<br />
The audio fingerprinting model plays an important role in copyright detection. It is useful for quickly locating or finding similar samples inside an audio database. Shazam is a popular music recognition application, which uses one of the most well-known fingerprinting models. Because of the three properties: temporal locality, translation invariance, and robustness, Shazam's algorithm is treated as a good fingerprinting algorithm. It shows strong robustness even in presence of noise by using local maximum in spectrogram to form hashes. Spectrograms are two-dimensional representations of audio frequency spectra over time. An example is shown below. <br />
<br />
<div style="text-align:center;">[[File:Spectrogram-19thC.png|Spectrogram-19thC|390px]]</div><br />
<div align="center"><span style="font-size:80%">Source:https://commons.wikimedia.org/wiki/File:Spectrogram-19thC.png</span></div><br />
<br />
=== Interpreting the fingerprint extractor as a CNN ===<br />
The intention of this section is to build a differentiable neural network whose function resembles that of an audio fingerprinting algorithm, which is well-known for its ability to identify the meta-data, i.e. song names, artists, and albums, while independent of an audio format (Group et al., 2005). The generic neural network model will then be used as an example of black-box attacks on many popular real-world systems, in this case, YouTube and audio tag. <br />
<br />
The generic neural network model consists of two convolutional layers and a max-pooling layer, which is used for dimension reduction. This is depicted in the figure below. As mentioned above, the convolutional neural network is well-known for its properties of temporal locality and transformational invariance. The purpose of this network is to generate audio fingerprinting signals that extract features that uniquely identify a signal, regardless of the starting and ending times of the inputs.<br />
<br />
[[File:cov network.png | thumb | center | 500px ]]<br />
<br />
While an audio sample enters the neural network, it is first transformed by the initial network layer, which can be described as a normalized Hann function. The form of the function is shown below, with N being the width of the Kernel. <br />
<br />
$$ f_{1}(n)=\frac {\sin^2(\frac{\pi n} {N})} {\sum_{i=0}^N \sin^2(\frac{\pi i}{N})} $$ <br />
<br />
The intention of the normalized Hann function is to smooth the adversarial perturbation of the input audio signal, which removes the discontinuity as well as the bad spectral properties. This transformation enhances the efficiency of black-box attacks that is later implemented.<br />
<br />
The next convolutional layer applies a Short Term Fourier Transformation to the input signal by computing the spectrogram of the waveform and converts the input into a feature representation. Once the input signal enters this network layer, it is being transformed by the convolutional function below. <br />
<br />
$$f_{2}(k,n)=e^{-i 2 \pi k n / N} $$<br />
where k <math>{\in}</math> 0,1,...,N-1 (output channel index) and n <math>{\in}</math> 0,1,...,N-1 (index of filter coefficient)<br />
<br />
The output of this layer is described as φ(x) (x being the input signal), a feature representation of the audio signal sample. <br />
However, this representation is flawed due to its vulnerability to noise and perturbation, as well as its difficulty to store and inspect. Therefore, a maximum pooling layer is being implemented to φ(x), in which the network computes a local maximum using a max-pooling function to become robust to changes in the position of the feature. This network layer outputs a binary fingerprint ψ (x) (x being the input signal) that will be used later to search for a signal against a database of previously processed signals.<br />
<br />
=== Formulating the adversarial loss function ===<br />
<br />
In the previous section, local maxima of the spectrogram are used to generate fingerprints by CNN, but a loss has not been quantified to compare how similar two fingerprints are. After the loss is found, standard gradient methods can be used to find a perturbation <math>{\delta}</math>, which can be added to a signal so that the copyright detection system will be tricked. Also, a bound is set to make sure the generated fingerprints are close enough to the original audio signal. <br />
$$\text{bound:}\ ||\delta||_p\le\epsilon$$<br />
<br />
where <math>{||\delta||_p\le\epsilon}</math> is the <math>{l_p}</math>-norm of the perturbation and <math>{\epsilon}</math> is the bound of the difference between the original file and the adversarial example. <br />
<br />
<br />
To compare how similar two binary fingerprints are, Hamming distance is employed. Hamming distance between two strings is the number of digits that are different (Hamming distance, 2020). For example, the Hamming distance between 101100 and 100110 is 2. <br />
<br />
Let <math>{\psi(x)}</math> and <math>{\psi(y)}</math> be two binary fingerprints outputted from the model, the number of peaks shared by <math>{x}</math> and <math>{y}</math> can be found through <math>{|\psi(x)\cdot\psi(y)|}</math>. Now, to get a differentiable loss function, the equation is found to be <br />
<br />
$$J(x,y)=|\phi(x)\cdot\psi(x)\cdot\psi(y)|$$<br />
<br />
<br />
This is effective for white-box attacks by knowing the fingerprinting system. However, the loss can be easily minimized by modifying the location of the peaks by one pixel, which would not be reliable to transfer to black-box industrial systems. To make it more transferable, a new loss function that involves more movements of the local maxima of the spectrogram is proposed. The idea is to move the locations of peaks in <math>{\psi(x)}</math> outside of neighborhood of the peaks of <math>{\psi(y)}</math>. In order to implement the model more efficiently, two max-pooling layers are used. One of the layers has a bigger width <math>{w_1}</math> while the other one has a smaller width <math>{w_2}</math>. For any location, if the output of <math>{w_1}</math> pooling is strictly greater than the output of <math>{w_2}</math> pooling, then it can be concluded that no peak is in that location with radius <math>{w_2}</math>. <br />
<br />
The loss function is as the following:<br />
<br />
$$J(x,y) = \sum_i\bigg(\text{ReLU}\bigg(c-\bigg(\underset{|j| \leq w_1}{\max}\phi(i+j;x)-\underset{|j| \leq w_2}{\max}\phi(i+j;x)\bigg)\bigg)\cdot\psi(i;y)\bigg)$$<br />
The equation above penalizes the peaks of <math>{x}</math> which are in neighborhood of peaks of <math>{y}</math> with radius of <math>{w_2}</math>. The activation function uses <math>{ReLU}</math>. <math>{c}</math> is the difference between the outputs of two max-pooling layers. <br />
<br />
<br />
Lastly, instead of the maximum operator, smoothed max function is used here:<br />
$$S_\alpha(x_1,x_2,...,x_n) = \frac{\sum_{i=1}^{n}x_ie^{\alpha x_i}}{\sum_{i=1}^{n}e^{\alpha x_i}}$$<br />
where <math>{\alpha}</math> is a smoothing hyper parameter. When <math>{\alpha}</math> approaches positive infinity, <math>{S_\alpha}</math> is closer to the actual max function. <br />
<br />
To summarize, the optimization problem can be formulated as the following:<br />
<br />
$$<br />
\underset{\delta}{\min}J(x+\delta,x)\\<br />
s.t.||\delta||_{\infty}\le\epsilon<br />
$$<br />
where <math>{x}</math> is the input signal, <math>{J}</math> is the loss function with the smoothed max function. Note that <math>||\delta||_\infty = \max_k |\delta_k|</math> (the largest component of <math>\delta</math>) which makes it relatively easy to satisfy the above constraint by clipping the values of each iteration of <math>\delta</math> to be less than <math>\epsilon</math>. However, since <math>||\delta||_\infty \leq \dots \leq ||\delta||_2 \leq ||\delta||_1</math> this choice of norm results in the loosest bound, and therefore admits a larger selection of perturbations <math>\delta</math> than might be permitted by other choices of norms.<br />
<br />
=== Remix adversarial examples===<br />
While solving the optimization problem, the resulted example would be able to fool the copyright detection system. But it could sound unnatural with the perturbations.<br />
<br />
Instead, the fingerprinting could be made in a more natural way (i.e., a different audio signal). <br />
<br />
By modifying the loss function, which switches the order of the max-pooling layers in the smooth maximum components in the loss function, this remix loss function is to make two signals x and y look as similar as possible.<br />
<br />
$$J_{remix}(x,y) = \sum_i\bigg(ReLU\bigg(c-\bigg(\underset{|j| \leq w_2}{\max}\phi(i+j;x)-\underset{|j| \leq w_1}{\max}\phi(i+j;x)\bigg)\bigg)\cdot\psi(i;y)\bigg)$$<br />
<br />
By adding this new loss function, a new optimization problem could be defined. <br />
<br />
$$<br />
\underset{\delta}{\min}J(x+\delta,x) + \lambda J_{remix}(x+\delta,y)\\<br />
s.t.||\delta||_{p}\le\epsilon<br />
$$<br />
<br />
where <math>{\lambda}</math> is a scalar parameter that controls the similarity of <math>{x+\delta}</math> and <math>{y}</math>.<br />
<br />
This optimization problem is able to generate an adversarial example from the selected source, and also enforce the adversarial example to be similar to another signal. The resulting adversarial example is called Remix adversarial example because it gets the references to its source signal and another signal.<br />
<br />
== Evaluating transfer attacks on industrial systems==<br />
The effectiveness of default and remix adversarial examples is tested through white-box attacks on the proposed model and black-box attacks on two real-world audio copyright detection systems - AudioTag and YouTube “Content ID” system. <math>{l_{\infty}}</math> norm and <math>{l_{2}}</math> norm of perturbations are two measures of modification. Both of them are calculated after normalizing the signals so that the samples could lie between 0 and 1.<br />
<br />
Before evaluating black-box attacks against real-world systems, white-box attacks against our own proposed model is used to provide the baseline of adversarial examples’ effectiveness. Loss function <math>{J(x,y)=|\phi(x)\cdot\psi(x)\cdot\psi(y)|}</math> is used to generate white-box attacks. The unnoticeable fingerprints of the audio with the noise can be changed or removed by optimizing the loss function.<br />
<br />
[[File:Table_1_White-box.jpg |center ]]<br />
<br />
<div align="center">Table 1: Norms of the perturbations for white-box attacks</div><br />
<br />
In black-box attacks, by applying random perturbations to the audio recordings, AudioTag’s claim of being robust to input<br />
distortions was verified. However, the AudioTag system is found to be relatively sensitive to the attacks since it can detect the songs with a benign signal while it failed to detect both default and remix adversarial examples. The architecture of the AudioTag fingerprint model and surrogate CNN model is guessed to be similar based on the experimental observations. <br />
<br />
Similar to AudioTag, the YouTube “Content ID” system also got the result with successful identification of benign songs but failure to detect adversarial examples. However, to fool the YouTube Content ID system, a larger value of the parameter <math>{\epsilon}</math> is required, which makes perturbations obvious although songs can still be recognized by humans. YouTube Content ID system has a more robust fingerprint model.<br />
<br />
<br />
[[File:Table_2_Black-box.jpg |center]]<br />
<br />
<div align="center">Table 2: Norms of the perturbations for black-box attacks</div><br />
<br />
[[File:YouTube_Figure.jpg |center]]<br />
<br />
<div align="center">Figure 2: YouTube’s copyright detection recall against the magnitude of noise</div><br />
<br />
== Conclusion ==<br />
In this paper, we obtain that many industrial copyright detection systems used in popular video and music websites, such as YouTube and AudioTag, are significantly vulnerable to adversarial attacks established in the existing literature. By building a simple music identification system resembling that of Shazam using a neural network and attack it by the well-known gradient method, this paper firmly proved the lack of robustness of the current online detector. <br />
<br />
Although the method used in the paper isn't optimal, the attacks can be strengthened using sharper technical. Also, we could transfer attacks using rudimentary surrogate models that rely on hand-crafted features, while commercial systems likely rely on trainable neural nets.<br />
<br />
Our goal here is not to facilitate but the intention of this paper is to raise the awareness of the vulnerability of the current online system to adversarial attacks and to emphasize the significance of copyright detection system. A number of mitigating approaches already exist, such as adversarial training, but they need to be further developed and examined in order to robustly protect against the threat of adversarial copyright attacks.<br />
<br />
== Appendix ==<br />
=== Feature Extraction Process in Audio-Fingerprinting System ===<br />
<br />
1. Preprocessing. In this step, the audio signal is digitalized and quantized at first.<br />
Then, it is converted to a mono signal by averaging two channels if necessary.<br />
Finally, it is resampled if the sampling rate is different from the target rate.<br />
<br />
2. Framing. Framing means dividing the audio signal into frames of equal length<br />
by a window function.<br />
<br />
3. Transformation. This step is designed to transform the set of frames into a new set<br />
of features, in order to reduce the redundancy. <br />
<br />
4. Feature Extraction. After transformation, final acoustic features are extracted<br />
from the time-frequency representation. The main purpose is to reduce the<br />
dimensionality and increase the robustness to distortions.<br />
<br />
5. Post-processing. To capture the temporal variations of the audio signal, higher<br />
order time derivatives are required sometimes.<br />
<br />
== Critiques ==<br />
- The experiments in this paper appear to be a proof-of-concept rather than a serious evaluation of a model. One problem is that the norm is used to evaluate the perturbation. Unlike the norm in image domains which can be visualized and easily understood, the perturbations in the audio domain are more difficult to comprehend. A cognitive study or something like a user study might need to be conducted in order to understand this. Another question related to this is that if the random noise is 2x bigger or 3x bigger in terms of the norm, does this make a huge difference when listening to it? Are these two perturbations both very obvious or unnoticeable? In addition, it seems that a dataset is built but the stats are missing. Third, no baseline methods are being compared to in this paper, not even an ablation study. The proposed two methods (default and remix) seem to perform similarly.<br />
<br />
- There could be an improvement in term of how to find the threshold in general, it mentioned how to measure the similarity of two pieces of content but have not discussed what threshold should we set for this model. In fact, it is always a challenge to determine the boundary of "Copyright Issue" or "Not Copyright Issue" and this is some important information that may be discussed in the paper.<br />
<br />
- The fingerprinting technique used in this paper seems rather elementary, which is a downfall in this context because the focus of this paper is adversarial attacks on these methods. A recent 2019 work (https://arxiv.org/pdf/1907.12956.pdf) proposed a deep fingerprinting algorithm along with some novel framing of the problem. There are several other older works in this area that also give useful insights that would have improved the algorithm in this paper.<br />
<br />
- Figure 1 clearly indicates the stricture of an audio fingerprinting model with 2 convolution layers and a max pooling layer. In the following paragraphs, it shows how and why the author choose to use each layer and what we could get at the output layer. This gives us a general thought of how this type of neural network is used to deal with data.<br />
<br />
- In the experiment section, authors could provide more background information on the dataset used. For example, the number of songs in the dataset and a brief introduction of different features.<br />
<br />
- Since the paper didn't go into details of how features are extracted in an audio-fingerprinting system, the details are listed out above in "Feature Extraction Process in Audio-Fingerprinting System"<br />
<br />
- In Shazam algorithm, the author should explore more on when the system will detect the copyright issue. Such as should the copyright issue be raised if there is only 5 seconds of melody or lyrics that are similar to other music?<br />
<br />
- In addition to Audio files, can this system detect copyright attacks on mixed data such as say, embedded audio files in the text? It will be interesting to see if it can do it. Nowadays, a lot of data is coming in mixed form, e.g. text+video, audio+text etc and therefore having a system to detect adversarial attacks on copyright detection systems for mixed data will be a useful development.<br />
<br />
- When introducing the types of copyright detection systems, there is a lack of clearness in explaining fingerprinting algorithms for audios and images. Besides a brief, one-line explanation, justifications regarding why audio fingerprinting algorithms are better compared to video fingerprinting algorithms were not provided. Moreover, discussions regarding image fingerprinting algorithms were missing. Hence, itt extent of the would be beneficial to add these two sections to enhance the readers' understanding of this summary.<br />
<br />
- It says: "Fingerprinting algorithms work by extracting the features of a source file as a hash", but it seems like they should have included more information about it so that readers could understand how feature extraction works in this case.<br />
<br />
== References ==<br />
<br />
Group, P., Cano, P., Group, M., Group, E., Batlle, E., Ton Kalker Philips Research Laboratories Eindhoven, . . . Authors: Pedro Cano Music Technology Group. (2005, November 01). A Review of Audio Fingerprinting. Retrieved November 13, 2020, from https://dl.acm.org/doi/10.1007/s11265-005-4151-3<br />
<br />
Hamming distance. (2020, November 1). In ''Wikipedia''. https://en.wikipedia.org/wiki/Hamming_distance<br />
<br />
Jovanovic. (2015, February 2). ''How does Shazam work? Music Recognition Algorithms, Fingerprinting, and Processing''. Toptal Engineering Blog. https://www.toptal.com/algorithms/shazam-it-music-processing-fingerprinting-and-recognition<br />
<br />
Saadatpanah, P., Shafahi, A., &amp; Goldstein, T. (2019, June 17). ''Adversarial attacks on copyright detection systems''. Retrieved November 13, 2020, from https://arxiv.org/abs/1906.07153.<br />
<br />
Saviaga, C. and Toxtli, C. ''Deepiracy: Video piracy detection system by using longest common subsequence and deep learning'', 2018. https://medium.com/hciwvu/piracy-detection-using-longestcommon-subsequence-and-neuralnetworks-a6f689a541a6<br />
<br />
Wang, A. et al. ''An industrial strength audio search algorithm''. In Ismir, volume 2003, pp. 7–13. Washington, DC, 2003.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Efficient_kNN_Classification_with_Different_Numbers_of_Nearest_Neighbors&diff=49218Efficient kNN Classification with Different Numbers of Nearest Neighbors2020-12-05T17:18:16Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Presented by == <br />
Cooper Brooke, Daniel Fagan, Maya Perelman<br />
<br />
== Introduction == <br />
Traditional model-based approaches for classification problem requires to train a model on training observations before predicting test samples. In contrast, the model-free k-Nearest Neighbors (KNNs) method classifies observations with a majority rule approach, labeling each piece of test data based on its k closest training observations (neighbors). This method has become very popular due to its relatively robust performance given how simple it is to implement.<br />
<br />
There are two main approaches to conduct kNN classification. The first is to use a fixed k value to classify all test samples, while the second is to use a different k value each time, for each test sample. The former, while easy to implement, has proven to be impractical in machine learning applications. Therefore, interest lies in developing an efficient way to apply a different optimal k value for each test sample. The authors of this paper presented the kTree and k*Tree methods to solve this research question.<br />
<br />
== Previous Work == <br />
<br />
Previous work on finding an optimal fixed k value for all test samples is well-studied. Zhang et al. [1] incorporated a certainty factor measure to solve for an optimal fixed k. This resulted in the conclusion that k should be <math>\sqrt{n}</math> (where n is the number of training samples) when n > 100. The method Song et al.[2] explored involved selecting a subset of the most informative samples from neighbourhoods. Vincent and Bengio [3] took the unique approach of designing a k-local hyperplane distance to solve for k. Premachandran and Kakarala [4] had the solution of selecting a robust k using the consensus of multiple rounds of kNNs. These fixed k methods are valuable however are impractical for data mining and machine learning applications. <br />
<br />
Finding an efficient approach to assigning varied k values has also been previously studied. Tuning approaches such as the ones taken by Zhu et al. as well as Sahugara et al. have been popular. Zhu et al. [5] determined that optimal k values should be chosen using cross validation while Sahugara et al. [6] proposed using Monte Carlo validation to select varied k parameters. Other learning approaches such as those taken by Zheng et al. and Góra and Wojna also show promise. Zheng et al. [7] applied a reconstruction framework to learn suitable k values. Góra and Wojna [8] proposed using rule induction and instance-based learning to learn optimal k-values for each test sample. While all these methods are valid, their processes of either learning varied k values or scanning all training samples are time-consuming.<br />
<br />
== Motivation == <br />
<br />
Due to the previously mentioned drawbacks of fixed-k and current varied-k kNN classification, the paper’s authors sought to design a new approach to solve for different k values. The kTree and k*Tree approach seeks to calculate optimal values of k while avoiding computationally costly steps such as cross-validation.<br />
<br />
A secondary motivation of this research was to ensure that the kTree method would perform better than kNN using fixed values of k given that running costs would be similar in this instance.<br />
<br />
== Approach == <br />
<br />
<br />
=== kTree Classification ===<br />
<br />
The proposed kTree method is illustrated by the following flow chart:<br />
<br />
[[File:Approach_Figure_1.png | center | 800x800px]]<br />
<br />
==== Reconstruction ====<br />
<br />
The first step is to use the training samples to reconstruct themselves. The goal of this is to find the matrix of correlations between the training samples themselves, <math>\textbf{W}</math>, such that the distance between an individual training sample and the corresponding correlation vector multiplied by the entire training set is minimized. This least square loss function where <math>\mathbf{X}\in \mathbb{R}^{d\times n} = [x_1,...,x_n]</math> represents the training set can be written as:<br />
<br />
$$\begin{aligned}<br />
\mathop{min}_{\textbf{W}} \sum_{i=1}^n ||Xw_i - x_i||^2<br />
\end{aligned}$$<br />
<br />
In addition, an <math>l_1</math> regularization term multiplied by a tuning parameter, <math>\rho_1</math>, is added to ensure that sparse results are generated as the objective is to minimize the number of training samples that will eventually be depended on by the test samples. <br />
<br />
$$\begin{aligned}<br />
\mathop{min}_{\textbf{W}} \sum_{i=1}^n ||Xw_i - x_i||^2 + \rho||\textbf{W}||^2_2<br />
\end{aligned}$$<br />
<br />
This is called ridge regression and it has a close solution where $$W = (X^TX+\rho I)^{-1}X^TX$$<br />
<br />
However, this objective function does not provide a sparse result, there we further employe a sparse objective function: <br />
<br />
$$W = (X^TX+\rho I)^{-1}X^TX, W >= 0$$<br />
<br />
<br />
The least square loss function is then further modified to account for samples that have similar values for certain features yielding similar results. It is penalized with the function: <br />
<br />
$$\frac{1}{2} \sum^{d}_{i,j} ||x^iW-x^jW||^2_2$$<br />
<br />
with sij denotes the relation between feature vectors. It uses a radial basis function kernel to calculate Sij. After some transformations, this second regularization term that has tuning parameter <math>\rho_2</math> is:<br />
<br />
$$\begin{aligned}<br />
R(W) = Tr(\textbf{W}^T \textbf{X}^T \textbf{LXW})<br />
\end{aligned}$$<br />
<br />
where <math>\mathbf{L}</math> is a Laplacian matrix that indicates the relationship between features. The Laplacian matrix, also called the graph Laplacian, is a matrix representation of a graph. <br />
<br />
This gives a final objective function of:<br />
<br />
$$\begin{aligned}<br />
\mathop{min}_{\textbf{W}} \sum_{i=1}^n ||Xw_i - x_i||^2 + \rho_1||\textbf{W}|| + \rho_2R(\textbf{W})<br />
\end{aligned}$$<br />
<br />
Since this is a convex function, an iterative method can be used to optimize it to find the optimal solution <math>\mathbf{W^*}</math>.<br />
<br />
==== Calculate ''k'' for training set ====<br />
<br />
Each element <math>w_{ij}</math> in <math>\textbf{W*}</math> represents the correlation between the ith and jth training sample so if a value is 0, it can be concluded that the jth training sample has no effect on the ith training sample which means that it should not be used in the prediction of the ith training sample. Consequently, all non-zero values in the <math>w_{.j}</math> vector would be useful in predicting the ith training sample which gives the result that the number of these non-zero elements for each sample is equal to the optimal ''k'' value for each sample.<br />
<br />
For example, if there was a 4x4 training set where <math>\textbf{W*}</math> had the form:<br />
<br />
[[File:Approach_Figure_2.png | center | 300x300px]]<br />
<br />
The optimal ''k'' value for training sample 1 would be 2 since the correlation between training sample 1 and both training samples 2 and 4 are non-zero.<br />
<br />
==== Train a Decision Tree using ''k'' as the label ====<br />
<br />
In a normal decision tree, the target data is the labels themselves. In contrast, in the kTree method, the target data is the optimal ''k'' value for each sample that was solved for in the previous step. So this decision tree has the following form:<br />
<br />
[[File:Approach_Figure_3.png | center | 300x300px]]<br />
<br />
==== Making Predictions for Test Data ====<br />
<br />
The optimal ''k'' values for each testing sample are easily obtainable using the kTree solved for in the previous step. The only remaining step is to predict the labels of the testing samples by finding the majority class of the optimal ''k'' nearest neighbors across '''all''' of the training data.<br />
<br />
=== k*Tree Classification ===<br />
<br />
The proposed k*Tree method is illustrated by the following flow chart:<br />
<br />
[[File:Approach_Figure_4.png | center | 1000x1000px]]<br />
<br />
Clearly, this is a very similar approach to the kTree as the k*Tree method attempts to sacrifice very little in predictive power in return for a substantial decrease in complexity when actually implementing the traditional kNN on the testing data once the optimal ''k'' values have been found.<br />
<br />
While all steps previous are the exact same, the difference comes from additional data stored in the leaf nodes. k*Tree method not only stores the optimal ''k'' value but also the following information:<br />
<br />
* The training samples that have the same optimal ''k''<br />
* The ''k'' nearest neighbours of the previously identified training samples<br />
* The nearest neighbor of each of the previously identified ''k'' nearest neighbours<br />
<br />
The data stored in each node is summarized in the following figure:<br />
<br />
[[File:Approach_Figure_5.png | center | 800x800px]]<br />
<br />
When testing, the constructed k*Tree is searched for its optimal k values well as its nearest neighbours in the leaf node. It then selects a number of its nearest neighbours from the subset of training samples and assigns the test sample with the majority label of these nearest neighbours.<br />
<br />
In the kTree method, predictions were made based on all of the training data, whereas in the k*Tree method, predicting the test labels will only be done using the samples stored in the applicable node of the tree.<br />
<br />
== Experiments == <br />
<br />
In order to assess the performance of the proposed method against existing methods, a number of experiments were performed to measure classification accuracy and run time. The experiments were run on twenty public datasets provided by the UCI Repository of Machine Learning Data, and contained a mix of data types varying in size, in dimensionality, in the number of classes, and in imbalanced nature of the data. Ten-fold cross-validation was used to measure classification accuracy, and the following methods were compared against:<br />
<br />
# k-Nearest Neighbor: The classical kNN approach with k set to k=1,5,10,20 and square root of the sample size [9]; the best result was reported.<br />
# kNN-Based Applicability Domain Approach (AD-kNN) [11]<br />
# kNN Method Based on Sparse Learning (S-kNN) [10]<br />
# kNN Based on Graph Sparse Reconstruction (GS-kNN) [7]<br />
# Filtered Attribute Subspace-based Bagging with Injected Randomness (FASBIR) [12], [13]<br />
# Landmark-based Spectral Clustering kNN (LC-kNN) [14]<br />
<br />
The experimental results were then assessed based on classification tasks that focused on different sample sizes, and tasks that focused on different numbers of features. <br />
<br />
<br />
'''A. Experimental Results on Different Sample Sizes'''<br />
<br />
The running cost and (cross-validation) classification accuracy based on experiments on ten UCI datasets can be seen in Table I below.<br />
<br />
[[File:Table_I_kNN.png | center | 1000x1000px]]<br />
<br />
The following key results are noted:<br />
* Regarding classification accuracy, the proposed methods (kTree and k*Tree) outperformed kNN, AD-KNN, FASBIR, and LC-kNN on all datasets by 1.5%-4.5%, but had no notable improvements compared to GS-kNN and S-kNN.<br />
* Classification methods which involved learning optimal k-values (for example the proposed kTree and k*Tree methods, or S-kNN, GS-kNN, AD-kNN) outperformed the methods with predefined k-values, such as traditional kNN.<br />
* The proposed k*Tree method had the lowest running cost of all methods. However, the k*Tree method was still outperformed in terms of classification accuracy by GS-kNN and S-kNN, but ran on average 15 000 times faster than either method. In addition, the kTree had the highest accuracy and it's running cost was lower than any other methods except the k*Tree method.<br />
<br />
<br />
'''B. Experimental Results on Different Feature Numbers'''<br />
<br />
The goal of this section was to evaluate the robustness of all methods under differing numbers of features; results can be seen in Table II below. The Fisher score, an algorithm that solves maximum likelihood equations numerically [15], was used to rank and select the most information features in the datasets. <br />
<br />
[[File:Table_II_kNN.png | center | 1000x1000px]]<br />
<br />
From Table II, the proposed kTree and k*Tree approaches outperformed kNN, AD-kNN, FASBIR and LC-KNN when tested for varying feature numbers. The S-kNN and GS-kNN approaches remained the best in terms of classification accuracy, but were greatly outperformed in terms of running cost by k*Tree. The cause for this is that k*Tree only scans a subsample of the training samples for kNN classification, while S-kNN and GS-kNN scan all training samples.<br />
<br />
== Conclusion == <br />
<br />
This paper introduced two novel approaches for kNN classification algorithms that can determine optimal k-values for each test sample. The proposed kTree and k*Tree methods can classify the test samples efficiently and effectively, by designing a training step that reduces the run time of the test stage and thus enhances the performance. Based on the experimental results for varying sample sizes and differing feature numbers, it was observed that the proposed methods outperformed existing ones in terms of running cost while still achieving similar or better classification accuracies. Future areas of investigation could focus on the improvement of kTree and k*Tree for data with large numbers of features.<br />
<br />
== Critiques == <br />
<br />
*The paper only assessed classification accuracy through cross-validation accuracy. However, it would be interesting to investigate how the proposed methods perform using different metrics, such as AUC, precision-recall curves, or in terms of holdout test data set accuracy. <br />
* The authors addressed that some of the UCI datasets contained imbalanced data (such as the Climate and German data sets) while others did not. However, the nature of the class imbalance was not extreme, and the effect of imbalanced data on algorithm performance was not discussed or assessed. Moreover, it would have been interesting to see how the proposed algorithms performed on highly imbalanced datasets in conjunction with common techniques to address imbalance (e.g. oversampling, undersampling, etc.). <br />
*While the authors contrast their kTree and k*Tree approach with different kNN methods, the paper could contrast their results with more of the approaches discussed in the Related Work section of their paper. For example, it would be interesting to see how the kTree and k*Tree results compared to Góra and Wojna varied optimal k method.<br />
<br />
* The paper conducted an experiment on kNN, AD-kNN, S-kNN, GS-kNN,FASBIR and LC-kNN with different sample sizes and feature numbers. It would be interesting to discuss why the running cost of FASBIR is between that of kTree and k*Tree in figure 21.<br />
<br />
* A different [https://iopscience.iop.org/article/10.1088/1757-899X/725/1/012133/pdf paper] also discusses optimizing the K value for the kNN algorithm in clustering. However, this paper suggests using the expectation-maximization algorithm as a means of finding the optimal k value.<br />
<br />
* It would be really helpful if kTrees method can be explained at the very beginning. The transition from KNN to kTrees is not very smooth.<br />
<br />
* It would be nice to have a comparison of the running costs of different methods to see how much faster kTree and k*Tree performed<br />
<br />
* It would be better to show the key result only on a summary rather than stacking up all results without screening.<br />
<br />
* In the results section, it was mentioned that in the experiment on data sets with different numbers of features, the kTree and k*Tree model did not achieve GS-kNN or S-kNN's accuracies, but was faster in terms of running cost. It might be helpful here if the authors add some more supporting arguments about the benefit of this tradeoff, which appears to be a minor decrease in accuracy for a large improvement in speed. This could further showcase the advantages of the kTree and k*Tree models. More quantitative analysis or real-life scenario examples could be some choices here.<br />
<br />
* An interesting thing to notice while solving for the optimal matrix <math>W^*</math> that minimizes the loss function is that <math>W^*</math> is not necessarily a symmetric matrix. That is, the correlation between the <math>i^{th}</math> entry and the <math>j^{th}</math> entry is different from that between the <math>j^{th}</math> entry and the <math>i^{th}</math> entry, which makes the resulting W* not really semantically meaningful. Therefore, it would be interesting if we may set a threshold on the allowing difference between the <math>ij^{th}</math> entry and the <math>ji^{th}</math> entry in <math>W^*</math> and see if this new configuration will give better or worse results compared to current ones, which will provide better insights of the algorithm.<br />
<br />
* It would be interesting to see how the proposed model works with highly non-linear datasets. In the event it does not work well, it would pose the question: would replacing the k*Tree with a SVM or a neural network improve the accuracy? There could be experiments to show if this variant would prove superior over the original models.<br />
<br />
* The key results are a little misleading - for example they claim "the kTree had the highest accuracy and it's running cost was lower than any other methods except the k*Tree method" is false. The kTree method had slightly lower accuracy than both GS-kNN and S-kNN and kTree was also slower than LC-kNN<br />
<br />
* I want to point to the discussion on k*Tree's structure. In order for k*Tree to work effectively, its leaf nodes needs to store additional information. In addition to the optimal k value, it also needs to store things like the training samples that have the optimal k, and the k nearest neighbours of the previously identified training samples. How big of am impact does this structure have on storage cost? Since the number of leaf nodes can be large, the storage cost may be large as well. This can potentially make k*tree ineffective to use in practice, especially for very large datasets.<br />
<br />
* It would be better if the author can explain more on KTree method and the similarity of KTree method and KNN method.<br />
<br />
* Even though we are given a table with averages on the accuracy and mean running cost, it would have been nice to see a direct visual comparison in the figures followed below. In addition to comparing to other algorithms, it would be helpful to see the average expected cost of these algorithms to show as control or rather a standard to accuracy and compute cost to assess the overall general expected cost of running such classification algorithm to fully assess its efficacy.<br />
<br />
* It doesn't clearly mention what's the definition/similarity/difference between Ktree and KNN methods. If the authors could put some detailed explanations in the beginning, the flow of this paper would have been much better.<br />
<br />
== References == <br />
<br />
[1] C. Zhang, Y. Qin, X. Zhu, and J. Zhang, “Clustering-based missing value imputation for data preprocessing,” in Proc. IEEE Int. Conf., Aug. 2006, pp. 1081–1086.<br />
<br />
[2] Y. Song, J. Huang, D. Zhou, H. Zha, and C. L. Giles, “IKNN: Informative K-nearest neighbor pattern classification,” in Knowledge Discovery in Databases. Berlin, Germany: Springer, 2007, pp. 248–264.<br />
<br />
[3] P. Vincent and Y. Bengio, “K-local hyperplane and convex distance nearest neighbor algorithms,” in Proc. NIPS, 2001, pp. 985–992.<br />
<br />
[4] V. Premachandran and R. Kakarala, “Consensus of k-NNs for robust neighborhood selection on graph-based manifolds,” in Proc. CVPR, Jun. 2013, pp. 1594–1601.<br />
<br />
[5] X. Zhu, S. Zhang, Z. Jin, Z. Zhang, and Z. Xu, “Missing value estimation for mixed-attribute data sets,” IEEE Trans. Knowl. Data Eng., vol. 23, no. 1, pp. 110–121, Jan. 2011.<br />
<br />
[6] F. Sahigara, D. Ballabio, R. Todeschini, and V. Consonni, “Assessing the validity of QSARS for ready biodegradability of chemicals: An applicability domain perspective,” Current Comput.-Aided Drug Design, vol. 10, no. 2, pp. 137–147, 2013.<br />
<br />
[7] S. Zhang, M. Zong, K. Sun, Y. Liu, and D. Cheng, “Efficient kNN algorithm based on graph sparse reconstruction,” in Proc. ADMA, 2014, pp. 356–369.<br />
<br />
[8] X. Zhu, L. Zhang, and Z. Huang, “A sparse embedding and least variance encoding approach to hashing,” IEEE Trans. Image Process., vol. 23, no. 9, pp. 3737–3750, Sep. 2014.<br />
<br />
[9] U. Lall and A. Sharma, “A nearest neighbor bootstrap for resampling hydrologic time series,” Water Resour. Res., vol. 32, no. 3, pp. 679–693, 1996.<br />
<br />
[10] D. Cheng, S. Zhang, Z. Deng, Y. Zhu, and M. Zong, “KNN algorithm with data-driven k value,” in Proc. ADMA, 2014, pp. 499–512.<br />
<br />
[11] F. Sahigara, D. Ballabio, R. Todeschini, and V. Consonni, “Assessing the validity of QSARS for ready biodegradability of chemicals: An applicability domain perspective,” Current Comput.-Aided Drug Design, vol. 10, no. 2, pp. 137–147, 2013. <br />
<br />
[12] Z. H. Zhou and Y. Yu, “Ensembling local learners throughmultimodal perturbation,” IEEE Trans. Syst. Man, B, vol. 35, no. 4, pp. 725–735, Apr. 2005.<br />
<br />
[13] Z. H. Zhou, Ensemble Methods: Foundations and Algorithms. London, U.K.: Chapman & Hall, 2012.<br />
<br />
[14] Z. Deng, X. Zhu, D. Cheng, M. Zong, and S. Zhang, “Efficient kNN classification algorithm for big data,” Neurocomputing, vol. 195, pp. 143–148, Jun. 2016.<br />
<br />
[15] K. Tsuda, M. Kawanabe, and K.-R. Müller, “Clustering with the fisher score,” in Proc. NIPS, 2002, pp. 729–736.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Task_Understanding_from_Confusing_Multi-task_Data&diff=49217Task Understanding from Confusing Multi-task Data2020-12-05T17:07:50Z<p>Wtjung: /* Critique */</p>
<hr />
<div>'''Presented By'''<br />
<br />
Qianlin Song, William Loh, Junyue Bai, Phoebe Choi<br />
<br />
= Introduction =<br />
<br />
Narrow AI is an artificial intelligence that outperforms humans in a narrowly defined task. The application of Narrow AI is becoming more and more common. For example, Narrow AI can be used for spam filtering, music recommendation services, assist doctors to make data-driven decisions, and even self-driving cars. One of the most famous integrated forms of Narrow AI is Apple's Siri. Siri has no self-awareness or genuine intelligence, and hence often has challenges performing tasks outside its range of abilities. However, the widespread use of Narrow AI in important infrastructure functions raises some concerns. Some people think that the characteristics of Narrow AI make it fragile, and when neural networks can be used to control important systems (such as power grids, financial transactions), alternatives may be more inclined to avoid risks. While these machines help companies improve efficiency and cut costs, the limitations of Narrow AI encouraged researchers to look into General AI. <br />
<br />
General AI is a machine that can apply its learning to different contexts, which closely resembles human intelligence. This paper attempts to generalize the multi-task learning system that learns from data from multiple classification tasks. For an isolated and very difficult task, the artificial intelligence may not learn it very well. For instance, a net with pixel dimension 1000*1000 is less likely to identify complicated objects in real-world situations on the time basis. However, if it could be learned simultaneously, it would be better as the tasks can share what they learned. It is easier for the learner to learn together instead of in isolation, for example, shapes, landmarks, textures, orientation and so on. This is called Multitask Learning. One application is image recognition. In figure 1, an image of an apple corresponds to 3 labels: “red”, “apple” and “sweet”. These labels correspond to 3 different classification tasks: color, fruit, and taste. <br />
<br />
[[File:CSLFigure1.PNG | 500px]]<br />
<br />
Currently, multi-task machines require researchers to construct a task definition. Otherwise, it will end up with different outputs with the same input value. Researchers manually assign tasks to each input in the sample to train the machine. See figure 1(a). This method incurs high annotation costs and restricts the machine’s ability to mirror the human recognition process. This paper is interested in developing an algorithm that understands task concepts and performs multi-task learning without manual task annotations. <br />
<br />
This paper proposed a new learning method called confusing supervised learning (CSL) which includes 2 functions: de-confusing function and mapping function. The de-confusing function allocates samples to respective tasks and the mapping function presents the relation from the input to its label within the allocated tasks. See figure 1(b). To implement the CSL, we use a risk functional to balance the effects of the de-confusing function and mapping function. <br />
<br />
However, simply combining the two functions or networks to a single architecture is impossible, since the one-hot constraint of the outputs for the de-confusing network makes the gradient back-propagation unfeasible. This difficulty is solved by alternatively performing training for the de-confusing net and mapping net optimization in the proposed architecture CLS-Net.<br />
<br />
Experiments for function regression and image recognition problems were constructed and compared with multi-task learning with complete information to test CSL-Net’s performance. Experiment results show that CSL-Net can learn multiple mappings for every task simultaneously and achieve the same cognition result as the current multi-task machine assigned with complete information.<br />
<br />
= Related Work =<br />
<br />
[[File:CSLFigure2.PNG | 700px]]<br />
<br />
==Latent variable learning==<br />
Latent variable learning aims to estimate the true function with mixed probability models. See '''figure 2a'''. In the multi-task learning problem without task annotations, we know that samples are generated from multiple distinct distributions instead of one distribution combining a mixture of multiple probability models. Thus, the latent variable learning can not fully distinguish labels into different tasks and different distributions, and it is insufficient to classify the multi-task confusing samples. <br />
<br />
==Multi-task learning==<br />
Multi-task learning aims to learn multiple tasks simultaneously using a shared feature representation. In multi-task learning, the task to which every sample belongs is known. By exploiting similarities and differences between tasks, the learning from one task can improve the learning of another task. (Caruana, 1997) This results in improved the overall learning efficiency, since the labels in different tasks are often correlated: improving the classfication result for one class also help with other classification tasks. In multi-task learning, the input-output mapping of every task can be represented by a unified function. However, these task definitions are manually constructed, and machines need manual task annotations to learn. If such manuual task annotation is abstent, then the algorithm can not be performed. <br />
<br />
==Multi-label learning==<br />
Multi-label learning aims to assign an input to a set of classes/labels. See '''figure 2b'''. It is a generalization of multi-class classification, which classifies an input into one class. In multi-label learning, an input can be classified into more than one class. Unlike multi-task learning, multi-label does not consider the relationship between different label judgments and it is assumed that each judgment is independent. An example where multi-label learning is applicable is the scenario where a website wants to automatically assign applicable tags/categories to an article. Since an article can be related to multiple categories (eg. an article can be tagged under the politics and business categories) multi-label learning is of primary concern here.<br />
<br />
= Confusing Supervised Learning =<br />
<br />
== Description of the Problem ==<br />
<br />
Confusing supervised learning (CSL) offers a solution to the issue at hand. A major area of improvement can be seen in the choice of risk measure. In traditional supervised learning, let <math> (x,y)</math> be the training samples from <math>y=f(x)</math>, which is an identical but unknown mapping relationship. Assuming the risk measure is mean squared error (MSE), the expected risk function is<br />
<br />
$$ R(g) = \int_x (f(x) - g(x))^2 p(x) \; \mathrm{d}x $$<br />
<br />
where <math>p(x)</math> is the data distribution of the input variable <math>x</math>. In practice, the methods select the optimal function by minimizing the empirical risk:<br />
<br />
$$ R_e(g) = \sum_{i=1}^n (y_i - g(x_i))^2 $$<br />
<br />
To minimize the risk function, the theoretically optimal solution is <math> f(x) </math>.<br />
<br />
When the problem involves different tasks, the model should optimize for each data point depending on the given task. Let <math>f_j(x)</math> be the true ground-truth function for each task <math> j </math>. Therefore, for some input variable <math> x_i </math>, an ideal model <math>g</math> would predict <math> g(x_i) = f_j(x_i) </math>. With this, the risk function can be modified to fit this new task for traditional supervised learning methods.<br />
<br />
$$ R(g) = \int_x \sum_{j=1}^n (f_j(x) - g(x))^2 p(f_j) p(x) \; \mathrm{d}x $$<br />
<br />
We call <math> (f_j(x) - g(x))^2 p(f_j) </math> the '''confusing multiple mappings'''. Then the optimal solution <math>g^*(x)</math> is <math>\bar{f}(x) = \sum_{j=1}^n p(f_j) f_j(x)</math>. However, the optimal solution is not conditional on the specific task at hand but rather on the entire ground-truth functions. The solution represents a mixed probably model instead of knowing the exact tasks and their correpsonding individual probability distribution. Therefore, for every non-trivial set of tasks where <math>f_u(x) \neq f_v(x)</math> for some input <math>x</math> and <math>u \neq v</math>, <math>R(g^*) > 0</math> which implies that there is an unavoidable confusion risk.<br />
<br />
== Learning Functions of CSL ==<br />
<br />
To overcome this issue, the authors introduce two types of learning functions:<br />
* '''Deconfusing function''' &mdash; allocation of which samples come from the same task<br />
* '''Mapping function''' &mdash; mapping relation from input to the output of every learned task<br />
<br />
Suppose there are <math>n</math> ground-truth mappings <math>\{f_j : 1 \leq j \leq n\}</math> that we wish to approximate with a set of mapping functions <math>\{g_k : 1 \leq k \leq l\}</math>. The authors define the deconfusing function as an indicator function <math>h(x, y, g_k) </math> which takes some sample <math>(x,y)</math> and determines whether the sample is assigned to task <math>g_k</math>. Under the CSL framework, the risk functional (using MSE loss) is <br />
<br />
$$ R(g,h) = \int_x \sum_{j,k} (f_j(x) - g_k(x))^2 \; h(x, f_j(x), g_k) \;p(f_j) \; p(x) \;\mathrm{d}x $$<br />
<br />
which can be estimated empirically with<br />
<br />
$$R_e(g,h) = \sum_{i=1}^m \sum_{k=1}^n |y_i - g_k(x_i)|^2 \cdot h(x_i, y_i, g_k) $$<br />
<br />
The risk metric of every sample affects only its assigned task.<br />
<br />
== Theoretical Results ==<br />
<br />
This novel framework yields some theoretical results to show the viability of its construction.<br />
<br />
'''Theorem 1 (Existence of Solution)'''<br />
''With the confusing supervised learning framework, there is an optimal solution''<br />
$$h^*(x, f_j(x), g_k) = \mathbb{I}[j=k]$$<br />
<br />
$$g_k^*(x) = f_k(x)$$<br />
<br />
''for each <math>k=1,..., n</math> that makes the expected risk function of the CSL problem zero.''<br />
<br />
However, necessity constraints are needed to avoid meaningless trivial solutions in all optimal risk solutions.<br />
<br />
'''Theorem 2 (Error Bound of CSL)'''<br />
''With probability at least <math>1 - \eta</math> simultaneously with finite VC dimension <math>\tau</math> of CSL learning framework, the risk measure is bounded by<br />
<br />
$$R(\alpha) \leq R_e(\alpha) + \frac{B\epsilon(m)}{2} \left(1 + \sqrt{1 + \frac{4R_e(\alpha)}{B\epsilon(m)}}\right)$$<br />
<br />
''where <math>\alpha</math> is the total parameters of learning functions <math>g, h</math>, <math>B</math> is the upper bound of one sample's risk, <math>m</math> is the size of training data and''<br />
$$\epsilon(m) = 4 \; \frac{\tau (\ln \frac{2m}{\tau} + 1) - \ln \eta / 4}{m}$$<br />
<br />
This theorem shows the method of empirical risk minimization is valid in the CSL framework. Moreover, the assumed number of tasks affects the VC dimension of the learning functions, which is positively related to the generalization error. Therefore, to make the training risk small, we need to choose the ''minimum number'' of tasks when determining the task.<br />
<br />
= CSL-Net =<br />
In this section, the authors describe how to implement and train a network for CSL, including the stucture of CSL-Net and Iterative deconfusing algorithm.<br />
<br />
== The Structure of CSL-Net ==<br />
Two neural networks, deconfusing-net and mapping-net are trained to implement two learning function variables in empirical risk. The optimization target of the training algorithm is:<br />
$$\min_{g, h} R_e = \sum_{i=1}^{m}\sum_{k=1}^{n} (y_i - g_k(x_i))^2 \cdot h(x_i, y_i; g_k)$$<br />
<br />
The mapping-net is corresponding to functions set <math>g_k</math>, where <math>y_k = g_k(x)</math> represents the output of one certain task. The deconfusing-net is corresponding to function h, whose input is a sample <math>(x,y)</math> and output is an n-dimensional one-hot vector. This output vector determines which task the sample <math>(x,y)</math> should be assigned to. The core difficulty of this algorithm is that the risk function cannot be optimized by gradient back-propagation due to the constraint of one-hot output from deconfusing-net. Approximation of softmax will lead the deconfusing-net output into a non-one-hot form, which results in meaningless trivial solutions.<br />
<br />
== Iterative Deconfusing Algorithm ==<br />
To overcome the training difficulty, the authors divide the empirical risk minimization into two local optimization problems. In each single-network optimization step, the parameters of one network are updated while the parameters of another remain fixed. With one network's parameters unchanged, the problem can be solved by a gradient descent method of neural networks. <br />
<br />
'''Training of Mapping-Net''': With function h from deconfusing-net being determined, the goal is to train every mapping function <math>g_k</math> with its corresponding sample <math>(x_i^k, y_i^k)</math>. The optimization problem becomes: <math>\displaystyle \min_{g_k} L_{map}(g_k) = \sum_{i=1}^{m_k} \mid y_i^k - g_k(x_i^k)\mid^2</math>. Back-propagation algorithm can be applied to solve this optimization problem.<br />
<br />
'''Training of Deconfusing-Net''': The task allocation is re-evaluated during the training phase while the parameters of the mapping-net remain fixed. To minimize the original risk, every sample <math>(x, y)</math> will be assigned to <math>g_k</math> that is closest to label y among all different <math>k</math>s. Mapping-net thus provides a temporary solution for deconfusing-net: <math>\hat{h}(x_i, y_i) = arg \displaystyle\min_{k} \mid y_i - g_k(x_i)\mid^2</math>. The optimization becomes: <math>\displaystyle \min_{h} L_{dec}(h) = \sum_{i=1}^{m} \mid {h}(x_i, y_i) - \hat{h}(x_i, y_i)\mid^2</math>. Similarly, the optimization problem can be solved by updating the deconfusing-net with a back-propagation algorithm.<br />
<br />
The two optimization stages are carried out alternately until the solution converges.<br />
<br />
=Experiment=<br />
==Setup==<br />
<br />
3 data sets are used to compare CSL to existing methods, 1 function regression task, and 2 image classification tasks. <br />
<br />
'''Function Regression''': The function regression data comes in the form of <math>(x_i,y_i),i=1,...,m</math> pairs. However, unlike typical regression problems, there are multiple <math>f_j(x),j=1,...,n</math> mapping functions, so the goal is to reproduce both the mapping functions <math>f_j</math> as well as determine which mapping function corresponds to each of the <math>m</math> observations. 3 scalar-valued, scalar-input functions that intersect at several points with each other have been chosen as the different tasks. <br />
<br />
'''Colorful-MNIST''': The first image classification data set consists of digit data in a range of 0 to 9, each of which is in a single color among the eight different colors. Each observation in this modified set consists of a colored image (<math>x_i</math>) and a label (<math>y_i</math>) that represents either the corresponding color, or the digit. The goal is to reproduce the classification task ("color" or "digit") for each observation and construct the 2 classifiers for both tasks. <br />
<br />
'''Kaggle Fashion Product''': The second image classification data set consists of several fashion-related objects labeled from any of the 3 criteria: “gender”, “category”, and “main color”, whose number of observations is larger than that of the "colored-MNIST" data set.<br />
<br />
==Use of Pre-Trained CNN Feature Layers==<br />
<br />
In the Kaggle Fashion Product experiment, CSL trains fully-connected layers that have been attached to feature-identifying layers from pre-trained Convolutional Neural Networks. The CSL methods autonomously learned three tasks which corresponded exactly to “Gender”,<br />
“Category”, and “Color” as we see it.<br />
<br />
==Metrics of Confusing Supervised Learning==<br />
<br />
There are two measures of accuracy used to evaluate and compare CSL to other methods, corresponding respectively to the accuracy of the task labeling and the accuracy of the learned mapping function. <br />
<br />
'''Task Prediction Accuracy''': <math>\alpha_T(j)</math> is the average number of times the learned deconfusing function <math>h</math> agrees with the task-assignment ability of humans <math>\tilde h</math> on whether each observation in the data "is" or "is not" in task <math>j</math>.<br />
<br />
$$ \alpha_T(j) = \operatorname{max}_k\frac{1}{m}\sum_{i=1}^m I[h(x_i,y_i;f_k),\tilde h(x_i,y_i;f_j)]$$<br />
<br />
The max over <math>k</math> is taken because we need to determine which learned task corresponds to which ground-truth task.<br />
<br />
'''Label Prediction Accuracy''': <math>\alpha_L(j)</math> again chooses <math>f_k</math>, the learned mapping function that is closest to the ground-truth of task <math>j</math>, and measures its average absolute accuracy compared to the ground-truth of task <math>j</math>, <math>f_j</math>, across all <math>m</math> observations.<br />
<br />
$$ \alpha_L(j) = \operatorname{max}_k\frac{1}{m}\sum_{i=1}^m 1-\dfrac{|g_k(x_i)-f_j(x_i)|}{|f_j(x_i)|}$$<br />
<br />
The purpose of this measure arises from the fact that, in addition to learning mapping allocations like humans, machines should be able to approximate all mapping functions accurately in order to provide corresponding labels. The Label Prediction Accuracy measure captures the exchange equivalence of the following task: each mapping contains its ground-truth output, and machines should be predicting the correct output that is close to the ground-truth. <br />
<br />
==Results==<br />
<br />
Given confusing data, CSL performs better than traditional supervised learning methods, Pseudo-Label(Lee, 2013), and SMiLE(Tan et al., 2017). This is demonstrated by CSL's <math>\alpha_L</math> scores of around 95%, compared to <math>\alpha_L</math> scores of under 50% for the other methods. This supports the assertion that traditional methods only learn the means of all the ground-truth mapping functions when presented with confusing data.<br />
<br />
'''Function Regression''': To "correctly" partition the observations into the correct tasks, a 5-shot warm-up was used. In this situation, the CSL methods work well in learning the ground-truth. That means the initialization of the neural network is set up properly.<br />
<br />
'''Image Classification''': Visualizations created through Spectral embedding confirm the task labelling proficiency of the deconfusing neural network <math>h</math>.<br />
<br />
The classification and function prediction accuracy of CSL are comparable to supervised learning programs that have been given access to the ground-truth labels.<br />
<br />
==Application of Multi-label Learning==<br />
<br />
CSL also had better accuracy than traditional supervised learning methods, Pseudo-Label(Lee, 2013), and SMiLE(Tan et al., 2017) when presented with partially labelled multi-label data <math>(x_i,y_i)</math>, where <math>y_i</math> is a <math>n</math>-long indicator vector for whether the image <math>(x_i,y_i)</math> corresponds to each of the <math>n</math> labels.<br />
<br />
Applications of multi-label classification include building a recommendation system, social media targeting, as well as detecting adverse drug reactions from the text.<br />
<br />
Multi-label can be used to improve the syndrome diagnosis of a patient by focusing on multiple syndromes instead of a single syndrome.<br />
<br />
==Limitations==<br />
<br />
'''Number of Tasks''': The number of tasks is determined by increasing the task numbers progressively and testing the performance. Ideally, a better way of deciding the number of tasks is expected rather than increasing it one by one and seeing which is the minimum number of tasks that gives the smallest risk. Adding low-quality constraints to deconfusing-net is a reasonable solution to this problem.<br />
<br />
'''Learning of Basic Features''': The CSL framework is not good at learning features. So far, a pre-trained CNN backbone is needed for complicated image classification problems. Even though the effectiveness of the proposed algorithm in learning confusing data based on pre-trained features hasn't been affected, the full-connect network can only be trained based on learned CNN features. It is still a challenge for the current algorithm to learn basic features directly through a CNN structure and understand tasks simultaneously.<br />
<br />
= Conclusion =<br />
<br />
This paper proposes the CSL method for tackling the multi-task learning problem without manual task annotations from basic input data. The model obtains a basic task concept by learning the minimum risk for confusing samples from differentiating multiple mappings. The paper also demonstrates that the CSL method is an important step to moving from Narrow AI towards General AI for multi-task learning.<br />
<br />
However, some limitations can be improved for future work:<br />
<br />
- The repeated training process of determining the lowest best task number that has the closest to zero causes inefficiency in the learning process; <br />
<br />
- The current algorithm is difficult to learn basic features directly through a CNN structure and understand tasks simultaneously by training a full-connect network. However, this limitation does not affect the effectiveness of our algorithm in learning confusing data based on pre-trained features.<br />
<br />
= Critique =<br />
<br />
The classification accuracy of CSL was made with algorithms not designed to deal with confusing data and which do not first classify the task of each observation.<br />
<br />
Human task annotation is also imperfect, so one additional application of CSL may be to attempt to flag task annotation errors made by humans, such as in sorting comments for items sold by online retailers; concerned customers, in particular, may not correctly label their comments as "refund", "order didn't arrive", "order damaged", "how good the item is" etc.<br />
<br />
Compared to the standard supervised learning, Multi-label learning can associate a training sample with multiple category tags at the same time. It can assign multiple labels to some hidden instances and can be reduced to standard supervised learning by limiting the number of class labels per instance. <br />
<br />
This algorithm will also have a huge issue in scaling, as the proposed method requires repeated training processes, so it might be too expensive for researchers to implement and improve on this algorithm.<br />
<br />
This research paper should have included a plot on loss (of both functions) against epochs in the paper. A common issue with fixing the parameters of one network and updating the other is the variability during training. This is prevalent in other algorithms with similar training methods such as generative adversarial networks (GAN). For instance, ''mode collapse'' is the issue of one network stuck in local minima and other networks that rely on this network may receive incorrect signals during backpropagation. In the case of CSL-Net, since the Deconfusing-Net directly relies on Mapping-Net for training labels, if the Mapping-Net is unable to sufficiently converge, the Deconfusing-Net may incorrectly learn the mapping from inputs to the task. For data with high noise, oscillations may severely prolong the time needed to converge because of the strong correlation in prediction between the two networks.<br />
<br />
- It would be interesting to see this implemented in more examples, to test the robustness of different types of data. The validation tasks chosen by data are all very simple, and CSL is actually not necessary for those tasks. For the colored MNIST data, a simple function can be written to distinguish the color label from the number label. The same problem applied to the Kaggle Fashion product dataset. The candidate label can be easily classified into different tasks by some wording analysis or meaning classification program or even manual classification. Even though the idea discussed by authors are interesting, the examples suggested by authors seem to suggest very limited or even unnecessary application. In most cases, it is more beneficial to treat the Confusing Multi-task Data problems separately into two distinct stages: we classify the tasks first according to the meaning of the label, and then we perform a multi-class/multi-label training process.<br />
<br />
Even though this paper has already included some examples when testing the CSL in experiments, it will be better to include more detailed examples for partial-label in the "Application of Multi-label Learning" section.<br />
<br />
When using this framework for classification, the order of the one-hot classification labels for each task will likely influence the relationships learned between each task, since the same output header is used for all tasks. This may be why this method fails to learn low-level representations and requires pretraining. I would like to see more explanation in the paper about why this isn't a problem if it was investigated.<br />
<br />
It would be a good idea to include comparison details in the summary to make the results and the conclusion more convincing. For instance, though the paper introduced the result generated using confusion data, and provide some applications for multi-label learning, these two sections still fell short and could use some technical details as supporting evidence.<br />
<br />
It is interesting to investigate if the order of adding tasks will influence the model performance.<br />
<br />
It would be interesting to see the effectiveness of applying CSL in face recognition, such that not only does the algorithm map the face to identity, it also categorizes the face based on other features like beard/no beard and glasses/no glasses simultaneously.<br />
<br />
It would be better for the researchers to compare the efficiency of this approach with other models.<br />
<br />
For pattern recognition,pre-trained features were used in the algorithm. It would be interesting to see how the effectiveness of the model changes if we train it with data directly from the CNN structure in the future.<br />
<br />
So basically given a confused dataset CSL finds the important tasks or labels from the dataset as can be seen from the fruit example. In the example, fruits are grouped under their names, their tastes, and their color, when CSL is given a mixed dataset. Hence given an unstructured data, unlabeled, confused dataset CSL helps in finding the labels, which in turn can help in cleaning the dataset and further in preparing high-quality training data set which is very important in different ML algorithms. Since at present preparing these dataset requires manual data annotations, CSL can save time in that process.<br />
<br />
For the Colorful-Mnist data set, the goal is to understand the concept of multiple classification tasks from these examples. All inputs have multiple classification tasks. Each observed sample only represents the classification result of one task, and the task from which the sample comes is unknown.<br />
<br />
It would be nice to know why the given metrics of confusing supervised learning are used. The authors should have used several different metrics and show that CSL's overall performs better than other methods. And what are "the other methods" referring to? algorithm<br />
<br />
In the Iterative Deconfusing algorithm section, the Training of Mapping-Net needs more explanation. The authors should specify what it is doing before showing its equations.<br />
<br />
For the results section, it would be more intuitive and stronger if the author provide more detail on these two methods and add a plot to support the claim. Based on the text, it might not be an obvious comparison.<br />
<br />
It will be interesting to see if this model can work for other datasets such as CIFAR-10, CIFAR-100, and ImageNet and how well it will perform on those datasets.<br />
<br />
= References =<br />
<br />
[1] Su, Xin, et al. "Task Understanding from Confusing Multi-task Data."<br />
<br />
[2] Caruana, R. (1997) "Multi-task learning"<br />
<br />
[3] Lee, D.-H. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. Workshop on challenges in representation learning, ICML, vol. 3, 2013, pp. 2–8. <br />
<br />
[4] Tan, Q., Yu, Y., Yu, G., and Wang, J. Semi-supervised multi-label classification using incomplete label information. Neurocomputing, vol. 260, 2017, pp. 192–202.<br />
<br />
[5] Chavdarova, Tatjana, and François Fleuret. "Sgan: An alternative training of generative adversarial networks." In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 9407-9415. 2018.<br />
<br />
[6] Guo-Ping Liu, Jian-Jun Yan, Yi-Qin Wang, Jing-Jing Fu, Zhao-Xia Xu, Rui Guo, Peng Qian, "Application of Multilabel Learning Using the Relevant Feature for Each Label in Chronic Gastritis Syndrome Diagnosis", Evidence-Based Complementary and Alternative Medicine, vol. 2012, Article ID 135387, 9 pages, 2012. https://doi.org/10.1155/2012/135387</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Yktan&diff=49216User:Yktan2020-12-05T17:02:37Z<p>Wtjung: /* Model Architecture and Algorithm */</p>
<hr />
<div><br />
== Introduction ==<br />
<br />
Much of the success in training deep neural networks (DNNs) is due to the collection of large datasets with human-annotated labels. However, human annotation is both a time-consuming and expensive task, especially for data that requires expertise such as medical data. Furthermore, certain datasets will be noisy due to the biases introduced by different annotators. Data obtained in large quantities through searching for images in search engines and data downloaded from social media sites (in a manner abiding by privacy and copyright laws) are especially noisy, since the labels are generally inferred from tags to save on human-annotation cost. <br />
<br />
There are a few existing approaches to use datasets with noisy labels. In learning with noisy labels (LNL), most methods take a loss correction approach. Other LNL methods estimate a noise transition matrix and employ it to correct the loss function. An example of a popular loss correction approach is the bootstrapping loss approach. Another approach to reduce annotation cost is semi-supervised learning (SSL), where the training data consists of labeled and unlabeled samples. The main limitation of these methods is that they do not perform well under high noise ratio and cause overfitting.<br />
<br />
This paper introduces DivideMix, which combines approaches from LNL and SSL. One unique thing about DivideMix is that it discards sample labels that are highly likely to be noisy and leverages these noisy samples as unlabeled data instead. This prevents the model from overfitting and improves generalization performance. Key contributions of this work are:<br />
1) Co-divide, which trains two networks simultaneously, aims to improve generalization and avoid confirmation bias.<br />
2) During the SSL phase, an improvement is made on an existing method (MixMatch) by combining it with another method (MixUp).<br />
3) Significant improvements to state-of-the-art results on multiple conditions are experimentally shown while using DivideMix. Extensive ablation study and qualitative results are also shown to examine the effect of different components.<br />
<br />
== Motivation ==<br />
<br />
While much has been achieved in training DNNs with noisy labels and SSL methods individually, not much progress has been made in exploring their underlying connections and building on top of the two approaches simultaneously. <br />
<br />
Existing LNL methods aim to correct the loss function by:<br />
<ol><br />
<li> Treating all samples equally and correcting loss explicitly or implicitly through relabelling of the noisy samples<br />
<li> Reweighting training samples or separating clean and noisy samples, which results in correction of the loss function<br />
</ol><br />
<br />
A few examples of LNL methods include:<br />
<ol><br />
<li> Estimating the noise transition matrix, which denotes the probability of clean labels flipping to noisy labels, to correct the loss function<br />
<li> Leveraging the predictions from DNNs to correct labels and using them to modify the loss<br />
<li> Reweighting samples so that noisy labels contribute less to the loss<br />
</ol><br />
<br />
However, these methods all have downsides: it is very challenging to correctly estimate the noise transition matrix in the first method; for the second method, DNNs tend to overfit to datasets with high noise ratio; and for the third method, we need to be able to identify clean samples, which has also proven to be challenging.<br />
<br />
On the other hand, SSL methods mostly leverage unlabeled data using regularization to improve model performance. A recently proposed method, MixMatch, incorporates the two classes of regularization. These classes are consistency regularization which enforces the model to produce consistent predictions on augmented input data, and entropy minimization which encourages the model to give high-confidence predictions on unlabeled data, as well as MixUp regularization. <br />
<br />
DivideMix partially adopts LNL in that it removes the labels that are highly likely to be noisy by using co-divide to avoid the confirmation bias problem. It then utilizes the noisy samples as unlabeled data and adopts an improved version of MixMatch (an SSL technique) which accounts for the label noise during the label co-refinement and co-guessing phase. By incorporating SSL techniques into LNL and taking the best of both worlds, DivideMix aims to produce highly promising results in training DNNs by better addressing the confirmation bias problem, more accurately distinguishing and utilizing noisy samples, and performing well under high levels of noise.<br />
<br />
== Model Architecture and Algorithm ==<br />
<br />
DivideMix leverages semi-supervised learning to achieve effective modeling. The sample is first split into a labeled set and an unlabeled set. This is achieved by fitting a Gaussian Mixture Model as a per-sample loss distribution. The unlabeled set is made up of data points with discarded labels deemed noisy. <br />
<br />
Then, to avoid confirmation bias, which is typical when a model is self-training, two models are being trained simultaneously to filter error for each other. This is done by dividing the data into clean labeled set (X)and a noisy unlabeled set (U) using one model and then training the other model on the data set U. This algorithm, known as Co-divide, keeps the two networks from converging when training, which avoids the bias from occurring. Gaussian Mixture Model is better at distinguishing X and U, whereas Beta Mixture Model produces flat distribution and fails to label correctly. Being diverged also offers the two networks distinct abilities to filter different types of error, making the model more robust to noise. However, the model could still have confirmation error where both model would prone to make and confirm the same mistake. Figure 1 describes the algorithm in graphical form.<br />
<br />
[[File:ModelArchitecture.PNG | center]]<br />
<br />
<div align="center">Figure 1: Model Architecture of DivideMix</div><br />
<br />
For each epoch, the network divides the dataset into a labeled set consisting of clean data, and an unlabeled set consisting of noisy data, which is then used as training data for the other network, where training is done in mini-batches. For each batch of the labelled samples, co-refinement is performed by using the ground truth label <math> y_b </math>, the predicted label <math> p_b </math>, and the posterior is used as the weight, <math> w_b </math>. <br />
<br />
<center><math> \bar{y}_b = w_b y_b + (1-w_b) p_b </math></center> <br />
<br />
Then, a sharpening function is implemented on this weighted sum to produce the estimate with reduced temperature, <math> \hat{y}_b </math>. <br />
<br />
<center><math> \hat{y}_b=Sharpen(\bar{y}_b,T)={\bar{y}^{c{\frac{1}{T}}}_b}/{\sum_{c=1}^C\bar{y}^{c{\frac{1}{T}}}_b} </math>, for <math>c = 1, 2,..,C</math></center><br />
<br />
Using all these predicted labels, the unlabeled samples will then be assigned a "co-guessed" label, which should produce a more accurate prediction. Having calculated all these labels, MixMatch is applied to the combined mini-batch of labeled, <math> \hat{X} </math> and unlabeled data, <math> \hat{U} </math>, where, for a pair of samples and their labels, one new sample and new label is produced. More specifically, for a pair of samples <math> (x_1,x_2) </math> and their labels <math> (p_1,p_2) </math>, the mixed sample <math> (x',p') </math> is:<br />
<br />
<center><br />
<math><br />
\begin{alignat}{2}<br />
<br />
\lambda &\sim Beta(\alpha, \alpha) \\<br />
\lambda ' &= max(\lambda, 1 - \lambda) \\<br />
x' &= \lambda ' x_1 + (1 - \lambda ' ) x_2 \\<br />
p' &= \lambda ' p_1 + (1 - \lambda ' ) p_2 \\<br />
<br />
\end{alignat}<br />
</math><br />
</center> <br />
<br />
MixMatch transforms <math> \hat{X} </math> and <math> \hat{U} </math> into <math> X' </math> and <math> U' </math>. Then, the loss on <math> X' </math>, <math> L_X </math> (Cross-entropy loss) and the loss on <math> U' </math>, <math> L_U </math> (Mean Squared Error) are calculated. A regularization term, <math> L_{reg} </math>, is introduced to regularize the model's average output across all samples in the mini-batch. Then, the total loss is calculated as:<br />
<br />
<center><math> L = L_X + \lambda_u L_U + \lambda_r L_{reg} </math></center> <br />
<br />
where <math> \lambda_r </math> is set to 1, and <math> \lambda_u </math> is used to control the unsupervised loss.<br />
<br />
Lastly, the stochastic gradient descent formula is updated with the calculated loss, <math> L </math>, and the estimated parameters, <math> \boldsymbol{ \theta } </math>.<br />
<br />
The full algorithm is shown below. [[File:dividemix.jpg|600px| | center]]<br />
<div align="center">Algorithm1: DivideMix. Line 4-8: co-divide; Line 17-18: label co-refinement; Line 20: co-guessing.</div><br />
<br />
Then, when the model is warmed up, it is trained on all data using standard cross-entropy to initially converge the model, but with a regulatory negative entropy term <math>\mathcal{H} = -\sum_{c}\text{p}^\text{c}_\text{model}(x;\theta)\log(\text{p}^\text{c}_\text{model}(x;\theta))</math>, where <math>\text{p}^\text{c}_\text{model}</math> is the softmax output probability for class c. This term penalizes confident predictions during the warm up to prevent overfitting to noise during the warm up, which can happen when there is asymmetric noise.<br />
<br />
== Results ==<br />
'''Applications'''<br />
<br />
The method was validated using four benchmark datasets: CIFAR-10, CIFAR100 (Krizhevsky & Hinton, 2009) which contain 50K training images and 10K test images of size 32 × 32), Clothing1M (Xiao et al., 2015), and WebVision (Li et al., 2017a).<br />
Two types of label noise are used in the experiments: symmetric and asymmetric.<br />
An 18-layer PreAct Resnet (He et al., 2016) is trained using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 300 epochs. The initial learning rate was set to 0.02 and reduced by a factor of 10 after 150 epochs. Before applying the Co-divide and MixMatch strategies, the models were first independently trained over the entire dataset using cross-entropy loss during a "warm-up" period. Initially, training the models in this way prepares a more regular distribution of losses to improve upon in subsequent epochs. The warm-up period is 10 epochs for CIFAR-10 and 30 epochs for CIFAR-100. For all CIFAR experiments, we use the same hyperparameters M = 2, T = 0.5, and α = 4. τ is set as 0.5 except for 90% noise ratio when it is set as 0.6.<br />
<br />
<br />
'''Comparison of State-of-the-Art Methods'''<br />
<br />
The effectiveness of DivideMix was shown by comparing the test accuracy with the most recent state-of-the-art methods: <br />
Meta-Learning (Li et al., 2019) proposes a gradient-based method to find model parameters that are more noise-tolerant; <br />
Joint-Optim (Tanaka et al., 2018) and P-correction (Yi & Wu, 2019) jointly optimize the sample labels and the network parameters;<br />
M-correction (Arazo et al., 2019) models sample loss with BMM and apply MixUp.<br />
The following are the results on CIFAR-10 and CIFAR-100 with different levels of symmetric label noise ranging from 20% to 90%. Both the best test accuracy across all epochs and the averaged test accuracy over the last 10 epochs were recorded in the following table:<br />
<br />
<br />
[[File:divideMixtable1.PNG | center]]<br />
<br />
From table 1, the author noticed that none of these methods can consistently outperform others across different datasets. M-correction excels at symmetric noise, whereas Meta-Learning performs better for asymmetric noise. DivideMix outperforms state-of-the-art methods by a large margin across all noise ratios. The improvement is substantial (∼10% of accuracy) for the more challenging CIFAR-100 with high noise ratios.<br />
<br />
DivideMix was compared with the state-of-the-art methods with the other two datasets: Clothing1M and WebVision. It also shows that DivideMix consistently outperforms state-of-the-art methods across all datasets with different types of label noise. For WebVision, DivideMix achieves more than 12% improvement in top-1 accuracy. <br />
<br />
<br />
'''Ablation Study'''<br />
<br />
The effect of removing different components to provide insights into what makes DivideMix successful. We analyze the results in Table 5 as follows.<br />
<br />
<br />
[[File:DivideMixtable5.PNG | center]]<br />
<br />
The authors combined self-divide with the original MixMatch as a naive baseline for using SLL in LNL.<br />
They also find that both label refinement and input augmentation are beneficial for DivideMix. ''Label refinement'' is important for high noise ratio due because samples that are noisier would be incorrectly divided into the labeled set. ''Augmentation'' upgrades model performance by creating more reliable predictions and by achieving consistent regularization. In addition, the performance drop was seen in the ''DivideMix w/o co-training'' highlights the disadvantage of self-training; the model still has dataset division, label refinement and label guessing, but they are all performed by the same model.<br />
<br />
== Conclusion ==<br />
<br />
This paper provides a new and effective algorithm for learning with noisy labels by using highly noisy data unlabelled data in a Semi-Supervised Learning framework. The DivideMix method trains two networks simultaneously and utilizes co-guessing and co-labeling effectively, therefore it is a robust approach to deal with noise in datasets. Also, the DivideMix method has been tested using various datasets with the results consistently being one of the best when compared to the state-of-the-art methods through extensive experiments.<br />
<br />
Future work of DivideMix is to create an adaptation for other applications such as Natural Language Processing, and incorporating the ideas of SSL and LNL into DivideMix architecture.<br />
<br />
== Critiques/ Insights ==<br />
<br />
1. While combining both models makes the result better, the author did not show the relative time increase using this new combined methodology, which is very crucial considering training a large amount of data, especially for images. In addition, it seems that the author did not perform much on hyperparameters tuning for the combined model.<br />
<br />
2. There is an interesting insight, which is when the noise ratio increases from 80% to 90%, the accuracy of DivideMix drops dramatically in both datasets.<br />
<br />
3. There should be a further explanation of why the learning rate drops by a factor of 10 after 150 epochs.<br />
<br />
4. It would be interesting to see the effectiveness of this method in other domains such as NLP. I am not aware of noisy training datasets available in NLP, but surely this is an important area to focus on, as much of the available data is collected from noisy sources from the web.<br />
<br />
5. The paper implicitly assumes that a Gaussian mixture model (GMM) is sufficiently capable of identifying noise. Given the nature of a GMM, it would work well for noise that is distributed by a Gaussian distribution but for all other noise, it would probably be only asymptotic. The paper should present theoretical results on the noise that are Exponential, Rayleigh, etc. This is particularly important because the experiments were done on massive datasets, but they do not directly address the case when there are not many data points. <br />
<br />
6. Comparing the training result on these benchmark datasets makes the algorithm quite comprehensive. This is a very insightful idea to maintain two networks to avoid bias from occurring.<br />
<br />
7. The current benchmark accuracy for CIFAR-10 is 99.7, CIFAR-100 is 96.08 using EffNet-L2 in 2020. In 2019, CIFAR-10 is 99.37, CIFAR-100 is 93.51 using BiT-L.(based on paperswithcode.com) As there exists better methods, it would be nice to know why the authors chose these state-of-the-art methods to compare the test accuracy.<br />
<br />
8. Another interesting observation is that DivideMix seems to maintain a similar accuracy while some methods give unstable results. That shows the reliability of the proposed algorithm.<br />
<br />
9. It would be interesting to see if the drop in accuracy from increasing the noise ratio to 90% is a result of a low porportion or low number of clean labels. That is, would increasing the size of the training set but keeping the noise ratio at 90% result in increased accuracy?<br />
<br />
10. For Ablation Study part, the paper also introduced a study on the Robustness of Testing Marking Methods Noise, including AUC for classification of clean/noisy samples of CIFAR-10 training data. And it shows that the method can effectively separate clean and noisy samples as training proceeds.<br />
<br />
11. It is interesting how unlike common methods, the method in this paper discards the labels that are highly likely to be<br />
noisy. It also utilizes the noisy samples as unlabeled data to regularize training in a SSL manner. This model can better distinguish and utilize noisy samples.<br />
<br />
12. In the result section, the author gives us a comprehensive understanding of this algorithm by introducing the applications and the comparison of it with respect to similar methods. It would be attractive if in the application part, the author could indicate how the application relative to our daily life.<br />
<br />
13. High quality data is very important for training Machine learning systems. Preparing the data to train ML systems requires data annotations which are prone to errors and are time-consuming. It is interesting to note how paper 14 and this paper aims to approach this problem from different perspectives. Paper 14 introduces CSL algorithm that learns from confused or Noisy data to find the tasks associated with them. And this paper proposes an algorithm that shows good performance when learning from noisy data. Hence both the papers seem to tackle similar problem and implementing the approaches described in both the papers when handling noisy data can be twice helpful.<br />
<br />
14. Noise exists in all big data, and big data is what we are dealing with in real life nowadays. Having an effective noise eliminating method such as Dividemix is important to us.<br />
<br />
15. The DivideMix consistently outperforms state-of-the-art methods across the given datasets, but how about some other potential datasets? If it can be given that it has advantages for a certain type of potential dataset, it will be a better discussion.<br />
<br />
16. It would be better if there was more information regarding four benchmark datasets (CIFAR-10, CIFAR100, Clothing1M, and WebVision) so that readers could be aware of the properties or differences of those datasets.<br />
<br />
== References ==<br />
[1] Eric Arazo, Diego Ortego, Paul Albert, Noel E. O’Connor, and Kevin McGuinness. Unsupervised<br />
label noise modeling and loss correction. In ICML, pp. 312–321, 2019.<br />
<br />
[2] David Berthelot, Nicholas Carlini, Ian J. Goodfellow, Nicolas Papernot, Avital Oliver, and Colin<br />
Raffel. Mixmatch: A holistic approach to semi-supervised learning. NeurIPS, 2019.<br />
<br />
[3] Yifan Ding, Liqiang Wang, Deliang Fan, and Boqing Gong. A semi-supervised two-stage approach<br />
to learning from noisy labels. In WACV, pp. 1215–1224, 2018.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Yktan&diff=49215User:Yktan2020-12-05T16:58:44Z<p>Wtjung: /* Critiques/ Insights */</p>
<hr />
<div><br />
== Introduction ==<br />
<br />
Much of the success in training deep neural networks (DNNs) is due to the collection of large datasets with human-annotated labels. However, human annotation is both a time-consuming and expensive task, especially for data that requires expertise such as medical data. Furthermore, certain datasets will be noisy due to the biases introduced by different annotators. Data obtained in large quantities through searching for images in search engines and data downloaded from social media sites (in a manner abiding by privacy and copyright laws) are especially noisy, since the labels are generally inferred from tags to save on human-annotation cost. <br />
<br />
There are a few existing approaches to use datasets with noisy labels. In learning with noisy labels (LNL), most methods take a loss correction approach. Other LNL methods estimate a noise transition matrix and employ it to correct the loss function. An example of a popular loss correction approach is the bootstrapping loss approach. Another approach to reduce annotation cost is semi-supervised learning (SSL), where the training data consists of labeled and unlabeled samples. The main limitation of these methods is that they do not perform well under high noise ratio and cause overfitting.<br />
<br />
This paper introduces DivideMix, which combines approaches from LNL and SSL. One unique thing about DivideMix is that it discards sample labels that are highly likely to be noisy and leverages these noisy samples as unlabeled data instead. This prevents the model from overfitting and improves generalization performance. Key contributions of this work are:<br />
1) Co-divide, which trains two networks simultaneously, aims to improve generalization and avoid confirmation bias.<br />
2) During the SSL phase, an improvement is made on an existing method (MixMatch) by combining it with another method (MixUp).<br />
3) Significant improvements to state-of-the-art results on multiple conditions are experimentally shown while using DivideMix. Extensive ablation study and qualitative results are also shown to examine the effect of different components.<br />
<br />
== Motivation ==<br />
<br />
While much has been achieved in training DNNs with noisy labels and SSL methods individually, not much progress has been made in exploring their underlying connections and building on top of the two approaches simultaneously. <br />
<br />
Existing LNL methods aim to correct the loss function by:<br />
<ol><br />
<li> Treating all samples equally and correcting loss explicitly or implicitly through relabelling of the noisy samples<br />
<li> Reweighting training samples or separating clean and noisy samples, which results in correction of the loss function<br />
</ol><br />
<br />
A few examples of LNL methods include:<br />
<ol><br />
<li> Estimating the noise transition matrix, which denotes the probability of clean labels flipping to noisy labels, to correct the loss function<br />
<li> Leveraging the predictions from DNNs to correct labels and using them to modify the loss<br />
<li> Reweighting samples so that noisy labels contribute less to the loss<br />
</ol><br />
<br />
However, these methods all have downsides: it is very challenging to correctly estimate the noise transition matrix in the first method; for the second method, DNNs tend to overfit to datasets with high noise ratio; and for the third method, we need to be able to identify clean samples, which has also proven to be challenging.<br />
<br />
On the other hand, SSL methods mostly leverage unlabeled data using regularization to improve model performance. A recently proposed method, MixMatch, incorporates the two classes of regularization. These classes are consistency regularization which enforces the model to produce consistent predictions on augmented input data, and entropy minimization which encourages the model to give high-confidence predictions on unlabeled data, as well as MixUp regularization. <br />
<br />
DivideMix partially adopts LNL in that it removes the labels that are highly likely to be noisy by using co-divide to avoid the confirmation bias problem. It then utilizes the noisy samples as unlabeled data and adopts an improved version of MixMatch (an SSL technique) which accounts for the label noise during the label co-refinement and co-guessing phase. By incorporating SSL techniques into LNL and taking the best of both worlds, DivideMix aims to produce highly promising results in training DNNs by better addressing the confirmation bias problem, more accurately distinguishing and utilizing noisy samples, and performing well under high levels of noise.<br />
<br />
== Model Architecture and Algorithm ==<br />
<br />
DivideMix leverages semi-supervised learning to achieve effective modeling. The sample is first split into a labeled set and an unlabeled set. This is achieved by fitting a Gaussian Mixture Model as a per-sample loss distribution. The unlabeled set is made up of data points with discarded labels deemed noisy. <br />
<br />
Then, to avoid confirmation bias, which is typical when a model is self-training, two models are being trained simultaneously to filter error for each other. This is done by dividing the data into clean labeled set (X)and a noisy unlabeled set (U) using one model and then training the other model on the data set U. This algorithm, known as Co-divide, keeps the two networks from converging when training, which avoids the bias from occurring. Gaussian Mixture Model is better at distinguishing X and U, whereas Beta Mixture Model produces flat distribution and fails to label correctly. Being diverged also offers the two networks distinct abilities to filter different types of error, making the model more robust to noise. However, the model could still have confirmation error where both model would prone to make and confirm the same mistake. Figure 1 describes the algorithm in graphical form.<br />
<br />
[[File:ModelArchitecture.PNG | center]]<br />
<br />
<div align="center">Figure 1: Model Architecture of DivideMix</div><br />
<br />
For each epoch, the network divides the dataset into a labeled set consisting of clean data, and an unlabeled set consisting of noisy data, which is then used as training data for the other network, where training is done in mini-batches. For each batch of the labelled samples, co-refinement is performed by using the ground truth label <math> y_b </math>, the predicted label <math> p_b </math>, and the posterior is used as the weight, <math> w_b </math>. <br />
<br />
<center><math> \bar{y}_b = w_b y_b + (1-w_b) p_b </math></center> <br />
<br />
Then, a sharpening function is implemented on this weighted sum to produce the estimate with reduced temperature, <math> \hat{y}_b </math>. <br />
<br />
<center><math> \hat{y}_b=Sharpen(\bar{y}_b,T)={\bar{y}^{c{\frac{1}{T}}}_b}/{\sum_{c=1}^C\bar{y}^{c{\frac{1}{T}}}_b} </math>, for <math>c = 1, 2,..,C</math></center><br />
<br />
Using all these predicted labels, the unlabeled samples will then be assigned a "co-guessed" label, which should produce a more accurate prediction. Having calculated all these labels, MixMatch is applied to the combined mini-batch of labeled, <math> \hat{X} </math> and unlabeled data, <math> \hat{U} </math>, where, for a pair of samples and their labels, one new sample and new label is produced. More specifically, for a pair of samples <math> (x_1,x_2) </math> and their labels <math> (p_1,p_2) </math>, the mixed sample <math> (x',p') </math> is:<br />
<br />
<center><br />
<math><br />
\begin{alignat}{2}<br />
<br />
\lambda &\sim Beta(\alpha, \alpha) \\<br />
\lambda ' &= max(\lambda, 1 - \lambda) \\<br />
x' &= \lambda ' x_1 + (1 - \lambda ' ) x_2 \\<br />
p' &= \lambda ' p_1 + (1 - \lambda ' ) p_2 \\<br />
<br />
\end{alignat}<br />
</math><br />
</center> <br />
<br />
MixMatch transforms <math> \hat{X} </math> and <math> \hat{U} </math> into <math> X' </math> and <math> U' </math>. Then, the loss on <math> X' </math>, <math> L_X </math> (Cross-entropy loss) and the loss on <math> U' </math>, <math> L_U </math> (Mean Squared Error) are calculated. A regularization term, <math> L_{reg} </math>, is introduced to regularize the model's average output across all samples in the mini-batch. Then, the total loss is calculated as:<br />
<br />
<center><math> L = L_X + \lambda_u L_U + \lambda_r L_{reg} </math></center> <br />
<br />
where <math> \lambda_r </math> is set to 1, and <math> \lambda_u </math> is used to control the unsupervised loss.<br />
<br />
Lastly, the stochastic gradient descent formula is updated with the calculated loss, <math> L </math>, and the estimated parameters, <math> \boldsymbol{ \theta } </math>.<br />
<br />
The full algorithm is shown below. [[File:dividemix.jpg|600px| | center]]<br />
<div align="center">Algorithm1: DivideMix. Line 4-8: co-divide; Line 17-18: label co-refinement; Line 20: co-guessing.</div><br />
<br />
The when the model is warmed up, it is trained on all data using standard cross-entropy to initially converge the model, but with a regulatory negative entropy term <math>\mathcal{H} = -\sum_{c}\text{p}^\text{c}_\text{model}(x;\theta)\log(\text{p}^\text{c}_\text{model}(x;\theta))</math>, where <math>\text{p}^\text{c}_\text{model}</math> is the softmax output probability for class c. This term penalizes confident predictions during the warm up to prevent overfitting to noise during the warm up, which can happen when there is asymmetric noise.<br />
<br />
== Results ==<br />
'''Applications'''<br />
<br />
The method was validated using four benchmark datasets: CIFAR-10, CIFAR100 (Krizhevsky & Hinton, 2009) which contain 50K training images and 10K test images of size 32 × 32), Clothing1M (Xiao et al., 2015), and WebVision (Li et al., 2017a).<br />
Two types of label noise are used in the experiments: symmetric and asymmetric.<br />
An 18-layer PreAct Resnet (He et al., 2016) is trained using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 300 epochs. The initial learning rate was set to 0.02 and reduced by a factor of 10 after 150 epochs. Before applying the Co-divide and MixMatch strategies, the models were first independently trained over the entire dataset using cross-entropy loss during a "warm-up" period. Initially, training the models in this way prepares a more regular distribution of losses to improve upon in subsequent epochs. The warm-up period is 10 epochs for CIFAR-10 and 30 epochs for CIFAR-100. For all CIFAR experiments, we use the same hyperparameters M = 2, T = 0.5, and α = 4. τ is set as 0.5 except for 90% noise ratio when it is set as 0.6.<br />
<br />
<br />
'''Comparison of State-of-the-Art Methods'''<br />
<br />
The effectiveness of DivideMix was shown by comparing the test accuracy with the most recent state-of-the-art methods: <br />
Meta-Learning (Li et al., 2019) proposes a gradient-based method to find model parameters that are more noise-tolerant; <br />
Joint-Optim (Tanaka et al., 2018) and P-correction (Yi & Wu, 2019) jointly optimize the sample labels and the network parameters;<br />
M-correction (Arazo et al., 2019) models sample loss with BMM and apply MixUp.<br />
The following are the results on CIFAR-10 and CIFAR-100 with different levels of symmetric label noise ranging from 20% to 90%. Both the best test accuracy across all epochs and the averaged test accuracy over the last 10 epochs were recorded in the following table:<br />
<br />
<br />
[[File:divideMixtable1.PNG | center]]<br />
<br />
From table 1, the author noticed that none of these methods can consistently outperform others across different datasets. M-correction excels at symmetric noise, whereas Meta-Learning performs better for asymmetric noise. DivideMix outperforms state-of-the-art methods by a large margin across all noise ratios. The improvement is substantial (∼10% of accuracy) for the more challenging CIFAR-100 with high noise ratios.<br />
<br />
DivideMix was compared with the state-of-the-art methods with the other two datasets: Clothing1M and WebVision. It also shows that DivideMix consistently outperforms state-of-the-art methods across all datasets with different types of label noise. For WebVision, DivideMix achieves more than 12% improvement in top-1 accuracy. <br />
<br />
<br />
'''Ablation Study'''<br />
<br />
The effect of removing different components to provide insights into what makes DivideMix successful. We analyze the results in Table 5 as follows.<br />
<br />
<br />
[[File:DivideMixtable5.PNG | center]]<br />
<br />
The authors combined self-divide with the original MixMatch as a naive baseline for using SLL in LNL.<br />
They also find that both label refinement and input augmentation are beneficial for DivideMix. ''Label refinement'' is important for high noise ratio due because samples that are noisier would be incorrectly divided into the labeled set. ''Augmentation'' upgrades model performance by creating more reliable predictions and by achieving consistent regularization. In addition, the performance drop was seen in the ''DivideMix w/o co-training'' highlights the disadvantage of self-training; the model still has dataset division, label refinement and label guessing, but they are all performed by the same model.<br />
<br />
== Conclusion ==<br />
<br />
This paper provides a new and effective algorithm for learning with noisy labels by using highly noisy data unlabelled data in a Semi-Supervised Learning framework. The DivideMix method trains two networks simultaneously and utilizes co-guessing and co-labeling effectively, therefore it is a robust approach to deal with noise in datasets. Also, the DivideMix method has been tested using various datasets with the results consistently being one of the best when compared to the state-of-the-art methods through extensive experiments.<br />
<br />
Future work of DivideMix is to create an adaptation for other applications such as Natural Language Processing, and incorporating the ideas of SSL and LNL into DivideMix architecture.<br />
<br />
== Critiques/ Insights ==<br />
<br />
1. While combining both models makes the result better, the author did not show the relative time increase using this new combined methodology, which is very crucial considering training a large amount of data, especially for images. In addition, it seems that the author did not perform much on hyperparameters tuning for the combined model.<br />
<br />
2. There is an interesting insight, which is when the noise ratio increases from 80% to 90%, the accuracy of DivideMix drops dramatically in both datasets.<br />
<br />
3. There should be a further explanation of why the learning rate drops by a factor of 10 after 150 epochs.<br />
<br />
4. It would be interesting to see the effectiveness of this method in other domains such as NLP. I am not aware of noisy training datasets available in NLP, but surely this is an important area to focus on, as much of the available data is collected from noisy sources from the web.<br />
<br />
5. The paper implicitly assumes that a Gaussian mixture model (GMM) is sufficiently capable of identifying noise. Given the nature of a GMM, it would work well for noise that is distributed by a Gaussian distribution but for all other noise, it would probably be only asymptotic. The paper should present theoretical results on the noise that are Exponential, Rayleigh, etc. This is particularly important because the experiments were done on massive datasets, but they do not directly address the case when there are not many data points. <br />
<br />
6. Comparing the training result on these benchmark datasets makes the algorithm quite comprehensive. This is a very insightful idea to maintain two networks to avoid bias from occurring.<br />
<br />
7. The current benchmark accuracy for CIFAR-10 is 99.7, CIFAR-100 is 96.08 using EffNet-L2 in 2020. In 2019, CIFAR-10 is 99.37, CIFAR-100 is 93.51 using BiT-L.(based on paperswithcode.com) As there exists better methods, it would be nice to know why the authors chose these state-of-the-art methods to compare the test accuracy.<br />
<br />
8. Another interesting observation is that DivideMix seems to maintain a similar accuracy while some methods give unstable results. That shows the reliability of the proposed algorithm.<br />
<br />
9. It would be interesting to see if the drop in accuracy from increasing the noise ratio to 90% is a result of a low porportion or low number of clean labels. That is, would increasing the size of the training set but keeping the noise ratio at 90% result in increased accuracy?<br />
<br />
10. For Ablation Study part, the paper also introduced a study on the Robustness of Testing Marking Methods Noise, including AUC for classification of clean/noisy samples of CIFAR-10 training data. And it shows that the method can effectively separate clean and noisy samples as training proceeds.<br />
<br />
11. It is interesting how unlike common methods, the method in this paper discards the labels that are highly likely to be<br />
noisy. It also utilizes the noisy samples as unlabeled data to regularize training in a SSL manner. This model can better distinguish and utilize noisy samples.<br />
<br />
12. In the result section, the author gives us a comprehensive understanding of this algorithm by introducing the applications and the comparison of it with respect to similar methods. It would be attractive if in the application part, the author could indicate how the application relative to our daily life.<br />
<br />
13. High quality data is very important for training Machine learning systems. Preparing the data to train ML systems requires data annotations which are prone to errors and are time-consuming. It is interesting to note how paper 14 and this paper aims to approach this problem from different perspectives. Paper 14 introduces CSL algorithm that learns from confused or Noisy data to find the tasks associated with them. And this paper proposes an algorithm that shows good performance when learning from noisy data. Hence both the papers seem to tackle similar problem and implementing the approaches described in both the papers when handling noisy data can be twice helpful.<br />
<br />
14. Noise exists in all big data, and big data is what we are dealing with in real life nowadays. Having an effective noise eliminating method such as Dividemix is important to us.<br />
<br />
15. The DivideMix consistently outperforms state-of-the-art methods across the given datasets, but how about some other potential datasets? If it can be given that it has advantages for a certain type of potential dataset, it will be a better discussion.<br />
<br />
16. It would be better if there was more information regarding four benchmark datasets (CIFAR-10, CIFAR100, Clothing1M, and WebVision) so that readers could be aware of the properties or differences of those datasets.<br />
<br />
== References ==<br />
[1] Eric Arazo, Diego Ortego, Paul Albert, Noel E. O’Connor, and Kevin McGuinness. Unsupervised<br />
label noise modeling and loss correction. In ICML, pp. 312–321, 2019.<br />
<br />
[2] David Berthelot, Nicholas Carlini, Ian J. Goodfellow, Nicolas Papernot, Avital Oliver, and Colin<br />
Raffel. Mixmatch: A holistic approach to semi-supervised learning. NeurIPS, 2019.<br />
<br />
[3] Yifan Ding, Liqiang Wang, Deliang Fan, and Boqing Gong. A semi-supervised two-stage approach<br />
to learning from noisy labels. In WACV, pp. 1215–1224, 2018.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Yktan&diff=49214User:Yktan2020-12-05T16:58:07Z<p>Wtjung: /* Critiques/ Insights */</p>
<hr />
<div><br />
== Introduction ==<br />
<br />
Much of the success in training deep neural networks (DNNs) is due to the collection of large datasets with human-annotated labels. However, human annotation is both a time-consuming and expensive task, especially for data that requires expertise such as medical data. Furthermore, certain datasets will be noisy due to the biases introduced by different annotators. Data obtained in large quantities through searching for images in search engines and data downloaded from social media sites (in a manner abiding by privacy and copyright laws) are especially noisy, since the labels are generally inferred from tags to save on human-annotation cost. <br />
<br />
There are a few existing approaches to use datasets with noisy labels. In learning with noisy labels (LNL), most methods take a loss correction approach. Other LNL methods estimate a noise transition matrix and employ it to correct the loss function. An example of a popular loss correction approach is the bootstrapping loss approach. Another approach to reduce annotation cost is semi-supervised learning (SSL), where the training data consists of labeled and unlabeled samples. The main limitation of these methods is that they do not perform well under high noise ratio and cause overfitting.<br />
<br />
This paper introduces DivideMix, which combines approaches from LNL and SSL. One unique thing about DivideMix is that it discards sample labels that are highly likely to be noisy and leverages these noisy samples as unlabeled data instead. This prevents the model from overfitting and improves generalization performance. Key contributions of this work are:<br />
1) Co-divide, which trains two networks simultaneously, aims to improve generalization and avoid confirmation bias.<br />
2) During the SSL phase, an improvement is made on an existing method (MixMatch) by combining it with another method (MixUp).<br />
3) Significant improvements to state-of-the-art results on multiple conditions are experimentally shown while using DivideMix. Extensive ablation study and qualitative results are also shown to examine the effect of different components.<br />
<br />
== Motivation ==<br />
<br />
While much has been achieved in training DNNs with noisy labels and SSL methods individually, not much progress has been made in exploring their underlying connections and building on top of the two approaches simultaneously. <br />
<br />
Existing LNL methods aim to correct the loss function by:<br />
<ol><br />
<li> Treating all samples equally and correcting loss explicitly or implicitly through relabelling of the noisy samples<br />
<li> Reweighting training samples or separating clean and noisy samples, which results in correction of the loss function<br />
</ol><br />
<br />
A few examples of LNL methods include:<br />
<ol><br />
<li> Estimating the noise transition matrix, which denotes the probability of clean labels flipping to noisy labels, to correct the loss function<br />
<li> Leveraging the predictions from DNNs to correct labels and using them to modify the loss<br />
<li> Reweighting samples so that noisy labels contribute less to the loss<br />
</ol><br />
<br />
However, these methods all have downsides: it is very challenging to correctly estimate the noise transition matrix in the first method; for the second method, DNNs tend to overfit to datasets with high noise ratio; and for the third method, we need to be able to identify clean samples, which has also proven to be challenging.<br />
<br />
On the other hand, SSL methods mostly leverage unlabeled data using regularization to improve model performance. A recently proposed method, MixMatch, incorporates the two classes of regularization. These classes are consistency regularization which enforces the model to produce consistent predictions on augmented input data, and entropy minimization which encourages the model to give high-confidence predictions on unlabeled data, as well as MixUp regularization. <br />
<br />
DivideMix partially adopts LNL in that it removes the labels that are highly likely to be noisy by using co-divide to avoid the confirmation bias problem. It then utilizes the noisy samples as unlabeled data and adopts an improved version of MixMatch (an SSL technique) which accounts for the label noise during the label co-refinement and co-guessing phase. By incorporating SSL techniques into LNL and taking the best of both worlds, DivideMix aims to produce highly promising results in training DNNs by better addressing the confirmation bias problem, more accurately distinguishing and utilizing noisy samples, and performing well under high levels of noise.<br />
<br />
== Model Architecture and Algorithm ==<br />
<br />
DivideMix leverages semi-supervised learning to achieve effective modeling. The sample is first split into a labeled set and an unlabeled set. This is achieved by fitting a Gaussian Mixture Model as a per-sample loss distribution. The unlabeled set is made up of data points with discarded labels deemed noisy. <br />
<br />
Then, to avoid confirmation bias, which is typical when a model is self-training, two models are being trained simultaneously to filter error for each other. This is done by dividing the data into clean labeled set (X)and a noisy unlabeled set (U) using one model and then training the other model on the data set U. This algorithm, known as Co-divide, keeps the two networks from converging when training, which avoids the bias from occurring. Gaussian Mixture Model is better at distinguishing X and U, whereas Beta Mixture Model produces flat distribution and fails to label correctly. Being diverged also offers the two networks distinct abilities to filter different types of error, making the model more robust to noise. However, the model could still have confirmation error where both model would prone to make and confirm the same mistake. Figure 1 describes the algorithm in graphical form.<br />
<br />
[[File:ModelArchitecture.PNG | center]]<br />
<br />
<div align="center">Figure 1: Model Architecture of DivideMix</div><br />
<br />
For each epoch, the network divides the dataset into a labeled set consisting of clean data, and an unlabeled set consisting of noisy data, which is then used as training data for the other network, where training is done in mini-batches. For each batch of the labelled samples, co-refinement is performed by using the ground truth label <math> y_b </math>, the predicted label <math> p_b </math>, and the posterior is used as the weight, <math> w_b </math>. <br />
<br />
<center><math> \bar{y}_b = w_b y_b + (1-w_b) p_b </math></center> <br />
<br />
Then, a sharpening function is implemented on this weighted sum to produce the estimate with reduced temperature, <math> \hat{y}_b </math>. <br />
<br />
<center><math> \hat{y}_b=Sharpen(\bar{y}_b,T)={\bar{y}^{c{\frac{1}{T}}}_b}/{\sum_{c=1}^C\bar{y}^{c{\frac{1}{T}}}_b} </math>, for <math>c = 1, 2,..,C</math></center><br />
<br />
Using all these predicted labels, the unlabeled samples will then be assigned a "co-guessed" label, which should produce a more accurate prediction. Having calculated all these labels, MixMatch is applied to the combined mini-batch of labeled, <math> \hat{X} </math> and unlabeled data, <math> \hat{U} </math>, where, for a pair of samples and their labels, one new sample and new label is produced. More specifically, for a pair of samples <math> (x_1,x_2) </math> and their labels <math> (p_1,p_2) </math>, the mixed sample <math> (x',p') </math> is:<br />
<br />
<center><br />
<math><br />
\begin{alignat}{2}<br />
<br />
\lambda &\sim Beta(\alpha, \alpha) \\<br />
\lambda ' &= max(\lambda, 1 - \lambda) \\<br />
x' &= \lambda ' x_1 + (1 - \lambda ' ) x_2 \\<br />
p' &= \lambda ' p_1 + (1 - \lambda ' ) p_2 \\<br />
<br />
\end{alignat}<br />
</math><br />
</center> <br />
<br />
MixMatch transforms <math> \hat{X} </math> and <math> \hat{U} </math> into <math> X' </math> and <math> U' </math>. Then, the loss on <math> X' </math>, <math> L_X </math> (Cross-entropy loss) and the loss on <math> U' </math>, <math> L_U </math> (Mean Squared Error) are calculated. A regularization term, <math> L_{reg} </math>, is introduced to regularize the model's average output across all samples in the mini-batch. Then, the total loss is calculated as:<br />
<br />
<center><math> L = L_X + \lambda_u L_U + \lambda_r L_{reg} </math></center> <br />
<br />
where <math> \lambda_r </math> is set to 1, and <math> \lambda_u </math> is used to control the unsupervised loss.<br />
<br />
Lastly, the stochastic gradient descent formula is updated with the calculated loss, <math> L </math>, and the estimated parameters, <math> \boldsymbol{ \theta } </math>.<br />
<br />
The full algorithm is shown below. [[File:dividemix.jpg|600px| | center]]<br />
<div align="center">Algorithm1: DivideMix. Line 4-8: co-divide; Line 17-18: label co-refinement; Line 20: co-guessing.</div><br />
<br />
The when the model is warmed up, it is trained on all data using standard cross-entropy to initially converge the model, but with a regulatory negative entropy term <math>\mathcal{H} = -\sum_{c}\text{p}^\text{c}_\text{model}(x;\theta)\log(\text{p}^\text{c}_\text{model}(x;\theta))</math>, where <math>\text{p}^\text{c}_\text{model}</math> is the softmax output probability for class c. This term penalizes confident predictions during the warm up to prevent overfitting to noise during the warm up, which can happen when there is asymmetric noise.<br />
<br />
== Results ==<br />
'''Applications'''<br />
<br />
The method was validated using four benchmark datasets: CIFAR-10, CIFAR100 (Krizhevsky & Hinton, 2009) which contain 50K training images and 10K test images of size 32 × 32), Clothing1M (Xiao et al., 2015), and WebVision (Li et al., 2017a).<br />
Two types of label noise are used in the experiments: symmetric and asymmetric.<br />
An 18-layer PreAct Resnet (He et al., 2016) is trained using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 300 epochs. The initial learning rate was set to 0.02 and reduced by a factor of 10 after 150 epochs. Before applying the Co-divide and MixMatch strategies, the models were first independently trained over the entire dataset using cross-entropy loss during a "warm-up" period. Initially, training the models in this way prepares a more regular distribution of losses to improve upon in subsequent epochs. The warm-up period is 10 epochs for CIFAR-10 and 30 epochs for CIFAR-100. For all CIFAR experiments, we use the same hyperparameters M = 2, T = 0.5, and α = 4. τ is set as 0.5 except for 90% noise ratio when it is set as 0.6.<br />
<br />
<br />
'''Comparison of State-of-the-Art Methods'''<br />
<br />
The effectiveness of DivideMix was shown by comparing the test accuracy with the most recent state-of-the-art methods: <br />
Meta-Learning (Li et al., 2019) proposes a gradient-based method to find model parameters that are more noise-tolerant; <br />
Joint-Optim (Tanaka et al., 2018) and P-correction (Yi & Wu, 2019) jointly optimize the sample labels and the network parameters;<br />
M-correction (Arazo et al., 2019) models sample loss with BMM and apply MixUp.<br />
The following are the results on CIFAR-10 and CIFAR-100 with different levels of symmetric label noise ranging from 20% to 90%. Both the best test accuracy across all epochs and the averaged test accuracy over the last 10 epochs were recorded in the following table:<br />
<br />
<br />
[[File:divideMixtable1.PNG | center]]<br />
<br />
From table 1, the author noticed that none of these methods can consistently outperform others across different datasets. M-correction excels at symmetric noise, whereas Meta-Learning performs better for asymmetric noise. DivideMix outperforms state-of-the-art methods by a large margin across all noise ratios. The improvement is substantial (∼10% of accuracy) for the more challenging CIFAR-100 with high noise ratios.<br />
<br />
DivideMix was compared with the state-of-the-art methods with the other two datasets: Clothing1M and WebVision. It also shows that DivideMix consistently outperforms state-of-the-art methods across all datasets with different types of label noise. For WebVision, DivideMix achieves more than 12% improvement in top-1 accuracy. <br />
<br />
<br />
'''Ablation Study'''<br />
<br />
The effect of removing different components to provide insights into what makes DivideMix successful. We analyze the results in Table 5 as follows.<br />
<br />
<br />
[[File:DivideMixtable5.PNG | center]]<br />
<br />
The authors combined self-divide with the original MixMatch as a naive baseline for using SLL in LNL.<br />
They also find that both label refinement and input augmentation are beneficial for DivideMix. ''Label refinement'' is important for high noise ratio due because samples that are noisier would be incorrectly divided into the labeled set. ''Augmentation'' upgrades model performance by creating more reliable predictions and by achieving consistent regularization. In addition, the performance drop was seen in the ''DivideMix w/o co-training'' highlights the disadvantage of self-training; the model still has dataset division, label refinement and label guessing, but they are all performed by the same model.<br />
<br />
== Conclusion ==<br />
<br />
This paper provides a new and effective algorithm for learning with noisy labels by using highly noisy data unlabelled data in a Semi-Supervised Learning framework. The DivideMix method trains two networks simultaneously and utilizes co-guessing and co-labeling effectively, therefore it is a robust approach to deal with noise in datasets. Also, the DivideMix method has been tested using various datasets with the results consistently being one of the best when compared to the state-of-the-art methods through extensive experiments.<br />
<br />
Future work of DivideMix is to create an adaptation for other applications such as Natural Language Processing, and incorporating the ideas of SSL and LNL into DivideMix architecture.<br />
<br />
== Critiques/ Insights ==<br />
<br />
1. While combining both models makes the result better, the author did not show the relative time increase using this new combined methodology, which is very crucial considering training a large amount of data, especially for images. In addition, it seems that the author did not perform much on hyperparameters tuning for the combined model.<br />
<br />
2. There is an interesting insight, which is when the noise ratio increases from 80% to 90%, the accuracy of DivideMix drops dramatically in both datasets.<br />
<br />
3. There should be a further explanation of why the learning rate drops by a factor of 10 after 150 epochs.<br />
<br />
4. It would be interesting to see the effectiveness of this method in other domains such as NLP. I am not aware of noisy training datasets available in NLP, but surely this is an important area to focus on, as much of the available data is collected from noisy sources from the web.<br />
<br />
5. The paper implicitly assumes that a Gaussian mixture model (GMM) is sufficiently capable of identifying noise. Given the nature of a GMM, it would work well for noise that is distributed by a Gaussian distribution but for all other noise, it would probably be only asymptotic. The paper should present theoretical results on the noise that are Exponential, Rayleigh, etc. This is particularly important because the experiments were done on massive datasets, but they do not directly address the case when there are not many data points. <br />
<br />
6. Comparing the training result on these benchmark datasets makes the algorithm quite comprehensive. This is a very insightful idea to maintain two networks to avoid bias from occurring.<br />
<br />
7. The current benchmark accuracy for CIFAR-10 is 99.7, CIFAR-100 is 96.08 using EffNet-L2 in 2020. In 2019, CIFAR-10 is 99.37, CIFAR-100 is 93.51 using BiT-L.(based on paperswithcode.com) As there exists better methods, it would be nice to know why the authors chose these state-of-the-art methods to compare the test accuracy.<br />
<br />
8. Another interesting observation is that DivideMix seems to maintain a similar accuracy while some methods give unstable results. That shows the reliability of the proposed algorithm.<br />
<br />
9. It would be interesting to see if the drop in accuracy from increasing the noise ratio to 90% is a result of a low porportion or low number of clean labels. That is, would increasing the size of the training set but keeping the noise ratio at 90% result in increased accuracy?<br />
<br />
10. For Ablation Study part, the paper also introduced a study on the Robustness of Testing Marking Methods Noise, including AUC for classification of clean/noisy samples of CIFAR-10 training data. And it shows that the method can effectively separate clean and noisy samples as training proceeds.<br />
<br />
11. It is interesting how unlike common methods, the method in this paper discards the labels that are highly likely to be<br />
noisy. It also utilizes the noisy samples as unlabeled data to regularize training in a SSL manner. This model can better distinguish and utilize noisy samples.<br />
<br />
12. In the result section, the author gives us a comprehensive understanding of this algorithm by introducing the applications and the comparison of it with respect to similar methods. It would be attractive if in the application part, the author could indicate how the application relative to our daily life.<br />
<br />
13. High quality data is very important for training Machine learning systems. Preparing the data to train ML systems requires data annotations which are prone to errors and are time-consuming. It is interesting to note how paper 14 and this paper aims to approach this problem from different perspectives. Paper 14 introduces CSL algorithm that learns from confused or Noisy data to find the tasks associated with them. And this paper proposes an algorithm that shows good performance when learning from noisy data. Hence both the papers seem to tackle similar problem and implementing the approaches described in both the papers when handling noisy data can be twice helpful.<br />
<br />
14. Noise exists in all big data, and big data is what we are dealing with in real life nowadays. Having an effective noise eliminating method such as Dividemix is important to us.<br />
<br />
15. The DivideMix consistently outperforms state-of-the-art methods across the given datasets, but how about some other potential datasets? If it can be given that it has advantages for a certain type of potential dataset, it will be a better discussion.<br />
<br />
16. It would be better if there was more information regarding four benchmark datasets: CIFAR-10; CIFAR100; Clothing1M; and WebVision so that readers could be aware of the properties or differences of those datasets.<br />
<br />
== References ==<br />
[1] Eric Arazo, Diego Ortego, Paul Albert, Noel E. O’Connor, and Kevin McGuinness. Unsupervised<br />
label noise modeling and loss correction. In ICML, pp. 312–321, 2019.<br />
<br />
[2] David Berthelot, Nicholas Carlini, Ian J. Goodfellow, Nicolas Papernot, Avital Oliver, and Colin<br />
Raffel. Mixmatch: A holistic approach to semi-supervised learning. NeurIPS, 2019.<br />
<br />
[3] Yifan Ding, Liqiang Wang, Deliang Fan, and Boqing Gong. A semi-supervised two-stage approach<br />
to learning from noisy labels. In WACV, pp. 1215–1224, 2018.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Deep_Learning_for_Cardiologist-level_Myocardial_Infarction_Detection_in_Electrocardiograms&diff=49191Deep Learning for Cardiologist-level Myocardial Infarction Detection in Electrocardiograms2020-12-05T04:35:08Z<p>Wtjung: /* Critiques */</p>
<hr />
<div><br />
== Presented by ==<br />
<br />
Zihui (Betty) Qin, Wenqi (Maggie) Zhao, Muyuan Yang, Amartya (Marty) Mukherjee<br />
<br />
== Introduction ==<br />
<br />
This paper presents ConvNetQuake, an approach on detecting heart disease from ECG signals by fine-tuning the deep learning neural network. For context, ConvNetQuake is a convolutional neural network, used by Perol, Gharbi, and Denolle [4], for Earthquake detection and location from a single waveform. A deep learning approach was used due to the model's ability to be trained using multiple GPUs and terabyte-sized datasets. This, in turn, creates a model that is robust against noise. The purpose of this paper is to provide detailed analyses of the contributions of the ECG leads on identifying heart disease, to show the use of multiple channels in ConvNetQuake enhances prediction accuracy, and to show that feature engineering is not necessary for any of the training, validation, or testing processes. In this area, the combination of data fusion and machine learning techniques exhibits great promise to healthcare innovation, and the analyses in this paper help further this realization. The benefits of translating knowledge between deep learning and its real-world applications in health are also illustrated.<br />
<br />
== Previous Work and Motivation ==<br />
<br />
The database used in previous works is the Physikalisch-Technische Bundesanstalt (PTB) database, which consists of ECG records. Previous papers used techniques, such as CNN, SVM, K-nearest neighbors, naïve Bayes classification, and ANN. From these instances, the paper observes several shortcomings in the previous papers. The first being the issue that most papers use feature selection on the raw ECG data before training the model. Dabanloo and Attarodi [2] used various techniques such as ANN, K-nearest neighbors, and Naïve Bayes. However, they extracted two features, the T-wave integral and the total integral, to aid in localizing and detecting heart disease. Sharma and Sunkaria [3] used SVM and K-nearest neighbors as their classifier, but extracted various features using stationary wavelet transforms to decompose the ECG signal into sub-bands. The second issue is that papers that do not use feature selection would arbitrarily pick ECG leads for classification without rationale. For example, Liu et al. [1] used a deep CNN that uses 3 seconds of ECG signal from lead II at a time as input. The decision for using lead II compared to the other leads was not explained. <br />
<br />
The issue with feature selection is that it can be time-consuming and impractical with large volumes of data. The second issue with the arbitrary selection of leads is that it does not offer insight into why the lead was chosen and the contributions of each lead in the identification of heart disease. Thus, this paper addresses these two issues through implementing a deep learning model that does not rely on feature selection of ECG data and to quantify the contributions of each ECG and Frank lead in identifying heart disease.<br />
<br />
== Model Architecture ==<br />
<br />
The dataset, which was used to train, validate, and test the neural network models, consists of 549 ECG records taken from 290 unique patients. Each ECG record has a mean length of over 100 seconds.<br />
<br />
This Deep Neural Network model was created by modifying the ConvNetQuake model by adding 1D batch normalization layers; this addition helps to combat overfitting. A second modification that was made was to introduce the use of label smoothing, which can help by discouraging the model from making overconfident predictions. Label smoothing refers to the method of relaxing the confidence on the model's prediction labels. The authors' experiments demonstrated that both of these modifications helped to increase model accuracy. <br />
<br />
During the training stage, a 10-second long two-channel input was fed into the neural network. In order to ensure that the two channels were weighted equally, both channels were normalized. Besides, time invariance was incorporated by selecting the 10-second long segment randomly from the entire signal. <br />
<br />
The input layer is a 10-second long ECG signal. There are 8 hidden layers in this model, each of which consists of a 1D convolution layer with the ReLu activation function followed by a batch normalization layer. The output layer is a one-dimensional layer that uses the Sigmoid activation function.<br />
<br />
This model is trained by using batches of size 10. The learning rate is 10^-4. The ADAM optimizer is used. The ADAM (adaptive moment estimation) optimizer is a stochastic gradient optimization method that uses adaptive learning rates for the parameters used in the estimating the gradient's first and second moments [5]. In training the model, the dataset is split into a train set, validation set, and test set with ratios 80-10-10.<br />
<br />
During the training process, the model was trained from scratch numerous times to avoid inserting unintended variation into the model by randomly initializing weights.<br />
<br />
The following images gives a visual representation of the model.<br />
<br />
[[File:architecture.png | thumb | center | 1000px | Model Architecture (Gupta et al., 2019)]]<br />
<br />
==Results== <br />
<br />
The paper first uses quantification of accuracies for single channels with 20-fold cross-validation, resulting in the highest individual accuracies: v5, v6, vx, vz, and ii. The researchers further investigated the accuracies for pairs of the top 5 highest individual channels using 20-fold cross-validation. They arrived at the conclusion of highest pairs accuracies to fed into a neural network is lead v6 and lead vz. They then use 100-fold cross validation on v6 and vz pair of channels, then compare outliers based on top 20, top 50 and total 100 performing models, finding that standard deviation is non-trivial and there are few models performed very poorly. <br />
<br />
Next, they discussed 2 factors affecting model performance evaluation: 1） Random train-val-test split might have effects on the performance of the model, but it can be improved by access with a larger data set and further discussion; and 2） random initialization of the weights of the neural network shows little effects on the performance of the model performance evaluation, because of showing high average results with a fixed train-val-test split. <br />
<br />
Comparing with other models in the other 12 papers, the model in this article has the highest accuracy, specificity, and precision. With concerns of patients' records affecting the training accuracy, they used 290 fold patient-wise split, resulting in the same highest accuracy of the pair v6 and vz same as record-wise split. The second best pair was ii and vz, which also contains the vz channel. Combining the two best pair channels into v6, vz, vii ultimately gave the best results over 10 trials which has an average of 97.83% in patient-wise split. Even though the patient-wise split might result in lower accuracy evaluation, however, it still maintains a very high average.<br />
<br />
==Conclusion & Discussion== <br />
<br />
The paper introduced a new architecture for heart condition classification based on raw ECG signals using multiple leads. It outperformed the state-of-art model by a large margin of 1 percent. This study finds that out of the 15 ECG channels(12 conventional ECG leads and 3 Frank Leads), channel v6, vz, and ii contain the most meaningful information for detecting myocardial infarction. Also, recent advances in machine learning can be leveraged to produce a model capable of classifying myocardial infraction with a cardiologist-level success rate. To further improve the performance of the models, access to a larger labeled data set is needed. The PTB database is small. It is difficult to test the true robustness of the model with a relatively small test set. If a larger data set can be found to help correctly identify other heart conditions beyond myocardial infraction, the research group plans to share the deep learning models and develop an open-source, computationally efficient app that can be readily used by cardiologists.<br />
<br />
A detailed analysis of the relative importance of each of the 15 ECG channels indicates that deep learning can identify myocardial infraction by processing only ten seconds of raw ECG data from the v6, vz, and ii leads and reaches a cardiologist-level success rate. Deep learning algorithms may be readily used as commodity software. The neural network model that was originally designed to identify earthquakes may be re-designed and tuned to identify myocardial infarction. Feature engineering of ECG data is not required to identify myocardial infraction in the PTB database. This model only required ten seconds of raw ECG data to identify this heart condition with cardiologist-level performance. Access to a larger database should be provided to deep learning researchers so they can work on detecting different types of heart conditions. Deep learning researchers and the cardiology community can work together to develop deep learning algorithms that provide trustworthy, real-time information regarding heart conditions with minimal computational resources.<br />
<br />
Fourier Transform (such as FFT) can be helpful when dealing with ECG signals. It transforms signals from the time domain to the frequency domain, which means some hidden features in frequency may be discovered.<br />
<br />
A limitation specified by the authors is the lack of labeled data. The use of a small dataset such as PTB makes it difficult to determine the robustness of the model due to the small size of the test set. Given a larger dataset, the model could be tested to see if it generalizes to identify heart conditions other than myocardial infarction.<br />
<br />
==Critiques==<br />
- The lack of large, labelled data sets is often a common problem in most applied deep learning studies. Since the PTB database is as small as you describe it to be, the robustness of the model which may be hard to gauge. There are very likely various other physical factors that may play a role in the study which the deep neural network may not be able to adjust for as well, since health data can be somewhat subjective at times and/or may be somewhat inaccurate, especially if machines are used to measurement. This might mean error was propagated forward in the study.<br />
<br />
- Additionally, there is a risk of confirmation bias, which may occur when a model is self-training, especially given the fact that the training set is small.<br />
<br />
- I feel that the results of deep learning models in medical settings where the consequences of misclassification can be severe should be evaluated by assigning weights to classification. In case if the misclassification can lead to severe consequences, then the network should be trained in such a way that it errs towards safety. For example, in case if heart disease, the consequences will be very high if the system says that there is no heart disease when in fact there is. So, the evaluation metric must be selected carefully.<br />
<br />
- This is a useful and meaningful application topic in machine learning. Using Deep Learning to detect heart disease can be very helpful if it is difficult to detect disease by looking at ECG by humans eys. This model also useful for doing statistics, such as calculating the percentage of people get heart disease. But I think the doctor should not 100% trust the result from the model, it is almost impossible to get 100% accuracy from a model. So, I think double-checking by human eyes is necessary if the result is weird. What is more, I think it will be interesting to discuss more applications in mediccal by using this method, such as detecting the Brainwave diagram to predict a person's mood and to diagnose mental diseases.<br />
<br />
- Compared to the dataset for other topics such as object recognition, the PTB database is pretty small with only 549 ECG records. And these are highly unbiased (Table 1) with 4 records for myocarditis and 148 for myocardial infarction. Medical datasets can only be labeled by specialists. This is why these datasets are related small. It would be great if there will be a larger, more comprehensive dataset.<br />
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- Only results using 20-fold cross validation were presented. It should be shown that the results could be reproduced using a more common number of folds like 5 or 10<br />
<br />
- There are potential issues with the inclusion of Frank leads. From a practitioner standpoint, ECGs taken with Frank leads are less common. This could prevent the use of this technique. Additionally, Frank leads are expressible as a linear combinations of the 12 traditional leads. The authors are not adding any fundamentally new information by including them and their inclusion could be viewed as a form of feature selection (going against the authors' original intentions).<br />
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- It will better if we can see how the model in this paper outperformed those methods that used feature selections. The details of the results are not enough.<br />
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- A new extended dataset for PTB dubbed [https://www.nature.com/articles/s41597-020-0495-6 PTB-XL], has 21837 records. Using this dataset could yield a more accurate result, since the original PTB's small dataset posed limitations on the deep learning model.<br />
<br />
- The paper mentions that it has better results, but by how much? what accuracy did the methods you compared to have? Also, what methods did you compare to? (Authors mentioned feature engineering methods but this is vague) Also how much were the labels smoothed? (i.e. 1 -> 0.99 or 1-> 0.95 for example) How much of a difference did the label smoothing make?<br />
<br />
- It is nice to see that the authors also considered training and testing the model on data via a patient-wise split, which gives more insights towards the cases when a patient has multiple records of diagnosis. Obviously and similar to what other critiques suggested, using a patient-wise split might disadvantage from the lack of training data, given that there are only 290 unique patients in the PTB database. Also, acquiring prior knowledge from professionals about correlations, such as causal relationships, between different diagnoses might be helpful for improving the model.<br />
<br />
- As mentioned above, the dataset is comparably small in the context of machine learning. While on the other hand, each record has a length of roughly 100 seconds, which is significantly large as a single input. Therefore, it might be helpful to apply data augmentation algorithms during data preprocessing sections so that there will be a more reasonable dataset than what we currently have so far, which has a high chance of being biased or overfitted.<br />
<br />
- There are several points from the Model Architecture section that can be improved. It mentions that both 1d batch normalization layers and label smoothing are used to improve the accuracy of the models, based on empirical experiment results. Yet, there's no breakdown of how each of these two method improves the accuracy. So it's left unclear whether each method is significant on its own, or the model simultaneously requires both methods in order to achieve improved accuracy. Some more data can be provided about this. It's mentioned that "models are trained from scratch numerous times." How many times is numerous times? Can we get the exact number? Training time about the models should also be provided. This is because if these models take a long time to train, then training them from scratch every time may cause issues with respect to runtime.<br />
<br />
- The authors should have indicated how much the accuracy has been improved by what method. It is a little unclear that how can we define "better results". Also, this paper could be more clear if they included the details about the Model Architecture such as how it was performed and how long was the training time for the model.<br />
<br />
== References ==<br />
<br />
[1] Na Liu et al. "A Simple and Effective Method for Detecting Myocardial Infarction Based on Deep Convolutional Neural Network". In: Journal of Medical Imaging and Health Informatics (Sept. 2018). doi: 10.1166/jmihi.2018.2463.<br />
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[2] Naser Safdarian, N.J. Dabanloo, and Gholamreza Attarodi. "A New Pattern Recognition Method for Detection and Localization of Myocardial Infarction Using T-Wave Integral and Total Integral as Extracted Features from One Cycle of ECG Signal". In: J. Biomedical Science and Engineering (Aug. 2014). doi: http://dx.doi.org/10.4236/jbise.2014.710081.<br />
<br />
[3] L.D. Sharma and R.K. Sunkaria. "Inferior myocardial infarction detection using stationary wavelet transform and machine learning approach." In: Signal, Image and Video Processing (July 2017). doi: https://doi.org/10.1007/s11760-017-1146-z.<br />
<br />
[4] Perol Thibaut, Gharbi Michaël, and Denolle Marin. "Convolutional neural network for earthquake detection and location". In: Science Advances (Feb. 2018). doi: 10.1126/sciadv.1700578<br />
<br />
[5] Kingma, D. and Ba, J., 2015. Adam: A Method for Stochastic Optimization. In: International Conference for Learning Representations. [online] San Diego: 3rd International Conference for Learning Representations, p.1. Available at: <https://arxiv.org/pdf/1412.6980.pdf> [Accessed 3 December 2020].</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Evaluating_Machine_Accuracy_on_ImageNet&diff=49190Evaluating Machine Accuracy on ImageNet2020-12-05T04:28:06Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Presented by == <br />
Siyuan Xia, Jiaxiang Liu, Jiabao Dong, Yipeng Du<br />
<br />
== Introduction == <br />
ImageNet is the most influential data set in machine learning with images and corresponding labels over 1000 classes. This paper intends to explore the causes for performance differences between human experts and machine learning models, more specifically, CNN, on ImageNet. <br />
<br />
Firstly, some images could belong to multiple classes. As a result, it is possible to underestimate the performance if we assign each image with only one label, which is what is being done in the top-1 metric. Therefore, we adopt both top-1 and top-5 metrics where the performances of models, unlike human labelers, are linearly correlated in both cases.<br />
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Secondly, in contrast to the uniform performance of models in classes, humans tend to achieve better performances on inanimate objects. Human labelers achieve similar overall accuracies as the models, which indicates spaces of improvements on specific classes for machines.<br />
<br />
Lastly, the setup of drawing training and test sets from the same distribution may favor models over human labelers. That is, the accuracy of multi-class prediction from models drops when the testing set is drawn from a different distribution than the training set, ImageNetV2. But this shift in distribution does not cause a problem for human labelers.<br />
<br />
== Experiment Setup ==<br />
=== Overview ===<br />
There are four main phases to the experiment, which are (i) initial multilabel annotation, (ii) human labeler training, (iii) human labeler evaluation, and (iv) final annotation overview. The five authors of the paper are the participants in the experiments. <br />
<br />
A brief overview of the four phases is as follows:<br />
[[File:Experiment Set Up.png |800px| center]]<br />
<br />
=== Initial multi-label annotation ===<br />
Three labelers A, B, and C provided multi-label annotations for a subset from the ImageNet validation set, and all images from the ImageNetV2 test sets. These experiences give A, B, and C extensive experience with the ImageNet dataset. <br />
<br />
=== Human Labeler Training === <br />
All five labelers trained on labeling a subset of the remaining ImageNet images. "Training" the human labelers consisted of teaching the humans the distinctions between very similar classes in the training set. For example, there are 118 classes of "dog" within ImageNet and typical human participants will not have working knowledge of the names of each breed of dog seen even if they can recognize and distinguish that breed from others.<br />
<br />
=== Human Labeler Evaluation ===<br />
Class-balanced random samples, which contains 1,000 images from the 20,000 annotated images are generated from both the ImageNet validation set and ImageNetV2. Five participants labeled these images over 28 days.<br />
<br />
=== Final annotation Review ===<br />
All labelers reviewed the additional annotations generated in the human labeler evaluation phase.<br />
<br />
== Multi-label annotations==<br />
[[File:Categories Multilabel.png|800px|center]]<br />
<div align="center">Figure 3</div><br />
<br />
===Top-1 accuracy===<br />
With Top-1 accuracy being the standard accuracy measure used in classification studies, it measures the proportions of examples for which the predicted label matches the single target label. As many images often contain more than one object for classification, for example, Figure 3a contains a desk, laptop, keyboard, space bar, and more. With Figure 3b showing a centered prominent figure yet labeled otherwise (people vs picket fence), it can be seen how a single target label is inaccurate for such a task since identifying the main objects in the image does not suffice due to its overly stringent and punishes predictions that are the main image yet does not match its label.<br />
===Top-5 accuracy===<br />
With Top-5 considers a classification correct if the object label is in the top 5 predicted labels, it partially resolves the problem with Top-1 labeling yet it is still not ideal since it can trivialize class distinctions. For instance, within the dataset, five turtle classes are given which is difficult to distinguish under such classification evaluations.<br />
===Multi-label accuracy===<br />
The paper then proposes that for every image, the image shall have a set of target labels and a prediction; if such prediction matches one of the labels, it will be considered as correct labeling. Due to the above-discussed limitations of Top-1 and Top-5 metrics, the paper claims it is necessary for rigorous accuracy evaluation on the dataset. <br />
<br />
===Types of Multi-label annotations===<br />
====Multiple objects or organisms====<br />
For the images containing more than one object or organism that corresponds to ImageNet, the paper proposed to add an additional target label for each entity in the image. With the discussed image in Figure 3b, the class groom, bow tie, suit, gown, and hoopskirt are all present in the foreground which is then subsequently added to the set of labels.<br />
====Synonym or subset relations====<br />
For similar classes, the paper considers them as under the same bigger class, that is, for two similarly labeled images, classification is considered correct if the produced label matches either one of the labels. For instance, warthog, African elephant, and Indian element all have prominent tusks, they will be considered subclasses of the tusker, Figure 3c shows a modification of labels to contain tusker as a correct label.<br />
====Unclear Image====<br />
In certain cases such as Figure 3d, there is a distinctive difficulty to determine whether a label was correct due to ambiguities in the class hierarchy.<br />
===Collecting multi-label annotations===<br />
Participants reviewed all predictions made by the models on the dataset ImageNet and ImageNet-V2, the participants then categorized every unique prediction made by the models on the dataset into correct and incorrect labels in order to allow all images to have multiple correct labels to satisfy the above-listed method.<br />
===The multi-label accuracy metric===<br />
One prediction is only correct if and only if it was marked correct by the expert reviewers during the annotation stage. As discussed in the experiment setup section, after human labelers have completed labeling, a second annotation stage is conducted. In Figure 4, a comparison of Top-1, Top-5, and multi-label accuracies showed higher Top-1 and Top-5 accuracy corresponds with higher multi-label accuracy as expected. With multi-label accuracies measures consistently higher than Top-1 yet lower than Top-5 which shows a high correlation between the three metrics, the paper concludes that multi-label metrics measures a semantically more meaningful notion of accuracy compared to its counterparts.<br />
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== Human Accuracy Measurement Process ==<br />
=== Bias Control ===<br />
Since three participants participated in the initial round of annotation, they did not look at the data for six months, and two additional annotators are introduced in the final evaluation phase to ensure fairness of the experiment. <br />
<br />
=== Human Labeler Training ===<br />
The three main difficulties encountered during human labeler training are fine-grained distinctions, class unawareness, and insufficient training images. Thus, three training regimens are provided to address the problems listed above, respectively. First, labelers will be assigned extra training tasks with immediate feedbacks on similar classes. Second, labelers will be provided access to search for specific classes during labeling. Finally, the training set will contain a reasonable amount of images for each class.<br />
<br />
=== Labeling Guide ===<br />
A labeling guide is constructed to distill class analysis learned during training into discriminative traits that could be used as a reference during the final labeling evaluation.<br />
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=== Final Evaluation and Review ===<br />
Two samples, each containing 1000 images, are sampled from ImageNet and ImageNetV2, respectively, They are sampled in a class-balanced manner and shuffled together. Over 28 days, all five participants labeled all images. They spent a median of 26 seconds per image. After labeling is completed, an additional multi-label annotation session was conducted, in which human predictions for all images are manually reviewed. Comparing to the initial round of labeling, 37% of the labels changes due to participants' greater familiarity with the classes.<br />
<br />
== Main Results ==<br />
[[File:Evaluating Machine Accuracy on ImageNet Figure 1.png | center]]<br />
<br />
<div align="center">Figure 1</div><br />
<br />
===Comparison of Human and Machine Accuracies on Image Net===<br />
From Figure 1, we can see that the difference in accuracies between the datasets is within 1% for all human participants. As hypothesized, human testers indeed performed better than the automated models on both datasets. It's worth noticing that labelers D and E, who did not participate in the initial annotation period, actually performed better than the best automated model.<br />
===Comparison of Human and Machine Accuracies on Image Net===<br />
Based on the results shown in Figure 1, we can see that the confidence interval of the best 4 human participants and 4 best model overlap; however, with a p-value of 0.037 using the McNemar's paired test, it rejects the hypothesis that the FixResNeXt model and Human E labeler have the same accuracy with respect to the ImageNet validation dataset. Figure 1 also shows that the confidence intervals of the labeling accuracies for human labelers C, D, E do not overlap with the confidence interval of the best model with respect to ImageNet-V2 and with the McNemar's test yielding a p-value of <math>2\times 10^{-4}</math>, it is clear that the hypothesis human and machined models have same robustness to model distribution shifts ought to be rejected.<br />
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== Other Observations ==<br />
<br />
[[File: Results_Summary_Table.png| 800px|center]]<br />
<br />
=== Difficult Images ===<br />
<br />
The experiment also shed some light on images that are difficult to label. 10 images were misclassified by all of the human labelers. Among those 10 images, there was 1 image of a monkey and 9 of dogs. In addition, 27 images, with 19 in object classes and 8 in organism classes, were misclassified by all 72 machine learning models in this experiment. Only 2 images were labeled wrong by all human labelers and models. Both images contained dogs. Researchers also noted that difficult images for models are mostly images of objects and exclusively images of animals for human labelers.<br />
<br />
=== Accuracies without dogs ===<br />
<br />
As previously discussed in the paper, machine learning models tend to outperform human labelers when classifying the 118 dog classes. To better understand to what extent does models outperform human labelers, researchers computed the accuracies again by excluding all the dog classes. Results showed a 0.6% increase in accuracy on the ImageNet images using the best model and a 1.1% increase on the ImageNet V2 images. In comparison, the mean increases in accuracy for human labelers are 1.9% and 1.8% on the ImageNet and ImageNet V2 images respectively. Researchers also conducted a simulation to demonstrate that the increase in human labeling accuracy on non-dog images is significant. This simulation was done by bootstrapping to estimate the changes in accuracy when only using data for the non-dog classes, and simulation results show smaller increases than in the experiment. <br />
<br />
In conclusion, it's more difficult for human labelers to classify images with dogs than it is for machine learning models.<br />
<br />
=== Accuracies on objects ===<br />
Researchers also computed machine and human labelers' accuracies on a subset of data with only objects, as opposed to organisms, to better illustrate the differences in performance. This test involved 590 object classes. As shown in the table above, there is a 3.3% and 3.4% increase in mean accuracies for human labelers on the ImageNet and ImageNet V2 images. In contrast, there is a 0.5% decrease in accuracy for the best model on both ImageNet and ImageNet V2. This indicates that human labelers are much better at classifying objects than these models are.<br />
<br />
=== Accuracies on fast images ===<br />
Unlike the CNN models, human labelers spent different amounts of time on different images, spanning from several seconds to 40 minutes. To further analyze the images that take human labelers less time to classify, researchers took a subset of images with median labeling time spent by human labelers of at most 60 seconds. These images were referred to as "fast images". There are 756 and 714 fast images from ImageNet and ImageNet V2 respectively, out of the total 2000 images used for evaluation. Accuracies of models and humans on the fast images increased significantly, especially for humans. <br />
<br />
This result suggests that human labelers know when an image is difficult to label and would spend more time on it. It also shows that the models are more likely to correctly label images that human labelers can label relatively quickly.<br />
<br />
== Related Work ==<br />
<br />
=== Human accuracy on ImageNet ===<br />
<br />
Russakovsky et al. (2015) studied two trained human labelers' accuracies on 1500 and 258 images in the context of the ImageNet challenge. The top-5 accuracy of the labeler who labeled 1500 images was the well-known human baseline on ImageNet. <br />
<br />
As introduced before, the researchers went beyond by using multi-label accuracy, using more labelers, and focusing on robustness to small distribution shifts. Although the researchers had some different findings, some results are also consistent with results from (Russakovsky et al., 2015). An example is that both experiments indicated that it takes human labelers around one minute to label an image. The time distribution also has a long tail, due to the difficult images as mentioned before.<br />
<br />
=== Human performance in computer vision broadly ===<br />
There are many examples of recent studies about humans in the area of computer vision, such as investigating human robustness to synthetic distribution change (Geirhos et al., 2017) and studying what characteristics do humans use to recognize objects (Geirhos et al., 2018). Other examples include the adversarial examples constructed to fool both machines and time-limited humans (Elsayed et al., 2018) and illustrating foreground/background objects' effects on human and machine performance (Zhu et al., 2016). <br />
<br />
=== Multi-label annotations ===<br />
Stock & Cissé (2017) also studied ImageNet's multi-label nature, which aligns with the researchers' study in this paper. According to Stock & Cissé (2017), the top-1 accuracy measure could underestimate multi-label by up to 13.2%.<br />
<br />
=== ImageNet inconsistencies and label error ===<br />
Researches have found and recorded some incorrectly labeled images from ImageNet and ImageNet V2 during this study. Earlier studies (Van Horn et al., 2015) also shown that at least 4% of the birds in ImageNet are misclassified. This work also noted that the inconsistent taxonomic structure in birds' classes could lead to weak class boundaries. Researchers also noted that the majority of the fine-grained organism classes also had similar taxonomic issues.<br />
<br />
=== Distribution shift ===<br />
There has been an increasing amount of studies in this area. One focus of the studies is distributionally robust optimization (DRO), which finds the model that has the smallest worst-case expected error over a set of probability distributions. Another focus is on finding the model with the lowest error rates on adversarial examples. Work in both areas has been productive, but none was shown to resolve the drop in accuracies between ImageNet and ImageNet V2. A recent [https://papers.nips.cc/paper/2019/file/8558cb408c1d76621371888657d2eb1d-Paper.pdf paper] also discusses quantifying uncertainty under a distribution shift, in other words whether the output of probabilistic deep learning models should or should not be trusted.<br />
<br />
== Conclusion and Future Work ==<br />
<br />
=== Conclusion ===<br />
Researchers noted that in order to achieve truly reliable machine learning, researchers need a deeper understanding of the range of parameters where the model still remain robust. Techniques from Combinatorics and sensitivity analysis, in particular, might yield fruitful results. This study has provided valuable insights into the desired robustness properties by comparing model performance to human performance. This is especially evident given the results of the experiment which show humans drastically outperforming machine learning in many cases and proposes the question of how much accuracy one is willing to give up in exchange for efficiency. The results have shown that current performance benchmarks are not addressing the robustness to small and natural distribution shifts, which are easily handled by humans.<br />
<br />
=== Future work ===<br />
Other than improving the robustness of models, researchers should consider investigating if less-trained human labelers can achieve a similar level of robustness to distributional shifts. In addition, researchers can study the robustness to temporal changes, which is another form of natural distribution shift (Gu et al., 2019; Shankar et al., 2019). Also, Convolutional Neural Network can be a candidate to improve the accuracy of classifying images.<br />
<br />
== Critiques ==<br />
<br />
# Table 1 simply showed a difference in ImageNet multi-label accuracy yet does not give an explicit reason as to why such a difference is present. Although the paper suggested the distribution shift has caused the difference, it does not give other factors to concretely explain why the distribution shift was the cause.<br />
# With the recommendation to future machine evaluations, the paper proposed to "Report performances on dogs, other animals, and inanimate objects separately.". Despite its intentions, it is narrowly specific and requires further generalization for it to be convincing. <br />
# With choosing human subjects as samplers, no further information was given as to how they are chosen nor there are any background information was given. As it is a classification problem involving many classes as specific to species, a biology student would give far more accurate results than a computer science student or a math student. <br />
# As explaining the importance of multi-label metrics using comparison to Top-5 metric, the turtle example falls within the overall similarity (simony) classification of the multi-label evaluation metric, as such, if the Top-5 evaluation suggests any one of the turtle species were selected, the algorithm is considered to produce a correct prediction which is the intention. The example does not convey the necessity of changing to the proposed metric over the Top-5 metric. <br />
# With the definition in the paper regarding multi-label metrics, it is hard to see why expanding the label set is different from a traditional Top-5 metric or rather necessary, ergo does not yield the claim which the proposed metric is necessary for rigorous accuracy evaluation on ImageNet.<br />
# When discussing the main results, the paper discusses the hypothesis on distribution shift having no effects on human and machine model accuracies; the presentation is poor at best with no clear centric to what they are trying to convey to how (in detail) they resulted in such claims.<br />
# In the experiment setup of the presentation, there are a lot of key terms without detailed description. For example, Human labeler training using a subset of the remaining 30,000 unannotated images in the ImageNet validation set, labelers A, B, C, D, and E underwent extensive training to understand the intricacies of fine-grained class distinctions in the ImageNet class hierarchy. Authors should clarify each key term in the presentation otherwise readers are hard to follow.<br />
# Not sure how the human samplers were determined and simply picking several people will have really high bias because the sample is too small and they have different background which will definitely affect the results a lot. Also, it will be better if there are more comparisons between the model introduced and other models.<br />
# Given the low amount of human participants, it is hard to take the results seriously (there is too much variance). Also it's not exactly clear how the authors determined that the multi-label accuracy metric measures a semantically more meaningful notion of accuracy compared to its counterparts. For example, one of the issues with top-5 accuracy that they mention is: "For instance, within the dataset, five turtle classes are given which is difficult to distinguish under such classification evaluations." But it's not clear how multi-label accuracy would be better in this instance.<br />
# It is unclear how well the human labeler can perform labeling after training. So the final result is not that trust-worthy.<br />
# In this experiment set up, label annotators are the same as participants of the experiments. Even if there's a break between the annotating and evaluating human labeler evaluation, the impact of the break in reducing bias is not clear. One potential human labeling data is google's "I'm not a robot" verification test. One variation of the verification test asks users to select all the photos from 9 images that are related to a certain keyword. This allows for a more accurate measurement of human performance vs ImageNet performance. In addition, it's going to reduce the biases from the small number of experiment participants.<br />
# Following Table 2, the authors appear to try and claim that the model is better than the human labelers, simply because the model experienced a better increase in classification following the removal of dog photos then the human labeler did, however, a quick look at the table shows that most human labelers still performed better than the best model. The authors should be making the claim that human labelers are better at labeling dogs than the modal, but are still better overall after removing the dogs dataset.<br />
# The reason why human labeler outperforms CNN could be human had much more training. It would be more convincing if the paper could provide a metric in order to measure human labelers' training data set size.<br />
# Actually, in the multi-label case, it is vague to determine whether the machine learning model or the human labellers were giving the correct label. The structure of the dataset is pretty essential in training a network, in which data with uncertain label (even determined by human) should be avoided.<br />
# I believe the authors needed to include more information about how they determined the samples such as human samplers, and also more details on how to define unclear images.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=what_game_are_we_playing&diff=49189what game are we playing2020-12-05T04:23:10Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Authors == <br />
Yuxin Wang, Evan Peters, Yifan Mou, Sangeeth Kalaichanthiran <br />
<br />
== Introduction ==<br />
Recently, there have been many different studies of methods using AI to solve the optimal solution for large-scale, zero-sum, extensive form problems. However, most of these works operate under the assumption that the parameters of the game are known, while this scenario is ideal. In real-world problems, most of the parameters are unknown prior to the game. This paper proposes a framework for finding an optimal solution using a primal-dual Newton Method and then using back-propagation to analytically compute the gradients of all the relevant game parameters.<br />
<br />
The approach to solving this problem is to consider ''quantal response equilibrium'' (QRE), which is a generalization of Nash equilibrium (NE) where the agents can make suboptimal decisions. It is shown that the solution to the QRE is a differentiable function of the payoff matrix. Consequently, back-propagation can be used to analytically solve for the payoff matrix (or other game parameters). This strategy has many future application areas as it allows for game-solving (both extensive and normal form) to be integrated as a module in a deep neural network.<br />
<br />
An example of architecture is presented below:<br />
<br />
[[File:Framework.png ]]<br />
<br />
Payoff matrix <math> P </math> is parameterized by a domain-dependent low dimensional vector <math> \phi </math>, where <math> \phi </math> depends on a differentiable function <math> M_1(x) </math>. Furthermore, <math> P </math> is applied to QRE to get the equilibrium strategies <math> (u^∗, v^∗) </math>. Lastly, the loss function is calculated after applying any differentiable <math> M_2(u^∗, v^∗) </math>.<br />
<br />
The effectiveness of this model is demonstrated using the games “Rock, Paper, Scissors”, one-card poker, and a security defense game. This could represent a substantial step forward in understanding how game-theoretic methods can be applied to uncertain settings by the ability to learn solely from observed actions and the relevant parameters.<br />
<br />
== Learning and Quantal Response in Normal Form Games ==<br />
<br />
The game-solving module provides all elements required in differentiable learning, which maps contextual features to payoff matrices, and computes equilibrium strategies under a set of contextual features. This paper will learn zero-sum games and start with normal form games since they have game solver and learning approach capturing much of intuition and basic methodology.<br />
<br />
=== Zero-Sum Normal Form Games ===<br />
<br />
In two-player zero-sum games there is a '''payoff matrix''' <math>P</math> that describes the rewards for two players employing specific strategies u and v respectively. The optimal strategy mixture may be found with a classic min-max formulation:<br />
$$\min_u \max_v \ u^T P v \\ subject \ to \ 1^T u =1, u \ge 0 \\ 1^T v =1, v \ge 0. \ $$<br />
<br />
Here, we consider the case where <math>P</math> is not known a priori. <math>P</math> could be a single fixed but unknown payoff<br />
matrix, or depend on some external context <math>x</math>. The solution <math> (u^*, v_0) </math> to this optimization and the solution <math> (u_0,v^*) </math> to the corresponding problem with inverse player order form the Nash equilibrium <math>(u^*,v^*) </math>. At this equilibrium, the players do not have anything to gain by changing their strategy, so this point is a stable state of the system. When the payoff matrix P is not known, we observe samples of actions <math> a^{(i)}, i =1,...,N </math> from one or both players, which depends on some external content <math> x </math>, sampled from the equilibrium strategies <math>(u^*,v^*) </math>, to recover the true underlying payoff matrix P or a function form P(x) depending on the current context.<br />
<br />
=== Quantal Response Equilibria ===<br />
<br />
However, NE is poorly suited because NEs are overly strict, discontinuous with respect to P, and may not be unique. To address these issues, model the players' actions with the '''quantal response equilibria''' (QRE), where noise is added to the payoff matric. Specifically, consider the ''logit'' equilibrium for zero-sum games that obeys the fixed point:<br />
$$<br />
u^* _i = \frac {exp(-Pv)_i}{\sum_{q \in [n]} exp (-Pv)_q}, \ v^* _j= \frac {exp(P^T u)_j}{\sum_{q \in [m]} exp (P^T u)_q} .\qquad \ (1)<br />
$$<br />
For a fixed opponent strategy, the logit equilibrium corresponding to a strategy is strictly convex, and thus the regularized best response is unique.<br />
<br />
=== End-to-End Learning ===<br />
<br />
Then to integrate a zero-sum solver, [1] introduced a method to solve the QRE and to differentiate through its solution.<br />
<br />
'''QRE solver''':<br />
To find the fixed point in (1), it is equivalent to solve the regularized min-max game:<br />
$$<br />
\min_{u \in \mathbb{R}^n} \max_{v \in \mathbb{R}^m} \ u^T P v -H(v) + H(u) \\<br />
\text{subject to } 1^T u =1, \ 1^T v =1, <br />
$$<br />
where H(y) is the Gibbs entropy <math> \sum_i y_i log y_i</math>.<br />
Entropy regularization guarantees the non-negative condition and makes the equilibrium continuous with respect to P, which means players are encouraged to play more randomly, and all actions have a non-zero probability. Moreover, this problem has a unique saddle point corresponding to <math> (u^*, v^*) </math>.<br />
<br />
Using a primal-dual Newton Method to solve the QRE for two-player zero-sum games, the KKT conditions for the problem are:<br />
$$ <br />
Pv + \log(u) + 1 +\mu 1 = 0 \\<br />
P^T v -\log(v) -1 +\nu 1 = 0 \\<br />
1^T u = 1, \ 1^T v = 1, <br />
$$<br />
where <math> (\mu, \nu) </math> are Lagrange multipliers for the equality constraints on u, v respectively. Then applying Newton's method gives the the update rule:<br />
$$<br />
Q \begin{bmatrix} \Delta u \\ \Delta v \\ \Delta \mu \\ \Delta \nu \\ \end{bmatrix} = - \begin{bmatrix} P v + \log u + 1 + \mu 1 \\ P^T u - \log v - 1 + \nu 1 \\ 1^T u - 1 \\ 1^T v - 1 \\ \end{bmatrix}, \qquad (2)<br />
$$<br />
where Q is the Hessian of the Lagrangian, given by <br />
$$ <br />
Q = \begin{bmatrix} <br />
diag(\frac{1}{u}) & P & 1 & 0 \\ <br />
P^T & -diag(\frac{1}{v}) & 0 & 1\\<br />
1^T & 0 & 0 & 0 \\<br />
0 & 1^T & 0 & 0 \\<br />
\end{bmatrix}. <br />
$$<br />
<br />
'''Differentiating Through QRE Solutions''':<br />
The QRE solver provides a method to compute the necessary Jacobian-vector products. Specifically, we compute the gradient of the loss given the solution <math> (u^*,v^*) </math> to the QRE, and some loss function <math> L(u^*,v^*) </math>: <br />
<br />
1. Take differentials of the KKT conditions: <br />
<math><br />
Q \begin{bmatrix} <br />
du & dv & d\mu & d\nu \\ <br />
\end{bmatrix} ^T = \begin{bmatrix} <br />
-dPv & -dP^Tu & 0 & 0 \\ <br />
\end{bmatrix}^T. \ <br />
</math><br />
<br />
2. For small changes du, dv, <br />
<math><br />
dL = \begin{bmatrix} <br />
v^TdP^T & u^TdP & 0 & 0 \\ <br />
\end{bmatrix} Q^{-1} \begin{bmatrix} <br />
-\nabla_u L & -\nabla_v L & 0 & 0 \\ <br />
\end{bmatrix}^T.<br />
</math><br />
<br />
3. Apply this to P, and take limits as dP is small:<br />
<math><br />
\nabla_P L = y_u v^T + u y_v^T, \qquad (3)<br />
</math> where <br />
<math><br />
\begin{bmatrix} <br />
y_u & y_v & y_{\mu} & y_{\nu}\\ <br />
\end{bmatrix}=Q^{-1}\begin{bmatrix} <br />
-\nabla_u L & -\nabla_v L & 0 & 0 \\ <br />
\end{bmatrix}^T.<br />
</math><br />
<br />
Hence, the forward pass is given by using the expression in (2) to solve for the logit equilibrium given P, and the backward pass is given by using <math> \nabla_u L </math> and <math> \nabla_v L </math> to obtain <math> \nabla_P L </math> using (3). There does not always exist a unique P which generates <math> u^*, v^* </math> under the logit QRE, and we cannot expect to recover P when under-constrained.<br />
<br />
== Learning Extensive form games ==<br />
<br />
The normal form representation for games where players have many choices quickly becomes intractable. For example, consider a chess game: One the first turn, player 1 has 20 possible moves and then player 2 has 20 possible responses. If in the following number of turns each player is estimated to have ~30 possible moves and if a typical game is 40 moves per player, the total number of strategies is roughly <math>10^{120} </math> per player (this is known as the Shannon number for game-tree complexity of chess) and so the payoff matrix for a typical game of chess must therefore have <math> O(10^{240}) </math> entries.<br />
<br />
Instead, it is much more useful to represent the game graphically as an "''' Extensive form game'''" (EFG). We'll also need to consider types of games where there is '''imperfect information''' - players do not necessarily have access to the full state of the game. An example of this is one-card poker: (1) Each player draws a single card from a 13-card deck (ignore suits) (2) Player 1 decides whether to bet/hold (3) Player 2 decides whether to call/raise (4) Player 1 must either call/fold if Player 2 raised. From this description, player 1 has <math> 2^{13} </math> possible first moves (all combinations of (card, raise/hold)) and has <math> 2^{13} </math> possible second moves (whenever player 1 gets a second move) for a total of <math> 2^{26} </math> possible strategies. In addition, Player 1 never knows what cards player 2 has and vice versa. So instead of representing the game with a huge payoff matrix, we can instead represent it as a simple decision tree (for a ''single'' drawn the card of player 1):<br />
<br />
<br />
<center> [[File:1cardpoker.PNG]] </center><br />
<br />
where player 1 is represented by "1", a node that has two branches corresponding to the allowed moves of player 1. However there must also be a notion of information available to either player: While this tree might correspond to say, player 1 holding a "9", it contains no information on what card player 2 is holding (and is much simpler because of this). This leads to the definition of an '''information set''': the set of all nodes belonging to a single player for which the other player cannot distinguish which node has been reached. The information set may therefore be treated as a node itself, for which actions stemming from the node must be chosen in ignorance of what the other player did immediately before arriving at the node. In the poker example, the full game tree consists of a much more complex version of the tree shown above (containing repetitions of the given tree for every possible combination of cards dealt) and the and an example of the information set for player 1 is the set of all of the nodes owned by player 2 that immediately follow player 1's decision to hold. In other words, if player 1 holds there are 13 possible nodes describing the responses of player 2 (raise/hold for player 2 having card = ace, 1, ... King), and all 13 of these nodes are indistinguishable to player 1, and so form the information set for player 1.<br />
<br />
The following is a review of important concepts for extensive form games first formalized in [2]. Let <math> \mathcal{I}_i </math> be the set of all information sets for player i, and for each <math> t \in \mathcal{I}_i </math> let <math> \sigma_t </math> be the actions taken by player i to arrive at <math> t </math> and <math> C_t </math> be the actions that player i can take from <math> u </math>. Then the set of all possible sequences that can be taken by player i is given by<br />
<br />
$$<br />
S_i = \{\emptyset \} \cup \{ \sigma_t c | u\in \mathcal{I}_i, c \in C_t \}<br />
$$<br />
<br />
So for the one-card poker we would have <math>S_1 = \{\emptyset, \text{raise}, \text{hold}, \text{hold-call}, \text{hold-fold\} }</math>. From the possible sequences follows two important concepts:<br />
<ol><br />
<li>The EFG '''payoff matrix''' <math> P </math> of size <math>|S_1| \times |S_2| </math> (this is all possible actions available to either player), is populated with rewards from each leaf of the tree (or "zero" for each <math> (s_1, s_2) </math> that is an invalid pair), and the expected payoff for realization plans <math> (u, v) </math> is given by <math> u^T P v </math> </li><br />
<li> A '''realization plan''' <math> u \in \mathbb{R}^{|S_1|} </math> for player 1 (<math> v \in \mathbb{R}^{|S_2|} </math> for player 2 ) will describe probabilities for players to carry out each possible sequence, and each realization plan must be constrained by (i) compatibility of sequences (e.g. "raise" is not compatible with "hold-call") and (ii) information sets available to the player. These constraints are linear:<br />
<br />
$$<br />
Eu = e \\<br />
Fv = f<br />
$$<br />
<br />
where <math> e = f = (1, 0, ..., 0)^T </math> and <math> E, F</math> contain entries in <math> {-1, 0, 1} </math> describing compatibility and information sets. </li><br />
<br />
</ol> <br />
<br />
<br />
The paper's main contribution is to develop a minimax problem for extensive form games:<br />
<br />
<br />
$$<br />
\min_u \max_v u^T P v + \sum_{t\in \mathcal{I}_1} \sum_{c \in C_t} u_c \log \frac{u_c}{u_{p_t}} - \sum_{t\in \mathcal{I}_2} \sum_{c \in C_t} v_c \log \frac{v_c}{v_{p_t}}<br />
$$<br />
<br />
where <math> p_t </math> is the action immediately preceding information set <math> t </math>. Intuitively, each sum resembles a cross-entropy over the distribution of probabilities in the realization plan comparing each probability to proceed from the information set to the probability to arrive at that information set. Importantly, these entropies are strictly convex or concave (for player 1 and player 2 respectively) [3] so that the min-max problem will have a unique solution and ''the objective function is continuous and continuously differentiable'' - this means there is a way to optimize the function. As noted in Theorem 1 of [1], the solution to this problem is equivalently a solution for the QRE of the game in reduced normal form.<br />
<br />
Minimax can also be seen from an algorithmic perspective. Referring to the above figure containing a tree, it contains a sequence of states and action which alternates between two or more competing players. The above formulation of the min-max problem essentially measures how well a decision rule is from the perspective of a single player. To describe it in terms of the tree, if it is player 1's turn, then it is a mutual recursion of player 1 choosing to maximize its payoff and player 2 choosing to minimize player 1's payoff.<br />
<br />
Having decided on a cost function, the method of Lagrange multipliers may be used to construct the Lagrangian that encodes the known constraints (<math> Eu = e \,, Fv = f </math>, and <math> u, v \geq 0</math>), and then optimize the Lagrangian using Newton's method (identically to the previous section). Accounting for the constraints, the Lagrangian becomes <br />
<br />
<br />
$$<br />
\mathcal{L} = g(u, v) + \sum_i \mu_i(Eu - e)_i + \sum_i \nu_i (Fv - f)_i<br />
$$<br />
<br />
where <math>g</math> is the argument from the minimax statement above and <math>u, v \geq 0</math> become KKT conditions. The general update rule for Newton's method may be written in terms of the derivatives of <math> \mathcal{L} </math> with respect to primal variables <math>u, v </math> and dual variables <math> \mu, \nu</math>, yielding:<br />
<br />
$$<br />
\nabla_{u,v,\mu,\nu}^2 \mathcal{L} \cdot (\Delta u, \Delta v, \Delta \mu, \Delta \nu)^T= - \nabla_{u,v,\mu,\nu} \mathcal{L}<br />
$$<br />
where <math>\nabla_{u,v,\mu,\nu}^2 \mathcal{L} </math> is the Hessian of the Lagrangian and <math>\nabla_{u,v,\mu,\nu} \mathcal{L} </math> is simply a column vector of the KKT stationarity conditions. Combined with the previous section, this completes the goal of the paper: To construct a differentiable problem for learning normal form and extensive form games.<br />
<br />
== Experiments ==<br />
<br />
The authors demonstrated learning on extensive form games in the presence of ''side information'', with ''partial observations'' using three experiments. In all cases, the goal was to maximize the likelihood of realizing an observed sequence from the player, assuming they act in accordance with the QRE. The authors found that the best way to implement the module was to use a medium to large batch size, RMSProp, or Adam optimizers with a learning rate between <math>\left[0.0001,0.01\right].</math> <br />
<br />
=== Rock, Paper, Scissors ===<br />
<br />
Rock, Paper, Scissors is a 2-player zero-sum game. For this game, the best strategy to reach a Nash equilibrium and a Quantal response equilibrium is to uniformly play each hand with equal odds.<br />
The first experiment was to learn a non-symmetric variant of Rock, Paper, Scissors with ''incomplete information'' with the following payoff matrix:<br />
<br />
{| class="wikitable" style="float:center; margin-left:1em; text-align:center;"<br />
|+ align="bottom"|''Payoff matrix of modified Rock-Paper-Scissors''<br />
! <br />
! ''Rock''<br />
! ''Paper''<br />
! ''Scissors''<br />
|-<br />
! ''Rock''<br />
| '''''0'''''<br />
| <math>-b_1</math><br />
| <math>b_2</math><br />
|-<br />
! ''Paper''<br />
| <math>b_1</math><br />
| '''''0'''''<br />
| <math>-b_3</math><br />
|-<br />
! ''Scissors''<br />
| <math>-b_2</math><br />
| <math>b_3</math><br />
| '''''0'''''<br />
|}<br />
<br />
where each of the <math> b </math> ’s are linear function of some features <math> x \in \mathbb{R}^{2} </math> (i.e., <math> b_y = x^Tw_y </math>, <math> y \in </math> {<math>1,2,3</math>} , where <math> w_y </math> are to be learned by the algorithm). Using many trials of random rewards the technique produced the following results for optimal strategies[1]: <br />
<br />
[[File:RPS Results.png|500px ]]<br />
<br />
From the graphs above, we can tell the following: 1) both parameters learned and predicted strategies improve with a larger dataset; 2) with a reasonably sized dataset, >1000 here, convergence is stable and is fairly quick.<br />
<br />
=== One-Card Poker ===<br />
<br />
Next they investigated extensive form games using the one-Card Poker (with ''imperfect information'') introduced in the previous section. In the experimental setup, they used a deck stacked non-uniformly (meaning repeat cards were allowed). Their goal was to learn this distribution of cards from observations of many rounds of the play. Different from the distribution of cards dealt, the method built in the paper is more suited to learn the player’s perceived or believed distribution of cards. It may even be a function of contextual features such as demographics of players. Three experiments were run with <math> n=4 </math>. Each experiment comprised 5 runs of training, with same weights but different training sets. Let <math> d \in \mathbb{R}^{n}, d \ge 0, \sum_{i} d_i = 1 </math> be the weights of the cards. The probability that the players are dealt cards <math> (i,j) </math> is <math> \frac{d_i d_j}{1-d_i} </math>. This distribution is asymmetric between players. Matrix <math> P, E, F </math> for the case <math> n=4 </math> are presented in [1]. With training for 2500 epochs, the mean squared error of learned parameters (card weights, <math> u, v </math> ) are averaged over all runs of and are presented as following [1]: <br />
<br />
<br />
[[File:One-card_Poker_Results.png|500px ]]<br />
<br />
=== Security Resource Allocation Game ===<br />
<br />
<br />
From Security Resource Allocation Game, they demonstrated the ability to learn from ''imperfect observations''. The defender possesses <math> k </math> indistinguishable and indivisible defensive resources which he splits among <math> n </math> targets, { <math> T_1, ……, T_n </math>}. The attacker chooses one target. If the attack succeeds, the attacker gets <math> R_i </math> reward and defender gets <math> -R_i </math>, otherwise zero payoff for both. If there are n defenders guarding <math> T_i </math>, probability of successful attack on <math> T_i </math> is <math> \frac{1}{2^n} </math>. The expected payoff matrix when <math> n = 2, k = 3 </math>, where the attackers are the row players is:<br />
<br />
{| class="wikitable" style="float:center; margin-left:1em; text-align:center;"<br />
|+ align="bottom"|''Payoff matrix when <math> n = 2, k = 3 </math>''<br />
! {#<math>D_1</math>,#<math>D_2</math>}<br />
! {0, 3}<br />
! {1, 2}<br />
! {2, 1}<br />
! {3, 0}<br />
|-<br />
! <math>T_1</math><br />
| <math>-R_1</math><br />
| <math>-\frac{1}{2}R_1</math><br />
| <math>-\frac{1}{4}R_1</math><br />
| <math>-\frac{1}{8}R_1</math><br />
|-<br />
! <math>T_2</math><br />
| <math>-\frac{1}{8}R_2</math><br />
| <math>-\frac{1}{4}R_2</math><br />
| <math>-\frac{1}{2}R_2</math><br />
| <math>-R_2</math><br />
|} <br />
<br />
<br />
For a multi-stage game the attacker can launch <math> t </math> attacks, one in each stage while defender can only stick with stage 1. The attacker may change target if the attack in stage 1 is failed. Three experiments are run with <math> n = 2, k = 5 </math> for games with single attack and double attack, i.e, <math> t = 1 </math> and <math> t = 2 </math>. The results of simulated experiments are shown below [1]:<br />
<br />
[[File:Security Game Results.png|500px ]]<br />
<br />
<br />
They learned <math> R_i </math> only based on observations of the defender’s actions and could still recover the game set by only observing the defender’s actions. Same as expectation, the larger dataset size improves the learned parameters. Two outliers are 1) Security Game, the green plot for when <math> t = 2 </math>; and 2) RPS, when comparing between training sizes of 2000 and 5000.<br />
<br />
== Conclusion ==<br />
Unsurprisingly, the results of this study show that in general, the quality of learned parameters improved as the number of observations increased. The network presented in this paper demonstrated improvement over the existing methodology. <br />
<br />
This paper presents an end-to-end framework for implementing a game solver, for both extensive and normal form, as a module in a deep neural network for zero-sum games. This method, unlike many previous works in this area, does not require the parameters of the game to be known to the agent prior to the start of the game. The two-part method analytically computes both the optimal solution and the parameters of the game. Future work involves taking advantage of the KKT matrix structure to increase computation speed and extensions to the area of learning general-sum games.<br />
<br />
== Critiques ==<br />
The proposed method appears to suffer from two flaws. Firstly, the assumption that players behave in accordance with the QRE severely limits the space of player strategies and is known to exhibit pathological behavior even in one-player settings. Second, the solvers are computationally inefficient and are unable to scale. For this setting, they used Newton's method and it is found that the second-order algorithms do not scale to large games as with Nash equilibrium [4]. <br />
<br />
In the one-card poker section, it might be better to write "Next, they investigated extensive form games ... It may even be a function of contextual features such as the demographics of players. ... 5 runs of training, with the same weights but different training sets."<br />
<br />
Zero-Sum Normal Form Games usually follow Gaussian distributions with two non-empty sets of strategies of player one and player two correspondingly, and also a payoff function defined on the set of possible realizations.<br />
<br />
This method of proposing that real players will follow a laid out scheme or playing strategy is almost always a drastic oversimplification of real-world scenarios and severely limits the applications. In order to better fit a model to the real players, there almost always has to be some sort of stochastic implementation which accounts for the presence of irrational users in the system. <br />
<br />
It might be a good idea to discuss the algorithm performance based on more complicated player reactions, for example, player 2 or NPC from the game might react based on player 1's action. If player 2 would react with a "clever" choice, we might be able to shrink the choice space, which might be helpful to accelerate the training time.<br />
<br />
The theory assumes rational players, which means that roughly speaking, the players make decisions based on increasing their respective payoffs (utility values, preferences,..). However, in real-life scenarios, this might not hold true. Thus, it would be a good idea to include considerations regarding this issue in this summary. Another area that is worth exploring is the need for things like “bluffing” in games with hidden information. It would be interesting to see whether the results generated in this paper are in support of "bluffing" or not.<br />
<br />
It is good that the authors of this paper provided analysis on different types of games, but it would be great if they could also provide some future insights on this method such as whether it can be applied to more complex games such as blackjack poker / it would work or not.<br />
<br />
== References ==<br />
<br />
[1] Ling, C. K., Fang, F., & Kolter, J. Z. (2018). What game are we playing? end-to-end learning in normal and extensive form games. arXiv preprint arXiv:1805.02777.<br />
<br />
[2] B. von Stengel. Efficient computation of behavior strategies.Games and Economics Behavior,14(0050):220–246, 1996.<br />
<br />
[3] Boyd, S., Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge university press.<br />
<br />
[4] Farina, G., Kroer, C., & Sandholm, T. (2019). Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01), 1917-1925. https://doi.org/10.1609/aaai.v33i01.33011917</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=what_game_are_we_playing&diff=49188what game are we playing2020-12-05T04:22:30Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Authors == <br />
Yuxin Wang, Evan Peters, Yifan Mou, Sangeeth Kalaichanthiran <br />
<br />
== Introduction ==<br />
Recently, there have been many different studies of methods using AI to solve the optimal solution for large-scale, zero-sum, extensive form problems. However, most of these works operate under the assumption that the parameters of the game are known, while this scenario is ideal. In real-world problems, most of the parameters are unknown prior to the game. This paper proposes a framework for finding an optimal solution using a primal-dual Newton Method and then using back-propagation to analytically compute the gradients of all the relevant game parameters.<br />
<br />
The approach to solving this problem is to consider ''quantal response equilibrium'' (QRE), which is a generalization of Nash equilibrium (NE) where the agents can make suboptimal decisions. It is shown that the solution to the QRE is a differentiable function of the payoff matrix. Consequently, back-propagation can be used to analytically solve for the payoff matrix (or other game parameters). This strategy has many future application areas as it allows for game-solving (both extensive and normal form) to be integrated as a module in a deep neural network.<br />
<br />
An example of architecture is presented below:<br />
<br />
[[File:Framework.png ]]<br />
<br />
Payoff matrix <math> P </math> is parameterized by a domain-dependent low dimensional vector <math> \phi </math>, where <math> \phi </math> depends on a differentiable function <math> M_1(x) </math>. Furthermore, <math> P </math> is applied to QRE to get the equilibrium strategies <math> (u^∗, v^∗) </math>. Lastly, the loss function is calculated after applying any differentiable <math> M_2(u^∗, v^∗) </math>.<br />
<br />
The effectiveness of this model is demonstrated using the games “Rock, Paper, Scissors”, one-card poker, and a security defense game. This could represent a substantial step forward in understanding how game-theoretic methods can be applied to uncertain settings by the ability to learn solely from observed actions and the relevant parameters.<br />
<br />
== Learning and Quantal Response in Normal Form Games ==<br />
<br />
The game-solving module provides all elements required in differentiable learning, which maps contextual features to payoff matrices, and computes equilibrium strategies under a set of contextual features. This paper will learn zero-sum games and start with normal form games since they have game solver and learning approach capturing much of intuition and basic methodology.<br />
<br />
=== Zero-Sum Normal Form Games ===<br />
<br />
In two-player zero-sum games there is a '''payoff matrix''' <math>P</math> that describes the rewards for two players employing specific strategies u and v respectively. The optimal strategy mixture may be found with a classic min-max formulation:<br />
$$\min_u \max_v \ u^T P v \\ subject \ to \ 1^T u =1, u \ge 0 \\ 1^T v =1, v \ge 0. \ $$<br />
<br />
Here, we consider the case where <math>P</math> is not known a priori. <math>P</math> could be a single fixed but unknown payoff<br />
matrix, or depend on some external context <math>x</math>. The solution <math> (u^*, v_0) </math> to this optimization and the solution <math> (u_0,v^*) </math> to the corresponding problem with inverse player order form the Nash equilibrium <math>(u^*,v^*) </math>. At this equilibrium, the players do not have anything to gain by changing their strategy, so this point is a stable state of the system. When the payoff matrix P is not known, we observe samples of actions <math> a^{(i)}, i =1,...,N </math> from one or both players, which depends on some external content <math> x </math>, sampled from the equilibrium strategies <math>(u^*,v^*) </math>, to recover the true underlying payoff matrix P or a function form P(x) depending on the current context.<br />
<br />
=== Quantal Response Equilibria ===<br />
<br />
However, NE is poorly suited because NEs are overly strict, discontinuous with respect to P, and may not be unique. To address these issues, model the players' actions with the '''quantal response equilibria''' (QRE), where noise is added to the payoff matric. Specifically, consider the ''logit'' equilibrium for zero-sum games that obeys the fixed point:<br />
$$<br />
u^* _i = \frac {exp(-Pv)_i}{\sum_{q \in [n]} exp (-Pv)_q}, \ v^* _j= \frac {exp(P^T u)_j}{\sum_{q \in [m]} exp (P^T u)_q} .\qquad \ (1)<br />
$$<br />
For a fixed opponent strategy, the logit equilibrium corresponding to a strategy is strictly convex, and thus the regularized best response is unique.<br />
<br />
=== End-to-End Learning ===<br />
<br />
Then to integrate a zero-sum solver, [1] introduced a method to solve the QRE and to differentiate through its solution.<br />
<br />
'''QRE solver''':<br />
To find the fixed point in (1), it is equivalent to solve the regularized min-max game:<br />
$$<br />
\min_{u \in \mathbb{R}^n} \max_{v \in \mathbb{R}^m} \ u^T P v -H(v) + H(u) \\<br />
\text{subject to } 1^T u =1, \ 1^T v =1, <br />
$$<br />
where H(y) is the Gibbs entropy <math> \sum_i y_i log y_i</math>.<br />
Entropy regularization guarantees the non-negative condition and makes the equilibrium continuous with respect to P, which means players are encouraged to play more randomly, and all actions have a non-zero probability. Moreover, this problem has a unique saddle point corresponding to <math> (u^*, v^*) </math>.<br />
<br />
Using a primal-dual Newton Method to solve the QRE for two-player zero-sum games, the KKT conditions for the problem are:<br />
$$ <br />
Pv + \log(u) + 1 +\mu 1 = 0 \\<br />
P^T v -\log(v) -1 +\nu 1 = 0 \\<br />
1^T u = 1, \ 1^T v = 1, <br />
$$<br />
where <math> (\mu, \nu) </math> are Lagrange multipliers for the equality constraints on u, v respectively. Then applying Newton's method gives the the update rule:<br />
$$<br />
Q \begin{bmatrix} \Delta u \\ \Delta v \\ \Delta \mu \\ \Delta \nu \\ \end{bmatrix} = - \begin{bmatrix} P v + \log u + 1 + \mu 1 \\ P^T u - \log v - 1 + \nu 1 \\ 1^T u - 1 \\ 1^T v - 1 \\ \end{bmatrix}, \qquad (2)<br />
$$<br />
where Q is the Hessian of the Lagrangian, given by <br />
$$ <br />
Q = \begin{bmatrix} <br />
diag(\frac{1}{u}) & P & 1 & 0 \\ <br />
P^T & -diag(\frac{1}{v}) & 0 & 1\\<br />
1^T & 0 & 0 & 0 \\<br />
0 & 1^T & 0 & 0 \\<br />
\end{bmatrix}. <br />
$$<br />
<br />
'''Differentiating Through QRE Solutions''':<br />
The QRE solver provides a method to compute the necessary Jacobian-vector products. Specifically, we compute the gradient of the loss given the solution <math> (u^*,v^*) </math> to the QRE, and some loss function <math> L(u^*,v^*) </math>: <br />
<br />
1. Take differentials of the KKT conditions: <br />
<math><br />
Q \begin{bmatrix} <br />
du & dv & d\mu & d\nu \\ <br />
\end{bmatrix} ^T = \begin{bmatrix} <br />
-dPv & -dP^Tu & 0 & 0 \\ <br />
\end{bmatrix}^T. \ <br />
</math><br />
<br />
2. For small changes du, dv, <br />
<math><br />
dL = \begin{bmatrix} <br />
v^TdP^T & u^TdP & 0 & 0 \\ <br />
\end{bmatrix} Q^{-1} \begin{bmatrix} <br />
-\nabla_u L & -\nabla_v L & 0 & 0 \\ <br />
\end{bmatrix}^T.<br />
</math><br />
<br />
3. Apply this to P, and take limits as dP is small:<br />
<math><br />
\nabla_P L = y_u v^T + u y_v^T, \qquad (3)<br />
</math> where <br />
<math><br />
\begin{bmatrix} <br />
y_u & y_v & y_{\mu} & y_{\nu}\\ <br />
\end{bmatrix}=Q^{-1}\begin{bmatrix} <br />
-\nabla_u L & -\nabla_v L & 0 & 0 \\ <br />
\end{bmatrix}^T.<br />
</math><br />
<br />
Hence, the forward pass is given by using the expression in (2) to solve for the logit equilibrium given P, and the backward pass is given by using <math> \nabla_u L </math> and <math> \nabla_v L </math> to obtain <math> \nabla_P L </math> using (3). There does not always exist a unique P which generates <math> u^*, v^* </math> under the logit QRE, and we cannot expect to recover P when under-constrained.<br />
<br />
== Learning Extensive form games ==<br />
<br />
The normal form representation for games where players have many choices quickly becomes intractable. For example, consider a chess game: One the first turn, player 1 has 20 possible moves and then player 2 has 20 possible responses. If in the following number of turns each player is estimated to have ~30 possible moves and if a typical game is 40 moves per player, the total number of strategies is roughly <math>10^{120} </math> per player (this is known as the Shannon number for game-tree complexity of chess) and so the payoff matrix for a typical game of chess must therefore have <math> O(10^{240}) </math> entries.<br />
<br />
Instead, it is much more useful to represent the game graphically as an "''' Extensive form game'''" (EFG). We'll also need to consider types of games where there is '''imperfect information''' - players do not necessarily have access to the full state of the game. An example of this is one-card poker: (1) Each player draws a single card from a 13-card deck (ignore suits) (2) Player 1 decides whether to bet/hold (3) Player 2 decides whether to call/raise (4) Player 1 must either call/fold if Player 2 raised. From this description, player 1 has <math> 2^{13} </math> possible first moves (all combinations of (card, raise/hold)) and has <math> 2^{13} </math> possible second moves (whenever player 1 gets a second move) for a total of <math> 2^{26} </math> possible strategies. In addition, Player 1 never knows what cards player 2 has and vice versa. So instead of representing the game with a huge payoff matrix, we can instead represent it as a simple decision tree (for a ''single'' drawn the card of player 1):<br />
<br />
<br />
<center> [[File:1cardpoker.PNG]] </center><br />
<br />
where player 1 is represented by "1", a node that has two branches corresponding to the allowed moves of player 1. However there must also be a notion of information available to either player: While this tree might correspond to say, player 1 holding a "9", it contains no information on what card player 2 is holding (and is much simpler because of this). This leads to the definition of an '''information set''': the set of all nodes belonging to a single player for which the other player cannot distinguish which node has been reached. The information set may therefore be treated as a node itself, for which actions stemming from the node must be chosen in ignorance of what the other player did immediately before arriving at the node. In the poker example, the full game tree consists of a much more complex version of the tree shown above (containing repetitions of the given tree for every possible combination of cards dealt) and the and an example of the information set for player 1 is the set of all of the nodes owned by player 2 that immediately follow player 1's decision to hold. In other words, if player 1 holds there are 13 possible nodes describing the responses of player 2 (raise/hold for player 2 having card = ace, 1, ... King), and all 13 of these nodes are indistinguishable to player 1, and so form the information set for player 1.<br />
<br />
The following is a review of important concepts for extensive form games first formalized in [2]. Let <math> \mathcal{I}_i </math> be the set of all information sets for player i, and for each <math> t \in \mathcal{I}_i </math> let <math> \sigma_t </math> be the actions taken by player i to arrive at <math> t </math> and <math> C_t </math> be the actions that player i can take from <math> u </math>. Then the set of all possible sequences that can be taken by player i is given by<br />
<br />
$$<br />
S_i = \{\emptyset \} \cup \{ \sigma_t c | u\in \mathcal{I}_i, c \in C_t \}<br />
$$<br />
<br />
So for the one-card poker we would have <math>S_1 = \{\emptyset, \text{raise}, \text{hold}, \text{hold-call}, \text{hold-fold\} }</math>. From the possible sequences follows two important concepts:<br />
<ol><br />
<li>The EFG '''payoff matrix''' <math> P </math> of size <math>|S_1| \times |S_2| </math> (this is all possible actions available to either player), is populated with rewards from each leaf of the tree (or "zero" for each <math> (s_1, s_2) </math> that is an invalid pair), and the expected payoff for realization plans <math> (u, v) </math> is given by <math> u^T P v </math> </li><br />
<li> A '''realization plan''' <math> u \in \mathbb{R}^{|S_1|} </math> for player 1 (<math> v \in \mathbb{R}^{|S_2|} </math> for player 2 ) will describe probabilities for players to carry out each possible sequence, and each realization plan must be constrained by (i) compatibility of sequences (e.g. "raise" is not compatible with "hold-call") and (ii) information sets available to the player. These constraints are linear:<br />
<br />
$$<br />
Eu = e \\<br />
Fv = f<br />
$$<br />
<br />
where <math> e = f = (1, 0, ..., 0)^T </math> and <math> E, F</math> contain entries in <math> {-1, 0, 1} </math> describing compatibility and information sets. </li><br />
<br />
</ol> <br />
<br />
<br />
The paper's main contribution is to develop a minimax problem for extensive form games:<br />
<br />
<br />
$$<br />
\min_u \max_v u^T P v + \sum_{t\in \mathcal{I}_1} \sum_{c \in C_t} u_c \log \frac{u_c}{u_{p_t}} - \sum_{t\in \mathcal{I}_2} \sum_{c \in C_t} v_c \log \frac{v_c}{v_{p_t}}<br />
$$<br />
<br />
where <math> p_t </math> is the action immediately preceding information set <math> t </math>. Intuitively, each sum resembles a cross-entropy over the distribution of probabilities in the realization plan comparing each probability to proceed from the information set to the probability to arrive at that information set. Importantly, these entropies are strictly convex or concave (for player 1 and player 2 respectively) [3] so that the min-max problem will have a unique solution and ''the objective function is continuous and continuously differentiable'' - this means there is a way to optimize the function. As noted in Theorem 1 of [1], the solution to this problem is equivalently a solution for the QRE of the game in reduced normal form.<br />
<br />
Minimax can also be seen from an algorithmic perspective. Referring to the above figure containing a tree, it contains a sequence of states and action which alternates between two or more competing players. The above formulation of the min-max problem essentially measures how well a decision rule is from the perspective of a single player. To describe it in terms of the tree, if it is player 1's turn, then it is a mutual recursion of player 1 choosing to maximize its payoff and player 2 choosing to minimize player 1's payoff.<br />
<br />
Having decided on a cost function, the method of Lagrange multipliers may be used to construct the Lagrangian that encodes the known constraints (<math> Eu = e \,, Fv = f </math>, and <math> u, v \geq 0</math>), and then optimize the Lagrangian using Newton's method (identically to the previous section). Accounting for the constraints, the Lagrangian becomes <br />
<br />
<br />
$$<br />
\mathcal{L} = g(u, v) + \sum_i \mu_i(Eu - e)_i + \sum_i \nu_i (Fv - f)_i<br />
$$<br />
<br />
where <math>g</math> is the argument from the minimax statement above and <math>u, v \geq 0</math> become KKT conditions. The general update rule for Newton's method may be written in terms of the derivatives of <math> \mathcal{L} </math> with respect to primal variables <math>u, v </math> and dual variables <math> \mu, \nu</math>, yielding:<br />
<br />
$$<br />
\nabla_{u,v,\mu,\nu}^2 \mathcal{L} \cdot (\Delta u, \Delta v, \Delta \mu, \Delta \nu)^T= - \nabla_{u,v,\mu,\nu} \mathcal{L}<br />
$$<br />
where <math>\nabla_{u,v,\mu,\nu}^2 \mathcal{L} </math> is the Hessian of the Lagrangian and <math>\nabla_{u,v,\mu,\nu} \mathcal{L} </math> is simply a column vector of the KKT stationarity conditions. Combined with the previous section, this completes the goal of the paper: To construct a differentiable problem for learning normal form and extensive form games.<br />
<br />
== Experiments ==<br />
<br />
The authors demonstrated learning on extensive form games in the presence of ''side information'', with ''partial observations'' using three experiments. In all cases, the goal was to maximize the likelihood of realizing an observed sequence from the player, assuming they act in accordance with the QRE. The authors found that the best way to implement the module was to use a medium to large batch size, RMSProp, or Adam optimizers with a learning rate between <math>\left[0.0001,0.01\right].</math> <br />
<br />
=== Rock, Paper, Scissors ===<br />
<br />
Rock, Paper, Scissors is a 2-player zero-sum game. For this game, the best strategy to reach a Nash equilibrium and a Quantal response equilibrium is to uniformly play each hand with equal odds.<br />
The first experiment was to learn a non-symmetric variant of Rock, Paper, Scissors with ''incomplete information'' with the following payoff matrix:<br />
<br />
{| class="wikitable" style="float:center; margin-left:1em; text-align:center;"<br />
|+ align="bottom"|''Payoff matrix of modified Rock-Paper-Scissors''<br />
! <br />
! ''Rock''<br />
! ''Paper''<br />
! ''Scissors''<br />
|-<br />
! ''Rock''<br />
| '''''0'''''<br />
| <math>-b_1</math><br />
| <math>b_2</math><br />
|-<br />
! ''Paper''<br />
| <math>b_1</math><br />
| '''''0'''''<br />
| <math>-b_3</math><br />
|-<br />
! ''Scissors''<br />
| <math>-b_2</math><br />
| <math>b_3</math><br />
| '''''0'''''<br />
|}<br />
<br />
where each of the <math> b </math> ’s are linear function of some features <math> x \in \mathbb{R}^{2} </math> (i.e., <math> b_y = x^Tw_y </math>, <math> y \in </math> {<math>1,2,3</math>} , where <math> w_y </math> are to be learned by the algorithm). Using many trials of random rewards the technique produced the following results for optimal strategies[1]: <br />
<br />
[[File:RPS Results.png|500px ]]<br />
<br />
From the graphs above, we can tell the following: 1) both parameters learned and predicted strategies improve with a larger dataset; 2) with a reasonably sized dataset, >1000 here, convergence is stable and is fairly quick.<br />
<br />
=== One-Card Poker ===<br />
<br />
Next they investigated extensive form games using the one-Card Poker (with ''imperfect information'') introduced in the previous section. In the experimental setup, they used a deck stacked non-uniformly (meaning repeat cards were allowed). Their goal was to learn this distribution of cards from observations of many rounds of the play. Different from the distribution of cards dealt, the method built in the paper is more suited to learn the player’s perceived or believed distribution of cards. It may even be a function of contextual features such as demographics of players. Three experiments were run with <math> n=4 </math>. Each experiment comprised 5 runs of training, with same weights but different training sets. Let <math> d \in \mathbb{R}^{n}, d \ge 0, \sum_{i} d_i = 1 </math> be the weights of the cards. The probability that the players are dealt cards <math> (i,j) </math> is <math> \frac{d_i d_j}{1-d_i} </math>. This distribution is asymmetric between players. Matrix <math> P, E, F </math> for the case <math> n=4 </math> are presented in [1]. With training for 2500 epochs, the mean squared error of learned parameters (card weights, <math> u, v </math> ) are averaged over all runs of and are presented as following [1]: <br />
<br />
<br />
[[File:One-card_Poker_Results.png|500px ]]<br />
<br />
=== Security Resource Allocation Game ===<br />
<br />
<br />
From Security Resource Allocation Game, they demonstrated the ability to learn from ''imperfect observations''. The defender possesses <math> k </math> indistinguishable and indivisible defensive resources which he splits among <math> n </math> targets, { <math> T_1, ……, T_n </math>}. The attacker chooses one target. If the attack succeeds, the attacker gets <math> R_i </math> reward and defender gets <math> -R_i </math>, otherwise zero payoff for both. If there are n defenders guarding <math> T_i </math>, probability of successful attack on <math> T_i </math> is <math> \frac{1}{2^n} </math>. The expected payoff matrix when <math> n = 2, k = 3 </math>, where the attackers are the row players is:<br />
<br />
{| class="wikitable" style="float:center; margin-left:1em; text-align:center;"<br />
|+ align="bottom"|''Payoff matrix when <math> n = 2, k = 3 </math>''<br />
! {#<math>D_1</math>,#<math>D_2</math>}<br />
! {0, 3}<br />
! {1, 2}<br />
! {2, 1}<br />
! {3, 0}<br />
|-<br />
! <math>T_1</math><br />
| <math>-R_1</math><br />
| <math>-\frac{1}{2}R_1</math><br />
| <math>-\frac{1}{4}R_1</math><br />
| <math>-\frac{1}{8}R_1</math><br />
|-<br />
! <math>T_2</math><br />
| <math>-\frac{1}{8}R_2</math><br />
| <math>-\frac{1}{4}R_2</math><br />
| <math>-\frac{1}{2}R_2</math><br />
| <math>-R_2</math><br />
|} <br />
<br />
<br />
For a multi-stage game the attacker can launch <math> t </math> attacks, one in each stage while defender can only stick with stage 1. The attacker may change target if the attack in stage 1 is failed. Three experiments are run with <math> n = 2, k = 5 </math> for games with single attack and double attack, i.e, <math> t = 1 </math> and <math> t = 2 </math>. The results of simulated experiments are shown below [1]:<br />
<br />
[[File:Security Game Results.png|500px ]]<br />
<br />
<br />
They learned <math> R_i </math> only based on observations of the defender’s actions and could still recover the game set by only observing the defender’s actions. Same as expectation, the larger dataset size improves the learned parameters. Two outliers are 1) Security Game, the green plot for when <math> t = 2 </math>; and 2) RPS, when comparing between training sizes of 2000 and 5000.<br />
<br />
== Conclusion ==<br />
Unsurprisingly, the results of this study show that in general, the quality of learned parameters improved as the number of observations increased. The network presented in this paper demonstrated improvement over the existing methodology. <br />
<br />
This paper presents an end-to-end framework for implementing a game solver, for both extensive and normal form, as a module in a deep neural network for zero-sum games. This method, unlike many previous works in this area, does not require the parameters of the game to be known to the agent prior to the start of the game. The two-part method analytically computes both the optimal solution and the parameters of the game. Future work involves taking advantage of the KKT matrix structure to increase computation speed and extensions to the area of learning general-sum games.<br />
<br />
== Critiques ==<br />
The proposed method appears to suffer from two flaws. Firstly, the assumption that players behave in accordance with the QRE severely limits the space of player strategies and is known to exhibit pathological behavior even in one-player settings. Second, the solvers are computationally inefficient and are unable to scale. For this setting, they used Newton's method and it is found that the second-order algorithms do not scale to large games as with Nash equilibrium [4]. <br />
<br />
In the one-card poker section, it might be better to write "Next, they investigated extensive form games ... It may even be a function of contextual features such as the demographics of players. ... 5 runs of training, with the same weights but different training sets."<br />
<br />
Zero-Sum Normal Form Games usually follow Gaussian distributions with two non-empty sets of strategies of player one and player two correspondingly, and also a payoff function defined on the set of possible realizations.<br />
<br />
This method of proposing that real players will follow a laid out scheme or playing strategy is almost always a drastic oversimplification of real-world scenarios and severely limits the applications. In order to better fit a model to the real players, there almost always has to be some sort of stochastic implementation which accounts for the presence of irrational users in the system. <br />
<br />
It might be a good idea to discuss the algorithm performance based on more complicated player reactions, for example, player 2 or NPC from the game might react based on player 1's action. If player 2 would react with a "clever" choice, we might be able to shrink the choice space, which might be helpful to accelerate the training time.<br />
<br />
The theory assumes rational players, which means that roughly speaking, the players make decisions based on increasing their respective payoffs (utility values, preferences,..). However, in real-life scenarios, this might not hold true. Thus, it would be a good idea to include considerations regarding this issue in this summary. Another area that is worth exploring is the need for things like “bluffing” in games with hidden information. It would be interesting to see whether the results generated in this paper are in support of "bluffing" or not.<br />
<br />
It is good that the authors of this paper provided analysis on different types of games, but it would be great if they could also provide some future insights on this method such as wether it can be applied to more complex games such as blackjack poker / it would work or not.<br />
<br />
== References ==<br />
<br />
[1] Ling, C. K., Fang, F., & Kolter, J. Z. (2018). What game are we playing? end-to-end learning in normal and extensive form games. arXiv preprint arXiv:1805.02777.<br />
<br />
[2] B. von Stengel. Efficient computation of behavior strategies.Games and Economics Behavior,14(0050):220–246, 1996.<br />
<br />
[3] Boyd, S., Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge university press.<br />
<br />
[4] Farina, G., Kroer, C., & Sandholm, T. (2019). Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01), 1917-1925. https://doi.org/10.1609/aaai.v33i01.33011917</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_ODEs&diff=49186Neural ODEs2020-12-05T04:16:36Z<p>Wtjung: /* Continuous Normalizing Flows */</p>
<hr />
<div>== Introduction ==<br />
Chen et al. propose a new class of neural networks called neural ordinary differential equations (ODEs) in their 2018 paper under the same title. Neural network models, such as residual or recurrent networks, can be generalized as a set of transformations through hidden states (a.k.a layers) <math>\mathbf{h}</math>, given by the equation <br />
<br />
<div style="text-align:center;"><math> \mathbf{h}_{t+1} = \mathbf{h}_t + f(\mathbf{h}_t,\theta_t) </math> (1) </div><br />
<br />
where <math>t \in \{0,...,T\}</math> and <math>\theta_t</math> corresponds to the set of parameters or weights in state <math>t</math>. It is important to note that it has been shown (Lu et al., 2017)(Haber<br />
and Ruthotto, 2017)(Ruthotto and Haber, 2018) that Equation 1 can be viewed as an Euler discretization. Given this Euler description, if the number of layers and step size between layers are taken to their limits, then Equation 1 can instead be described continuously in the form of the ODE, <br />
<br />
<div style="text-align:center;"><math> \frac{d\mathbf{h}(t)}{dt} = f(\mathbf{h}(t),t,\theta) </math> (2). </div><br />
<br />
Equation 2 now describes a network where the output layer <math>\mathbf{h}(T)</math> is generated by solving for the ODE at time <math>T</math>, given the initial value at <math>t=0</math>, where <math>\mathbf{h}(0)</math> is the input layer of the network. <br />
<br />
With a vast amount of theory and research in the field of solving ODEs numerically, there are a number of benefits to formulating the hidden state dynamics this way. One major advantage is that a continuous description of the network allows for the calculation of <math>f</math> at arbitrary intervals and locations. The authors provide an example in section five of how the neural ODE network outperforms the discretized version i.e. residual networks, by taking advantage of the continuity of <math>f</math>. A depiction of this distinction is shown in the figure below. <br />
<br />
<div style="text-align:center;"> [[File:NeuralODEs_Fig1.png|350px]] </div><br />
<br />
In section four the authors show that the single-unit bottleneck of normalizing flows can be overcome by constructing a new class of density models that incorporates the neural ODE network formulation.<br />
The next section on automatic differentiation will describe how utilizing ODE solvers allows for the calculation of gradients of the loss function without storing any of the hidden state information. This results in a very low memory requirement for neural ODE networks in comparison to traditional networks that rely on intermediate hidden state quantities for backpropagation.<br />
<br />
== Reverse-mode Automatic Differentiation of ODE Solutions ==<br />
Like most neural networks, optimizing the weight parameters <math>\theta</math> for a neural ODE network involves finding the gradient of a loss function with respect to those parameters. Differentiating in the forward direction is a simple task, however, this method is very computationally expensive and unstable, as it introduces additional numerical error. Instead, the authors suggest that the gradients can be calculated in the reverse-mode with the adjoint sensitivity method (Pontryagin et al., 1962). This "backpropagation" method solves an augmented version of the forward ODE problem but in reverse, which is something that all ODE solvers are capable of. Section 3 provides results showing that this method gives very desirable memory costs and numerical stability. <br />
<br />
The authors provide an example of the adjoint method by considering the minimization of the scalar-valued loss function <math>L</math>, which takes the solution of the ODE solver as its argument.<br />
<br />
<div style="text-align:center;">[[File:NeuralODEs_Eq1.png|700px]],</div> <br />
This minimization problem requires the calculation of <math>\frac{\partial L}{\partial \mathbf{z}(t_0)}</math> and <math>\frac{\partial L}{\partial \theta}</math>.<br />
<br />
The adjoint itself is defined as <math>\mathbf{a}(t) = \frac{\partial L}{\partial \mathbf{z}(t)}</math>, which describes the gradient of the loss with respect to the hidden state <math>\mathbf{z}(t)</math>. By taking the first derivative of the adjoint, another ODE arises in the form of,<br />
<br />
<div style="text-align:center;"><math>\frac{d \mathbf{a}(t)}{dt} = -\mathbf{a}(t)^T \frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \mathbf{z}}</math> (3).</div> <br />
<br />
Since the value <math>\mathbf{a}(t_0)</math> is required to minimize the loss, the ODE in equation 3 must be solved backwards in time from <math>\mathbf{a}(t_1)</math>. Solving this problem is dependent on the knowledge of the hidden state <math>\mathbf{z}(t)</math> for all <math>t</math>, which an neural ODE does not save on the forward pass. Luckily, both <math>\mathbf{a}(t)</math> and <math>\mathbf{z}(t)</math> can be calculated in reverse, at the same time, by setting up an augmented version of the dynamics and is shown in the final algorithm. Finally, the derivative <math>dL/d\theta</math> can be expressed in terms of the adjoint and the hidden state as, <br />
<br />
<div style="text-align:center;"><math> \frac{dL}{d\theta} -\int_{t_1}^{t_0} \mathbf{a}(t)^T\frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \theta}dt</math> (4).</div><br />
<br />
To obtain very inexpensive calculations of <math>\frac{\partial f}{\partial z}</math> and <math>\frac{\partial f}{\partial \theta}</math> in equation 3 and 4, automatic differentiation can be utilized. The authors present an algorithm to calculate the gradients of <math>L</math> and their dependent quantities with only one call to an ODE solver and is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Algorithm1.png|850px]]</div><br />
<br />
If the loss function has a stronger dependence on the hidden states for <math>t \neq t_0,t_1</math>, then Algorithm 1 can be modified to handle multiple calls to the ODESolve step since most ODE solvers have the capability to provide <math>z(t)</math> at arbitrary times. A visual depiction of this scenario is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODES Fig2.png|350px]]</div><br />
<br />
Please see the [https://arxiv.org/pdf/1806.07366.pdf#page=13 appendix] for extended versions of Algorithm 1 and detailed derivations of each equation in this section.<br />
<br />
== Replacing Residual Networks with ODEs for Supervised Learning ==<br />
Section three of the paper investigates an application of the reverse-mode differentiation described in section two, for the training of neural ODE networks on the MNIST digit data set. To solve for the forward pass in the neural ODE network, the following experiment used Adams-Moulton (AM) method, which is an implicit ODE solver. Although it has a marked improvement over explicit ODE solvers in numerical accuracy, integrating backward through the network for backpropagation is still not preferred and the adjoint sensitivity method is used to perform efficient weight optimization. The network with this "backpropagation" technique is referred to as ODE-Net in this section. <br />
<br />
=== Implementation ===<br />
A residual network (ResNet), studied by He et al. (2016), with six standard residual blocks was used as a comparative model for this experiment. The competing model, ODE-net, replaces the residual blocks of the ResNet with the AM solver. As a hybrid of the two models ResNet and ODE-net, a third network was created called RK-Net, which solves the weight optimization of the neural ODE network explicitly through backward Runge-Kutta integration. The following table shows the training and performance results of each network. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Table1.png|400px]]</div><br />
<br />
Note that <math>L</math> and <math>\tilde{L}</math> are the number of layers in ResNet and the number of function calls that the AM method makes for the two ODE networks and are effectively analogous quantities. As shown in Table 1, both of the ODE networks achieve comparable performance to that of the ResNet with a notable decrease in memory cost for ODE-net.<br />
<br />
<br />
Another interesting component of ODE networks is the ability to control the tolerance in the ODE solver used and subsequently the numerical error in the solution. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Fig3.png|700px]]</div><br />
<br />
The tolerance of the ODE solver is represented by the color bar in Figure 3 above and notice that a variety of effects arise from adjusting this parameter. Primarily, if one was to treat the tolerance as a hyperparameter of sorts, you could tune it such that you find a balance between accuracy (Figure 3a) and computational complexity (Figure 3b). Figure 3c also provides further evidence for the benefits of the adjoint method for the backward pass in ODE-nets since there is a nearly 1:0.5 ratio of forward to backward function calls. In the ResNet and RK-Net examples, this ratio is 1:1.<br />
<br />
Additionally, the authors loosely define the concept of depth in a neural ODE network by referring to Figure 3d. Here it's evident that as you continue to train an ODE network, the number of function evaluations the ODE solver performs increases. As previously mentioned, this quantity is comparable to the network depth of a discretized network. However, as the authors note, this result should be seen as the progression of the network's complexity over training epochs, which is something we expect to increase over time.<br />
<br />
== Continuous Normalizing Flows ==<br />
<br />
Section four tackles the implementation of continuous-depth Neural Networks, but to do so, in the first part of section four the authors discuss theoretically how to establish this kind of network through the use of normalizing flows. The authors use a change of variables method presented in other works (Rezende and Mohamed, 2015), (Dinh et al., 2014), to compute the change of a probability distribution if sample points are transformed through a bijective function, <math>f</math>.<br />
<br />
<div style="text-align:center;"><math>z_1=f(z_0) \Rightarrow \log(p(z_1))=\log(p(z_0))-\log|\det\frac{\partial f}{\partial z_0}|</math></div><br />
<br />
Where p(z) is the probability distribution of the samples and <math>det\frac{\partial f}{\partial z_0}</math> is the determinant of the Jacobian which has a cubic cost in the dimension of '''z''' or the number of hidden units in the network. The authors discovered however that transforming the discrete set of hidden layers in the normalizing flow network to continuous transformations simplifies the computations significantly, due primarily to the following theorem:<br />
<br />
'''''Theorem 1:''' (Instantaneous Change of Variables). Let z(t) be a finite continuous random variable with probability p(z(t)) dependent on time. Let dz/dt=f(z(t),t) be a differential equation describing a continuous-in-time transformation of z(t). Assuming that f is uniformly Lipschitz continuous in z and continuous in t, then the change in log probability also follows a differential equation:''<br />
<br />
<div style="text-align:center;"><math>\frac{\partial \log(p(z(t)))}{\partial t}=-tr\left(\frac{df}{dz(t)}\right)</math></div><br />
<br />
The biggest advantage of using this theorem is that the trace function is a linear function, so if the dynamics of the problem, f, is represented by a sum of functions, then so is the log density. This essentially means that you can now compute flow models with only a linear cost with respect to the number of hidden units, <math>M</math>. In standard normalizing flow models, the cost is <math>O(M^3)</math>, so they will generally fit many layers with a single hidden unit in each layer.<br />
<br />
Finally the authors use these realizations to construct Continuous Normalizing Flow networks (CNFs) by specifying the parameters of the flow as a function of ''t'', ie, <math>f(z(t),t)</math>. They also use a gating mechanism for each hidden unit, <math>\frac{dz}{dt}=\sum_n \sigma_n(t)f_n(z)</math> where <math>\sigma_n(t)\in (0,1)</math> is a separate neural network which learns when to apply each dynamic <math>f_n</math>.<br />
<br />
===Implementation===<br />
<br />
The authors construct two separate types of neural networks to compare against each other, the first is the standard planar Normalizing Flow network (NF) using 64 layers of single hidden units, and the second is their new CNF with 64 hidden units. The NF model is trained over 500,000 iterations using RMSprop, and the CNF network is trained over 10,000 iterations using Adam(algorithm for first-order gradient-based optimization of stochastic objective functions). The loss function is <math>KL(q(x)||p(x))</math> where <math>q(x)</math> is the flow model and <math>p(x)</math> is the target probability density.<br />
<br />
One of the biggest advantages when implementing CNF is that you can train the flow parameters just by performing maximum likelihood estimation on <math>\log(q(x))</math> given <math>p(x)</math>, where <math>q(x)</math> is found via the theorem above, and then reversing the CNF to generate random samples from <math>q(x)</math>. This reversal of the CNF is done with about the same cost of the forward pass which is not able to be done in an NF network. The following two figures demonstrate the ability of CNF to generate more expressive and accurate output data as compared to standard NF networks.<br />
<br />
<div style="text-align:center;"><br />
[[Image:CNFcomparisons.png]]<br />
<br />
[[Image:CNFtransitions.png]]<br />
</div><br />
<br />
Figure 4 clearly shows that the CNF structure exhibits significantly lower loss functions than NF. In figure 5 both networks were tasked with transforming a standard Gaussian distribution into a target distribution, not only was the CNF network more accurate on the two moons target, but also the steps it took along the way were much more intuitive than the output from NF.<br />
<br />
== A Generative Latent Function Time-Series Model ==<br />
<br />
One of the largest issues at play in terms of Neural ODE networks is the fact that in many instances, data points are either very sparsely distributed, or irregularly-sampled. The latent dynamics are discretized and the observations are in the bins of fixed duration. This creates issues with missing data and ill-defined latent variables. An example of this is medical records which are only updated when a patient visits a doctor or the hospital. To solve this issue the authors had to create a generative time-series model which would be able to fill in the gaps of missing data. The authors consider each time series as a latent trajectory stemming from the initial local state <math>z_{t_0 }</math> and determined from a global set of latent parameters. Given a set of observation times and initial state, the generative model constructs points via the following sample procedure:<br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_0}∼p(z_{t_0}) <br />
</math><br />
</div> <br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_1},z_{t_2},\dots,z_{t_N}={\rm ODESolve}(z_{t_0},f,θ_f,t_0,...,t_N)<br />
</math><br />
</div><br />
<br />
<div style="text-align:center;"><br />
each <br />
<math><br />
x_{t_i}∼p(x│z_{t_i},θ_x)<br />
</math><br />
</div><br />
<br />
<math>f</math> is a function which outputs the gradient <math>\frac{\partial z(t)}{\partial t}=f(z(t),θ_f)</math> which is parameterized via a neural net. In order to train this latent variable model, the authors had to first encode their given data and observation times using an RNN encoder, construct the new points using the trained parameters, then decode the points back into the original space. The following figure describes this process:<br />
<br />
<div style="text-align:center;"><br />
[[Image:EncodingFigure.png]]<br />
</div><br />
<br />
Another variable which could affect the latent state of a time-series model is how often an event actually occurs. The authors solved this by parameterizing the rate of events in terms of a Poisson process. They described the set of independent observation times in an interval <math>\left[t_{start},t_{end}\right]</math> as:<br />
<br />
<div style="text-align:center;"> <br />
<math><br />
{\rm log}(p(t_1,t_2,\dots,t_N ))=\sum_{i=1}^N{\rm log}(\lambda(z(t_i)))-\int_{t_{start}}^{t_{end}}λ(z(t))dt<br />
</math><br />
</div><br />
<br />
where <math>\lambda(*)</math> is parameterized via another neural network.<br />
<br />
===Implementation===<br />
<br />
To test the effectiveness of the Latent time-series ODE model (LODE), they fit the encoder with 25 hidden units, parametrize function f with a one-layer 20 hidden unit network, and the decoder as another neural network with 20 hidden units. They compare this against a standard recurrent neural net (RNN) with 25 hidden units trained to minimize Gaussian log-likelihood. The authors tested both of these network systems on a dataset of 2-dimensional spirals which either rotated clockwise or counter-clockwise and sampled the positions of each spiral at 100 equally spaced time steps. They can then simulate irregularly timed data by taking random amounts of points without replacement from each spiral. The next two figures show the outcome of these experiments:<br />
<br />
<div style="text-align:center;"><br />
[[Image:LODEtestresults.png]] [[Image:SpiralFigure.png|The blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model]]<br />
</div><br />
<br />
In the figure on the right the blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model. It is noted that the LODE performs significantly better than the standard RNN model, especially on smaller sets of data points.<br />
<br />
== Scope and Limitations ==<br />
<br />
This part mainly discusses the scope and limitations of the paper. Firstly, while "batching" the training data is a useful step in standard neural nets and can still be applied here by combining the ODEs associated with each batch, the authors found that controlling the error, in this case, may increase the number of calculations required. In practice, however, the number of calculations did not increase significantly.<br />
<br />
So long as the model proposed in this paper uses finite weights and Lipschitz nonlinearities, Picard's existence theorem (Coddington and Levinson, 1955) applies, which guarantees that the solution to the IVP exists and is unique. This theorem holds for the model presented above when the network has finite weights and uses nonlinearities in the Lipshitz class.<br />
<br />
In controlling the error amount in the model, the authors could only reduce tolerances to approximately 10−3 and 10−5 in classification and density estimation, respectively, without also degrading the computational performance.<br />
<br />
The authors believe that reconstructing state trajectories by running the dynamics backward can introduce extra numerical error. They address a possible solution to this problem by checkpointing specific time steps and storing intermediate values of z on the forward pass. Then while reconstructing, it does each part individually between checkpoints. The authors acknowledged that they informally checked this method's validity since they do not consider it a practical problem.<br />
<br />
There remain, however, areas where standard neural networks may perform better than Neural ODEs. Firstly, conventional nets can fit non-homeomorphic functions. Examples of non-homeomorphic functions are functions whose output has a smaller dimension than their input or that change the input space's topology. However, this could be handled by composing ODE nets with standard network layers. In addition, conventional nets that can be evaluated precisely with a fixed amount of computation are typically faster to train. Also, they do not require an error tolerance for a solver.<br />
<br />
== Conclusions and Critiques ==<br />
<br />
We covered the use of black-box ODE solvers as a model component and their application to initial value problems constructed from real applications. Neural ODE Networks show promising gains in computational cost without large sacrifices in accuracy when applied to certain problems. A drawback of some of these implementations is that the ODE Neural Networks are limited by the underlying distributions of the problems they are trying to solve (requirement of Lipschitz continuity, etc.). There are plenty of further advances to be made in this field as hundreds of years of ODE theory and literature is available, so this is currently an important area of research.<br />
<br />
ODEs indeed represent an important area of applied mathematics where neural networks can be used to solve them numerically. Perhaps, a parallel area of investigation can be PDEs (Partial Differential Equations). PDEs are also widely encountered in many areas of applied mathematics, physics, social sciences, and many other fields. It will be interesting to see how neural networks can be used to solve PDEs.<br />
<br />
== More Critiques ==<br />
Table 1 shows a comparison between different implementations which is very helpful. We can see from the table that the 1-Layer MLP has the largest test error and the one with the best performance should be ODE-Net. Although it doesn't have the lowest test error (the test error of ODE-Net is 0.42% and the lowest test error is 0.41% for ResNet), it still has the least number of parameters, memory, and time. This convinced us that it can be widely used in other applications. <br />
<br />
For the last paragraph in the scope and limitations section, I guess the author wants to use the word "than" instead of using "that" in the sentence "for example, functions whose output has a smaller dimension that their input, or that change the topology of the input space."<br />
<br />
This paper covers the memory efficiency of Neural ODE Networks, but does not address runtime. In practice, most systems are bound by latency requirements more-so than memory requirements (except in edge device cases). Though it may be unreasonable to expect the authors to produce a performance-optimized implementation, it would be insightful to understand the computational bottlenecks so existing frameworks can take steps to address them. This model looks promising and practical performance is the key to enabling future research in this.<br />
<br />
The above critique also questions the need for a neural network for such a problem. This problem was studied by Brunel et al. and they presented their solution in their paper ''Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions''. While this solution also requires iteratively solving a complex optimization problem, they did not require the massive memory and runtime overhead of a neural network. For the neural network solution to demonstrate its potential, it should be including experimental comparisons with specialized ordinary differential equation algorithms instead of simply comparing with a general recurrent neural network.<br />
<br />
Table 2 shows that potential ODEs have lower predicted RMSE, and more relevant information should be provided. For example, the reason of setting the n to 30/50/100 can be simply described. And it is good to talk more about the performance of Latent ODE since the change of n does not have much impact on its RMSE.<br />
<br />
== References ==<br />
Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. ''arXiv preprint arXiv'':1710.10121, 2017.<br />
<br />
Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks. ''Inverse Problems'', 34 (1):014004, 2017.<br />
<br />
Lars Ruthotto and Eldad Haber. Deep neural networks motivated by partial differential equations. ''arXiv preprint arXiv'':1804.04272, 2018.<br />
<br />
Lev Semenovich Pontryagin, EF Mishchenko, VG Boltyanskii, and RV Gamkrelidze. ''The mathematical theory of optimal processes''. 1962.<br />
<br />
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In ''European conference on computer vision'', pages 630–645. Springer, 2016b.<br />
<br />
Earl A Coddington and Norman Levinson. ''Theory of ordinary differential equations''. Tata McGrawHill Education, 1955.<br />
<br />
Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. ''arXiv preprint arXiv:1505.05770'', 2015.<br />
<br />
Laurent Dinh, David Krueger, and Yoshua Bengio. NICE: Non-linear independent components estimation. ''arXiv preprint arXiv:1410.8516'', 2014.<br />
<br />
Brunel, N. J., Clairon, Q., & d’Alché-Buc, F. (2014). Parametric estimation of ordinary differential equations with orthogonality conditions. ''Journal of the American Statistical Association'', 109(505), 173-185.<br />
<br />
A. Iserles. A first course in the numerical analysis of differential equations. Cambridge Texts in Applied Mathematics, second edition, 2009</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_ODEs&diff=49185Neural ODEs2020-12-05T04:14:46Z<p>Wtjung: Undo revision 49184 by Wtjung (talk)</p>
<hr />
<div>== Introduction ==<br />
Chen et al. propose a new class of neural networks called neural ordinary differential equations (ODEs) in their 2018 paper under the same title. Neural network models, such as residual or recurrent networks, can be generalized as a set of transformations through hidden states (a.k.a layers) <math>\mathbf{h}</math>, given by the equation <br />
<br />
<div style="text-align:center;"><math> \mathbf{h}_{t+1} = \mathbf{h}_t + f(\mathbf{h}_t,\theta_t) </math> (1) </div><br />
<br />
where <math>t \in \{0,...,T\}</math> and <math>\theta_t</math> corresponds to the set of parameters or weights in state <math>t</math>. It is important to note that it has been shown (Lu et al., 2017)(Haber<br />
and Ruthotto, 2017)(Ruthotto and Haber, 2018) that Equation 1 can be viewed as an Euler discretization. Given this Euler description, if the number of layers and step size between layers are taken to their limits, then Equation 1 can instead be described continuously in the form of the ODE, <br />
<br />
<div style="text-align:center;"><math> \frac{d\mathbf{h}(t)}{dt} = f(\mathbf{h}(t),t,\theta) </math> (2). </div><br />
<br />
Equation 2 now describes a network where the output layer <math>\mathbf{h}(T)</math> is generated by solving for the ODE at time <math>T</math>, given the initial value at <math>t=0</math>, where <math>\mathbf{h}(0)</math> is the input layer of the network. <br />
<br />
With a vast amount of theory and research in the field of solving ODEs numerically, there are a number of benefits to formulating the hidden state dynamics this way. One major advantage is that a continuous description of the network allows for the calculation of <math>f</math> at arbitrary intervals and locations. The authors provide an example in section five of how the neural ODE network outperforms the discretized version i.e. residual networks, by taking advantage of the continuity of <math>f</math>. A depiction of this distinction is shown in the figure below. <br />
<br />
<div style="text-align:center;"> [[File:NeuralODEs_Fig1.png|350px]] </div><br />
<br />
In section four the authors show that the single-unit bottleneck of normalizing flows can be overcome by constructing a new class of density models that incorporates the neural ODE network formulation.<br />
The next section on automatic differentiation will describe how utilizing ODE solvers allows for the calculation of gradients of the loss function without storing any of the hidden state information. This results in a very low memory requirement for neural ODE networks in comparison to traditional networks that rely on intermediate hidden state quantities for backpropagation.<br />
<br />
== Reverse-mode Automatic Differentiation of ODE Solutions ==<br />
Like most neural networks, optimizing the weight parameters <math>\theta</math> for a neural ODE network involves finding the gradient of a loss function with respect to those parameters. Differentiating in the forward direction is a simple task, however, this method is very computationally expensive and unstable, as it introduces additional numerical error. Instead, the authors suggest that the gradients can be calculated in the reverse-mode with the adjoint sensitivity method (Pontryagin et al., 1962). This "backpropagation" method solves an augmented version of the forward ODE problem but in reverse, which is something that all ODE solvers are capable of. Section 3 provides results showing that this method gives very desirable memory costs and numerical stability. <br />
<br />
The authors provide an example of the adjoint method by considering the minimization of the scalar-valued loss function <math>L</math>, which takes the solution of the ODE solver as its argument.<br />
<br />
<div style="text-align:center;">[[File:NeuralODEs_Eq1.png|700px]],</div> <br />
This minimization problem requires the calculation of <math>\frac{\partial L}{\partial \mathbf{z}(t_0)}</math> and <math>\frac{\partial L}{\partial \theta}</math>.<br />
<br />
The adjoint itself is defined as <math>\mathbf{a}(t) = \frac{\partial L}{\partial \mathbf{z}(t)}</math>, which describes the gradient of the loss with respect to the hidden state <math>\mathbf{z}(t)</math>. By taking the first derivative of the adjoint, another ODE arises in the form of,<br />
<br />
<div style="text-align:center;"><math>\frac{d \mathbf{a}(t)}{dt} = -\mathbf{a}(t)^T \frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \mathbf{z}}</math> (3).</div> <br />
<br />
Since the value <math>\mathbf{a}(t_0)</math> is required to minimize the loss, the ODE in equation 3 must be solved backwards in time from <math>\mathbf{a}(t_1)</math>. Solving this problem is dependent on the knowledge of the hidden state <math>\mathbf{z}(t)</math> for all <math>t</math>, which an neural ODE does not save on the forward pass. Luckily, both <math>\mathbf{a}(t)</math> and <math>\mathbf{z}(t)</math> can be calculated in reverse, at the same time, by setting up an augmented version of the dynamics and is shown in the final algorithm. Finally, the derivative <math>dL/d\theta</math> can be expressed in terms of the adjoint and the hidden state as, <br />
<br />
<div style="text-align:center;"><math> \frac{dL}{d\theta} -\int_{t_1}^{t_0} \mathbf{a}(t)^T\frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \theta}dt</math> (4).</div><br />
<br />
To obtain very inexpensive calculations of <math>\frac{\partial f}{\partial z}</math> and <math>\frac{\partial f}{\partial \theta}</math> in equation 3 and 4, automatic differentiation can be utilized. The authors present an algorithm to calculate the gradients of <math>L</math> and their dependent quantities with only one call to an ODE solver and is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Algorithm1.png|850px]]</div><br />
<br />
If the loss function has a stronger dependence on the hidden states for <math>t \neq t_0,t_1</math>, then Algorithm 1 can be modified to handle multiple calls to the ODESolve step since most ODE solvers have the capability to provide <math>z(t)</math> at arbitrary times. A visual depiction of this scenario is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODES Fig2.png|350px]]</div><br />
<br />
Please see the [https://arxiv.org/pdf/1806.07366.pdf#page=13 appendix] for extended versions of Algorithm 1 and detailed derivations of each equation in this section.<br />
<br />
== Replacing Residual Networks with ODEs for Supervised Learning ==<br />
Section three of the paper investigates an application of the reverse-mode differentiation described in section two, for the training of neural ODE networks on the MNIST digit data set. To solve for the forward pass in the neural ODE network, the following experiment used Adams-Moulton (AM) method, which is an implicit ODE solver. Although it has a marked improvement over explicit ODE solvers in numerical accuracy, integrating backward through the network for backpropagation is still not preferred and the adjoint sensitivity method is used to perform efficient weight optimization. The network with this "backpropagation" technique is referred to as ODE-Net in this section. <br />
<br />
=== Implementation ===<br />
A residual network (ResNet), studied by He et al. (2016), with six standard residual blocks was used as a comparative model for this experiment. The competing model, ODE-net, replaces the residual blocks of the ResNet with the AM solver. As a hybrid of the two models ResNet and ODE-net, a third network was created called RK-Net, which solves the weight optimization of the neural ODE network explicitly through backward Runge-Kutta integration. The following table shows the training and performance results of each network. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Table1.png|400px]]</div><br />
<br />
Note that <math>L</math> and <math>\tilde{L}</math> are the number of layers in ResNet and the number of function calls that the AM method makes for the two ODE networks and are effectively analogous quantities. As shown in Table 1, both of the ODE networks achieve comparable performance to that of the ResNet with a notable decrease in memory cost for ODE-net.<br />
<br />
<br />
Another interesting component of ODE networks is the ability to control the tolerance in the ODE solver used and subsequently the numerical error in the solution. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Fig3.png|700px]]</div><br />
<br />
The tolerance of the ODE solver is represented by the color bar in Figure 3 above and notice that a variety of effects arise from adjusting this parameter. Primarily, if one was to treat the tolerance as a hyperparameter of sorts, you could tune it such that you find a balance between accuracy (Figure 3a) and computational complexity (Figure 3b). Figure 3c also provides further evidence for the benefits of the adjoint method for the backward pass in ODE-nets since there is a nearly 1:0.5 ratio of forward to backward function calls. In the ResNet and RK-Net examples, this ratio is 1:1.<br />
<br />
Additionally, the authors loosely define the concept of depth in a neural ODE network by referring to Figure 3d. Here it's evident that as you continue to train an ODE network, the number of function evaluations the ODE solver performs increases. As previously mentioned, this quantity is comparable to the network depth of a discretized network. However, as the authors note, this result should be seen as the progression of the network's complexity over training epochs, which is something we expect to increase over time.<br />
<br />
== Continuous Normalizing Flows ==<br />
<br />
Section four tackles the implementation of continuous-depth Neural Networks, but to do so, in the first part of section four the authors discuss theoretically how to establish this kind of network through the use of normalizing flows. The authors use a change of variables method presented in other works (Rezende and Mohamed, 2015), (Dinh et al., 2014), to compute the change of a probability distribution if sample points are transformed through a bijective function, <math>f</math>.<br />
<br />
<div style="text-align:center;"><math>z_1=f(z_0) \Rightarrow \log(p(z_1))=\log(p(z_0))-\log|\det\frac{\partial f}{\partial z_0}|</math></div><br />
<br />
Where p(z) is the probability distribution of the samples and <math>det\frac{\partial f}{\partial z_0}</math> is the determinant of the Jacobian which has a cubic cost in the dimension of '''z''' or the number of hidden units in the network. The authors discovered however that transforming the discrete set of hidden layers in the normalizing flow network to continuous transformations simplifies the computations significantly, due primarily to the following theorem:<br />
<br />
'''''Theorem 1:''' (Instantaneous Change of Variables). Let z(t) be a finite continuous random variable with probability p(z(t)) dependent on time. Let dz/dt=f(z(t),t) be a differential equation describing a continuous-in-time transformation of z(t). Assuming that f is uniformly Lipschitz continuous in z and continuous in t, then the change in log probability also follows a differential equation:''<br />
<br />
<div style="text-align:center;"><math>\frac{\partial \log(p(z(t)))}{\partial t}=-tr\left(\frac{df}{dz(t)}\right)</math></div><br />
<br />
The biggest advantage to using this theorem is that the trace function is a linear function, so if the dynamics of the problem, f, is represented by a sum of functions, then so is the log density. This essentially means that you can now compute flow models with only a linear cost with respect to the number of hidden units, <math>M</math>. In standard normalizing flow models, the cost is <math>O(M^3)</math>, so they will generally fit many layers with a single hidden unit in each layer.<br />
<br />
Finally the authors use these realizations to construct Continuous Normalizing Flow networks (CNFs) by specifying the parameters of the flow as a function of ''t'', ie, <math>f(z(t),t)</math>. They also use a gating mechanism for each hidden unit, <math>\frac{dz}{dt}=\sum_n \sigma_n(t)f_n(z)</math> where <math>\sigma_n(t)\in (0,1)</math> is a separate neural network which learns when to apply each dynamic <math>f_n</math>.<br />
<br />
===Implementation===<br />
<br />
The authors construct two separate types of neural networks to compare against each other, the first is the standard planar Normalizing Flow network (NF) using 64 layers of single hidden units, and the second is their new CNF with 64 hidden units. The NF model is trained over 500,000 iterations using RMSprop, and the CNF network is trained over 10,000 iterations using Adam(algorithm for first-order gradient-based optimization of stochastic objective functions). The loss function is <math>KL(q(x)||p(x))</math> where <math>q(x)</math> is the flow model and <math>p(x)</math> is the target probability density.<br />
<br />
One of the biggest advantages when implementing CNF is that you can train the flow parameters just by performing maximum likelihood estimation on <math>\log(q(x))</math> given <math>p(x)</math>, where <math>q(x)</math> is found via the theorem above, and then reversing the CNF to generate random samples from <math>q(x)</math>. This reversal of the CNF is done with about the same cost of the forward pass which is not able to be done in an NF network. The following two figures demonstrate the ability of CNF to generate more expressive and accurate output data as compared to standard NF networks.<br />
<br />
<div style="text-align:center;"><br />
[[Image:CNFcomparisons.png]]<br />
<br />
[[Image:CNFtransitions.png]]<br />
</div><br />
<br />
Figure 4 shows clearly that the CNF structure exhibits significantly lower loss functions than NF. In figure 5 both networks were tasked with transforming a standard Gaussian distribution into a target distribution, not only was the CNF network more accurate on the two moons target, but also the steps it took along the way are much more intuitive than the output from NF.<br />
<br />
== A Generative Latent Function Time-Series Model ==<br />
<br />
One of the largest issues at play in terms of Neural ODE networks is the fact that in many instances, data points are either very sparsely distributed, or irregularly-sampled. The latent dynamics are discretized and the observations are in the bins of fixed duration. This creates issues with missing data and ill-defined latent variables. An example of this is medical records which are only updated when a patient visits a doctor or the hospital. To solve this issue the authors had to create a generative time-series model which would be able to fill in the gaps of missing data. The authors consider each time series as a latent trajectory stemming from the initial local state <math>z_{t_0 }</math> and determined from a global set of latent parameters. Given a set of observation times and initial state, the generative model constructs points via the following sample procedure:<br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_0}∼p(z_{t_0}) <br />
</math><br />
</div> <br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_1},z_{t_2},\dots,z_{t_N}={\rm ODESolve}(z_{t_0},f,θ_f,t_0,...,t_N)<br />
</math><br />
</div><br />
<br />
<div style="text-align:center;"><br />
each <br />
<math><br />
x_{t_i}∼p(x│z_{t_i},θ_x)<br />
</math><br />
</div><br />
<br />
<math>f</math> is a function which outputs the gradient <math>\frac{\partial z(t)}{\partial t}=f(z(t),θ_f)</math> which is parameterized via a neural net. In order to train this latent variable model, the authors had to first encode their given data and observation times using an RNN encoder, construct the new points using the trained parameters, then decode the points back into the original space. The following figure describes this process:<br />
<br />
<div style="text-align:center;"><br />
[[Image:EncodingFigure.png]]<br />
</div><br />
<br />
Another variable which could affect the latent state of a time-series model is how often an event actually occurs. The authors solved this by parameterizing the rate of events in terms of a Poisson process. They described the set of independent observation times in an interval <math>\left[t_{start},t_{end}\right]</math> as:<br />
<br />
<div style="text-align:center;"> <br />
<math><br />
{\rm log}(p(t_1,t_2,\dots,t_N ))=\sum_{i=1}^N{\rm log}(\lambda(z(t_i)))-\int_{t_{start}}^{t_{end}}λ(z(t))dt<br />
</math><br />
</div><br />
<br />
where <math>\lambda(*)</math> is parameterized via another neural network.<br />
<br />
===Implementation===<br />
<br />
To test the effectiveness of the Latent time-series ODE model (LODE), they fit the encoder with 25 hidden units, parametrize function f with a one-layer 20 hidden unit network, and the decoder as another neural network with 20 hidden units. They compare this against a standard recurrent neural net (RNN) with 25 hidden units trained to minimize Gaussian log-likelihood. The authors tested both of these network systems on a dataset of 2-dimensional spirals which either rotated clockwise or counter-clockwise and sampled the positions of each spiral at 100 equally spaced time steps. They can then simulate irregularly timed data by taking random amounts of points without replacement from each spiral. The next two figures show the outcome of these experiments:<br />
<br />
<div style="text-align:center;"><br />
[[Image:LODEtestresults.png]] [[Image:SpiralFigure.png|The blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model]]<br />
</div><br />
<br />
In the figure on the right the blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model. It is noted that the LODE performs significantly better than the standard RNN model, especially on smaller sets of data points.<br />
<br />
== Scope and Limitations ==<br />
<br />
This part mainly discusses the scope and limitations of the paper. Firstly, while "batching" the training data is a useful step in standard neural nets and can still be applied here by combining the ODEs associated with each batch, the authors found that controlling the error, in this case, may increase the number of calculations required. In practice, however, the number of calculations did not increase significantly.<br />
<br />
So long as the model proposed in this paper uses finite weights and Lipschitz nonlinearities, Picard's existence theorem (Coddington and Levinson, 1955) applies, which guarantees that the solution to the IVP exists and is unique. This theorem holds for the model presented above when the network has finite weights and uses nonlinearities in the Lipshitz class.<br />
<br />
In controlling the error amount in the model, the authors could only reduce tolerances to approximately 10−3 and 10−5 in classification and density estimation, respectively, without also degrading the computational performance.<br />
<br />
The authors believe that reconstructing state trajectories by running the dynamics backward can introduce extra numerical error. They address a possible solution to this problem by checkpointing specific time steps and storing intermediate values of z on the forward pass. Then while reconstructing, it does each part individually between checkpoints. The authors acknowledged that they informally checked this method's validity since they do not consider it a practical problem.<br />
<br />
There remain, however, areas where standard neural networks may perform better than Neural ODEs. Firstly, conventional nets can fit non-homeomorphic functions. Examples of non-homeomorphic functions are functions whose output has a smaller dimension than their input or that change the input space's topology. However, this could be handled by composing ODE nets with standard network layers. In addition, conventional nets that can be evaluated precisely with a fixed amount of computation are typically faster to train. Also, they do not require an error tolerance for a solver.<br />
<br />
== Conclusions and Critiques ==<br />
<br />
We covered the use of black-box ODE solvers as a model component and their application to initial value problems constructed from real applications. Neural ODE Networks show promising gains in computational cost without large sacrifices in accuracy when applied to certain problems. A drawback of some of these implementations is that the ODE Neural Networks are limited by the underlying distributions of the problems they are trying to solve (requirement of Lipschitz continuity, etc.). There are plenty of further advances to be made in this field as hundreds of years of ODE theory and literature is available, so this is currently an important area of research.<br />
<br />
ODEs indeed represent an important area of applied mathematics where neural networks can be used to solve them numerically. Perhaps, a parallel area of investigation can be PDEs (Partial Differential Equations). PDEs are also widely encountered in many areas of applied mathematics, physics, social sciences, and many other fields. It will be interesting to see how neural networks can be used to solve PDEs.<br />
<br />
== More Critiques ==<br />
Table 1 shows a comparison between different implementations which is very helpful. We can see from the table that the 1-Layer MLP has the largest test error and the one with the best performance should be ODE-Net. Although it doesn't have the lowest test error (the test error of ODE-Net is 0.42% and the lowest test error is 0.41% for ResNet), it still has the least number of parameters, memory, and time. This convinced us that it can be widely used in other applications. <br />
<br />
For the last paragraph in the scope and limitations section, I guess the author wants to use the word "than" instead of using "that" in the sentence "for example, functions whose output has a smaller dimension that their input, or that change the topology of the input space."<br />
<br />
This paper covers the memory efficiency of Neural ODE Networks, but does not address runtime. In practice, most systems are bound by latency requirements more-so than memory requirements (except in edge device cases). Though it may be unreasonable to expect the authors to produce a performance-optimized implementation, it would be insightful to understand the computational bottlenecks so existing frameworks can take steps to address them. This model looks promising and practical performance is the key to enabling future research in this.<br />
<br />
The above critique also questions the need for a neural network for such a problem. This problem was studied by Brunel et al. and they presented their solution in their paper ''Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions''. While this solution also requires iteratively solving a complex optimization problem, they did not require the massive memory and runtime overhead of a neural network. For the neural network solution to demonstrate its potential, it should be including experimental comparisons with specialized ordinary differential equation algorithms instead of simply comparing with a general recurrent neural network.<br />
<br />
Table 2 shows that potential ODEs have lower predicted RMSE, and more relevant information should be provided. For example, the reason of setting the n to 30/50/100 can be simply described. And it is good to talk more about the performance of Latent ODE since the change of n does not have much impact on its RMSE.<br />
<br />
== References ==<br />
Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. ''arXiv preprint arXiv'':1710.10121, 2017.<br />
<br />
Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks. ''Inverse Problems'', 34 (1):014004, 2017.<br />
<br />
Lars Ruthotto and Eldad Haber. Deep neural networks motivated by partial differential equations. ''arXiv preprint arXiv'':1804.04272, 2018.<br />
<br />
Lev Semenovich Pontryagin, EF Mishchenko, VG Boltyanskii, and RV Gamkrelidze. ''The mathematical theory of optimal processes''. 1962.<br />
<br />
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In ''European conference on computer vision'', pages 630–645. Springer, 2016b.<br />
<br />
Earl A Coddington and Norman Levinson. ''Theory of ordinary differential equations''. Tata McGrawHill Education, 1955.<br />
<br />
Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. ''arXiv preprint arXiv:1505.05770'', 2015.<br />
<br />
Laurent Dinh, David Krueger, and Yoshua Bengio. NICE: Non-linear independent components estimation. ''arXiv preprint arXiv:1410.8516'', 2014.<br />
<br />
Brunel, N. J., Clairon, Q., & d’Alché-Buc, F. (2014). Parametric estimation of ordinary differential equations with orthogonality conditions. ''Journal of the American Statistical Association'', 109(505), 173-185.<br />
<br />
A. Iserles. A first course in the numerical analysis of differential equations. Cambridge Texts in Applied Mathematics, second edition, 2009</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_ODEs&diff=49184Neural ODEs2020-12-05T04:12:55Z<p>Wtjung: /* Continuous Normalizing Flows */</p>
<hr />
<div>== Introduction ==<br />
Chen et al. propose a new class of neural networks called neural ordinary differential equations (ODEs) in their 2018 paper under the same title. Neural network models, such as residual or recurrent networks, can be generalized as a set of transformations through hidden states (a.k.a layers) <math>\mathbf{h}</math>, given by the equation <br />
<br />
<div style="text-align:center;"><math> \mathbf{h}_{t+1} = \mathbf{h}_t + f(\mathbf{h}_t,\theta_t) </math> (1) </div><br />
<br />
where <math>t \in \{0,...,T\}</math> and <math>\theta_t</math> corresponds to the set of parameters or weights in state <math>t</math>. It is important to note that it has been shown (Lu et al., 2017)(Haber<br />
and Ruthotto, 2017)(Ruthotto and Haber, 2018) that Equation 1 can be viewed as an Euler discretization. Given this Euler description, if the number of layers and step size between layers are taken to their limits, then Equation 1 can instead be described continuously in the form of the ODE, <br />
<br />
<div style="text-align:center;"><math> \frac{d\mathbf{h}(t)}{dt} = f(\mathbf{h}(t),t,\theta) </math> (2). </div><br />
<br />
Equation 2 now describes a network where the output layer <math>\mathbf{h}(T)</math> is generated by solving for the ODE at time <math>T</math>, given the initial value at <math>t=0</math>, where <math>\mathbf{h}(0)</math> is the input layer of the network. <br />
<br />
With a vast amount of theory and research in the field of solving ODEs numerically, there are a number of benefits to formulating the hidden state dynamics this way. One major advantage is that a continuous description of the network allows for the calculation of <math>f</math> at arbitrary intervals and locations. The authors provide an example in section five of how the neural ODE network outperforms the discretized version i.e. residual networks, by taking advantage of the continuity of <math>f</math>. A depiction of this distinction is shown in the figure below. <br />
<br />
<div style="text-align:center;"> [[File:NeuralODEs_Fig1.png|350px]] </div><br />
<br />
In section four the authors show that the single-unit bottleneck of normalizing flows can be overcome by constructing a new class of density models that incorporates the neural ODE network formulation.<br />
The next section on automatic differentiation will describe how utilizing ODE solvers allows for the calculation of gradients of the loss function without storing any of the hidden state information. This results in a very low memory requirement for neural ODE networks in comparison to traditional networks that rely on intermediate hidden state quantities for backpropagation.<br />
<br />
== Reverse-mode Automatic Differentiation of ODE Solutions ==<br />
Like most neural networks, optimizing the weight parameters <math>\theta</math> for a neural ODE network involves finding the gradient of a loss function with respect to those parameters. Differentiating in the forward direction is a simple task, however, this method is very computationally expensive and unstable, as it introduces additional numerical error. Instead, the authors suggest that the gradients can be calculated in the reverse-mode with the adjoint sensitivity method (Pontryagin et al., 1962). This "backpropagation" method solves an augmented version of the forward ODE problem but in reverse, which is something that all ODE solvers are capable of. Section 3 provides results showing that this method gives very desirable memory costs and numerical stability. <br />
<br />
The authors provide an example of the adjoint method by considering the minimization of the scalar-valued loss function <math>L</math>, which takes the solution of the ODE solver as its argument.<br />
<br />
<div style="text-align:center;">[[File:NeuralODEs_Eq1.png|700px]],</div> <br />
This minimization problem requires the calculation of <math>\frac{\partial L}{\partial \mathbf{z}(t_0)}</math> and <math>\frac{\partial L}{\partial \theta}</math>.<br />
<br />
The adjoint itself is defined as <math>\mathbf{a}(t) = \frac{\partial L}{\partial \mathbf{z}(t)}</math>, which describes the gradient of the loss with respect to the hidden state <math>\mathbf{z}(t)</math>. By taking the first derivative of the adjoint, another ODE arises in the form of,<br />
<br />
<div style="text-align:center;"><math>\frac{d \mathbf{a}(t)}{dt} = -\mathbf{a}(t)^T \frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \mathbf{z}}</math> (3).</div> <br />
<br />
Since the value <math>\mathbf{a}(t_0)</math> is required to minimize the loss, the ODE in equation 3 must be solved backwards in time from <math>\mathbf{a}(t_1)</math>. Solving this problem is dependent on the knowledge of the hidden state <math>\mathbf{z}(t)</math> for all <math>t</math>, which an neural ODE does not save on the forward pass. Luckily, both <math>\mathbf{a}(t)</math> and <math>\mathbf{z}(t)</math> can be calculated in reverse, at the same time, by setting up an augmented version of the dynamics and is shown in the final algorithm. Finally, the derivative <math>dL/d\theta</math> can be expressed in terms of the adjoint and the hidden state as, <br />
<br />
<div style="text-align:center;"><math> \frac{dL}{d\theta} -\int_{t_1}^{t_0} \mathbf{a}(t)^T\frac{\partial f(\mathbf{z}(t),t,\theta)}{\partial \theta}dt</math> (4).</div><br />
<br />
To obtain very inexpensive calculations of <math>\frac{\partial f}{\partial z}</math> and <math>\frac{\partial f}{\partial \theta}</math> in equation 3 and 4, automatic differentiation can be utilized. The authors present an algorithm to calculate the gradients of <math>L</math> and their dependent quantities with only one call to an ODE solver and is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Algorithm1.png|850px]]</div><br />
<br />
If the loss function has a stronger dependence on the hidden states for <math>t \neq t_0,t_1</math>, then Algorithm 1 can be modified to handle multiple calls to the ODESolve step since most ODE solvers have the capability to provide <math>z(t)</math> at arbitrary times. A visual depiction of this scenario is shown below. <br />
<br />
<div style="text-align:center;">[[File:NeuralODES Fig2.png|350px]]</div><br />
<br />
Please see the [https://arxiv.org/pdf/1806.07366.pdf#page=13 appendix] for extended versions of Algorithm 1 and detailed derivations of each equation in this section.<br />
<br />
== Replacing Residual Networks with ODEs for Supervised Learning ==<br />
Section three of the paper investigates an application of the reverse-mode differentiation described in section two, for the training of neural ODE networks on the MNIST digit data set. To solve for the forward pass in the neural ODE network, the following experiment used Adams-Moulton (AM) method, which is an implicit ODE solver. Although it has a marked improvement over explicit ODE solvers in numerical accuracy, integrating backward through the network for backpropagation is still not preferred and the adjoint sensitivity method is used to perform efficient weight optimization. The network with this "backpropagation" technique is referred to as ODE-Net in this section. <br />
<br />
=== Implementation ===<br />
A residual network (ResNet), studied by He et al. (2016), with six standard residual blocks was used as a comparative model for this experiment. The competing model, ODE-net, replaces the residual blocks of the ResNet with the AM solver. As a hybrid of the two models ResNet and ODE-net, a third network was created called RK-Net, which solves the weight optimization of the neural ODE network explicitly through backward Runge-Kutta integration. The following table shows the training and performance results of each network. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Table1.png|400px]]</div><br />
<br />
Note that <math>L</math> and <math>\tilde{L}</math> are the number of layers in ResNet and the number of function calls that the AM method makes for the two ODE networks and are effectively analogous quantities. As shown in Table 1, both of the ODE networks achieve comparable performance to that of the ResNet with a notable decrease in memory cost for ODE-net.<br />
<br />
<br />
Another interesting component of ODE networks is the ability to control the tolerance in the ODE solver used and subsequently the numerical error in the solution. <br />
<br />
<div style="text-align:center;">[[File:NeuralODEs Fig3.png|700px]]</div><br />
<br />
The tolerance of the ODE solver is represented by the color bar in Figure 3 above and notice that a variety of effects arise from adjusting this parameter. Primarily, if one was to treat the tolerance as a hyperparameter of sorts, you could tune it such that you find a balance between accuracy (Figure 3a) and computational complexity (Figure 3b). Figure 3c also provides further evidence for the benefits of the adjoint method for the backward pass in ODE-nets since there is a nearly 1:0.5 ratio of forward to backward function calls. In the ResNet and RK-Net examples, this ratio is 1:1.<br />
<br />
Additionally, the authors loosely define the concept of depth in a neural ODE network by referring to Figure 3d. Here it's evident that as you continue to train an ODE network, the number of function evaluations the ODE solver performs increases. As previously mentioned, this quantity is comparable to the network depth of a discretized network. However, as the authors note, this result should be seen as the progression of the network's complexity over training epochs, which is something we expect to increase over time.<br />
<br />
== Continuous Normalizing Flows ==<br />
<br />
Section four tackles the implementation of continuous-depth Neural Networks, but to do so, in the first part of section four the authors discuss theoretically how to establish this kind of network through the use of normalizing flows. The authors use a change of variables method presented in other works (Rezende and Mohamed, 2015), (Dinh et al., 2014), to compute the change of a probability distribution if sample points are transformed through a bijective function, <math>f</math>.<br />
<br />
<div style="text-align:center;"><math>z_1=f(z_0) \Rightarrow \log(p(z_1))=\log(p(z_0))-\log|\det\frac{\partial f}{\partial z_0}|</math></div><br />
<br />
Where p(z) is the probability distribution of the samples and <math>det\frac{\partial f}{\partial z_0}</math> is the determinant of the Jacobian which has a cubic cost in the dimension of '''z''' or the number of hidden units in the network. The authors discovered however that transforming the discrete set of hidden layers in the normalizing flow network to continuous transformations simplifies the computations significantly, due primarily to the following theorem:<br />
<br />
'''''Theorem 1:''' (Instantaneous Change of Variables). Let z(t) be a finite continuous random variable with probability p(z(t)) dependent on time. Let dz/dt=f(z(t),t) be a differential equation describing a continuous-in-time transformation of z(t). Assuming that f is uniformly Lipschitz continuous in z and continuous in t, then the change in log probability also follows a differential equation:''<br />
<br />
<div style="text-align:center;"><math>\frac{\partial \log(p(z(t)))}{\partial t}=-tr\left(\frac{df}{dz(t)}\right)</math></div><br />
<br />
The biggest advantage OG using this theorem is that the trace function is a linear function, so if the dynamics of the problem, f, is represented by a sum of functions, then so is the log density. This essentially means that you can now compute flow models with only a linear cost with respect to the number of hidden units, <math>M</math>. In standard normalizing flow models, the cost is <math>O(M^3)</math>, so they will generally fit many layers with a single hidden unit in each layer.<br />
<br />
Finally, the authors use these realizations to construct Continuous Normalizing Flow networks (CNFs) by specifying the parameters of the flow as a function of ''t'', ie, <math>f(z(t),t)</math>. They also use a gating mechanism for each hidden unit, <math>\frac{dz}{dt}=\sum_n \sigma_n(t)f_n(z)</math> where <math>\sigma_n(t)\in (0,1)</math> is a separate neural network which learns when to apply each dynamic <math>f_n</math>.<br />
<br />
===Implementation===<br />
<br />
The authors construct two separate types of neural networks to compare against each other, the first is the standard planar Normalizing Flow network (NF) using 64 layers of single hidden units, and the second is their new CNF with 64 hidden units. The NF model is trained over 500,000 iterations using RMSprop, and the CNF network is trained over 10,000 iterations using Adam(algorithm for first-order gradient-based optimization of stochastic objective functions). The loss function is <math>KL(q(x)||p(x))</math> where <math>q(x)</math> is the flow model and <math>p(x)</math> is the target probability density.<br />
<br />
One of the biggest advantages when implementing CNF is that you can train the flow parameters just by performing maximum likelihood estimation on <math>\log(q(x))</math> given <math>p(x)</math>, where <math>q(x)</math> is found via the theorem above, and then reversing the CNF to generate random samples from <math>q(x)</math>. This reversal of the CNF is done with about the same cost of the forward pass which is not able to be done in an NF network. The following two figures demonstrate the ability of CNF to generate more expressive and accurate output data as compared to standard NF networks.<br />
<br />
<div style="text-align:center;"><br />
[[Image:CNFcomparisons.png]]<br />
<br />
[[Image:CNFtransitions.png]]<br />
</div><br />
<br />
Figure 4 shows clearly that the CNF structure exhibits significantly lower loss functions than NF. In figure 5 both networks were tasked with transforming a standard Gaussian distribution into a target distribution, not only was the CNF network more accurate on the two moons target, but also the steps it took along the way are much more intuitive than the output from NF.<br />
<br />
== A Generative Latent Function Time-Series Model ==<br />
<br />
One of the largest issues at play in terms of Neural ODE networks is the fact that in many instances, data points are either very sparsely distributed, or irregularly-sampled. The latent dynamics are discretized and the observations are in the bins of fixed duration. This creates issues with missing data and ill-defined latent variables. An example of this is medical records which are only updated when a patient visits a doctor or the hospital. To solve this issue the authors had to create a generative time-series model which would be able to fill in the gaps of missing data. The authors consider each time series as a latent trajectory stemming from the initial local state <math>z_{t_0 }</math> and determined from a global set of latent parameters. Given a set of observation times and initial state, the generative model constructs points via the following sample procedure:<br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_0}∼p(z_{t_0}) <br />
</math><br />
</div> <br />
<br />
<div style="text-align:center;"><br />
<math><br />
z_{t_1},z_{t_2},\dots,z_{t_N}={\rm ODESolve}(z_{t_0},f,θ_f,t_0,...,t_N)<br />
</math><br />
</div><br />
<br />
<div style="text-align:center;"><br />
each <br />
<math><br />
x_{t_i}∼p(x│z_{t_i},θ_x)<br />
</math><br />
</div><br />
<br />
<math>f</math> is a function which outputs the gradient <math>\frac{\partial z(t)}{\partial t}=f(z(t),θ_f)</math> which is parameterized via a neural net. In order to train this latent variable model, the authors had to first encode their given data and observation times using an RNN encoder, construct the new points using the trained parameters, then decode the points back into the original space. The following figure describes this process:<br />
<br />
<div style="text-align:center;"><br />
[[Image:EncodingFigure.png]]<br />
</div><br />
<br />
Another variable which could affect the latent state of a time-series model is how often an event actually occurs. The authors solved this by parameterizing the rate of events in terms of a Poisson process. They described the set of independent observation times in an interval <math>\left[t_{start},t_{end}\right]</math> as:<br />
<br />
<div style="text-align:center;"> <br />
<math><br />
{\rm log}(p(t_1,t_2,\dots,t_N ))=\sum_{i=1}^N{\rm log}(\lambda(z(t_i)))-\int_{t_{start}}^{t_{end}}λ(z(t))dt<br />
</math><br />
</div><br />
<br />
where <math>\lambda(*)</math> is parameterized via another neural network.<br />
<br />
===Implementation===<br />
<br />
To test the effectiveness of the Latent time-series ODE model (LODE), they fit the encoder with 25 hidden units, parametrize function f with a one-layer 20 hidden unit network, and the decoder as another neural network with 20 hidden units. They compare this against a standard recurrent neural net (RNN) with 25 hidden units trained to minimize Gaussian log-likelihood. The authors tested both of these network systems on a dataset of 2-dimensional spirals which either rotated clockwise or counter-clockwise and sampled the positions of each spiral at 100 equally spaced time steps. They can then simulate irregularly timed data by taking random amounts of points without replacement from each spiral. The next two figures show the outcome of these experiments:<br />
<br />
<div style="text-align:center;"><br />
[[Image:LODEtestresults.png]] [[Image:SpiralFigure.png|The blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model]]<br />
</div><br />
<br />
In the figure on the right the blue lines represent the test data learned curves and the red lines represent the extrapolated curves predicted by each model. It is noted that the LODE performs significantly better than the standard RNN model, especially on smaller sets of data points.<br />
<br />
== Scope and Limitations ==<br />
<br />
This part mainly discusses the scope and limitations of the paper. Firstly, while "batching" the training data is a useful step in standard neural nets and can still be applied here by combining the ODEs associated with each batch, the authors found that controlling the error, in this case, may increase the number of calculations required. In practice, however, the number of calculations did not increase significantly.<br />
<br />
So long as the model proposed in this paper uses finite weights and Lipschitz nonlinearities, Picard's existence theorem (Coddington and Levinson, 1955) applies, which guarantees that the solution to the IVP exists and is unique. This theorem holds for the model presented above when the network has finite weights and uses nonlinearities in the Lipshitz class.<br />
<br />
In controlling the error amount in the model, the authors could only reduce tolerances to approximately 10−3 and 10−5 in classification and density estimation, respectively, without also degrading the computational performance.<br />
<br />
The authors believe that reconstructing state trajectories by running the dynamics backward can introduce extra numerical error. They address a possible solution to this problem by checkpointing specific time steps and storing intermediate values of z on the forward pass. Then while reconstructing, it does each part individually between checkpoints. The authors acknowledged that they informally checked this method's validity since they do not consider it a practical problem.<br />
<br />
There remain, however, areas where standard neural networks may perform better than Neural ODEs. Firstly, conventional nets can fit non-homeomorphic functions. Examples of non-homeomorphic functions are functions whose output has a smaller dimension than their input or that change the input space's topology. However, this could be handled by composing ODE nets with standard network layers. In addition, conventional nets that can be evaluated precisely with a fixed amount of computation are typically faster to train. Also, they do not require an error tolerance for a solver.<br />
<br />
== Conclusions and Critiques ==<br />
<br />
We covered the use of black-box ODE solvers as a model component and their application to initial value problems constructed from real applications. Neural ODE Networks show promising gains in computational cost without large sacrifices in accuracy when applied to certain problems. A drawback of some of these implementations is that the ODE Neural Networks are limited by the underlying distributions of the problems they are trying to solve (requirement of Lipschitz continuity, etc.). There are plenty of further advances to be made in this field as hundreds of years of ODE theory and literature is available, so this is currently an important area of research.<br />
<br />
ODEs indeed represent an important area of applied mathematics where neural networks can be used to solve them numerically. Perhaps, a parallel area of investigation can be PDEs (Partial Differential Equations). PDEs are also widely encountered in many areas of applied mathematics, physics, social sciences, and many other fields. It will be interesting to see how neural networks can be used to solve PDEs.<br />
<br />
== More Critiques ==<br />
Table 1 shows a comparison between different implementations which is very helpful. We can see from the table that the 1-Layer MLP has the largest test error and the one with the best performance should be ODE-Net. Although it doesn't have the lowest test error (the test error of ODE-Net is 0.42% and the lowest test error is 0.41% for ResNet), it still has the least number of parameters, memory, and time. This convinced us that it can be widely used in other applications. <br />
<br />
For the last paragraph in the scope and limitations section, I guess the author wants to use the word "than" instead of using "that" in the sentence "for example, functions whose output has a smaller dimension that their input, or that change the topology of the input space."<br />
<br />
This paper covers the memory efficiency of Neural ODE Networks, but does not address runtime. In practice, most systems are bound by latency requirements more-so than memory requirements (except in edge device cases). Though it may be unreasonable to expect the authors to produce a performance-optimized implementation, it would be insightful to understand the computational bottlenecks so existing frameworks can take steps to address them. This model looks promising and practical performance is the key to enabling future research in this.<br />
<br />
The above critique also questions the need for a neural network for such a problem. This problem was studied by Brunel et al. and they presented their solution in their paper ''Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions''. While this solution also requires iteratively solving a complex optimization problem, they did not require the massive memory and runtime overhead of a neural network. For the neural network solution to demonstrate its potential, it should be including experimental comparisons with specialized ordinary differential equation algorithms instead of simply comparing with a general recurrent neural network.<br />
<br />
Table 2 shows that potential ODEs have lower predicted RMSE, and more relevant information should be provided. For example, the reason of setting the n to 30/50/100 can be simply described. And it is good to talk more about the performance of Latent ODE since the change of n does not have much impact on its RMSE.<br />
<br />
== References ==<br />
Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. ''arXiv preprint arXiv'':1710.10121, 2017.<br />
<br />
Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks. ''Inverse Problems'', 34 (1):014004, 2017.<br />
<br />
Lars Ruthotto and Eldad Haber. Deep neural networks motivated by partial differential equations. ''arXiv preprint arXiv'':1804.04272, 2018.<br />
<br />
Lev Semenovich Pontryagin, EF Mishchenko, VG Boltyanskii, and RV Gamkrelidze. ''The mathematical theory of optimal processes''. 1962.<br />
<br />
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In ''European conference on computer vision'', pages 630–645. Springer, 2016b.<br />
<br />
Earl A Coddington and Norman Levinson. ''Theory of ordinary differential equations''. Tata McGrawHill Education, 1955.<br />
<br />
Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. ''arXiv preprint arXiv:1505.05770'', 2015.<br />
<br />
Laurent Dinh, David Krueger, and Yoshua Bengio. NICE: Non-linear independent components estimation. ''arXiv preprint arXiv:1410.8516'', 2014.<br />
<br />
Brunel, N. J., Clairon, Q., & d’Alché-Buc, F. (2014). Parametric estimation of ordinary differential equations with orthogonality conditions. ''Journal of the American Statistical Association'', 109(505), 173-185.<br />
<br />
A. Iserles. A first course in the numerical analysis of differential equations. Cambridge Texts in Applied Mathematics, second edition, 2009</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Mask_RCNN&diff=49182Mask RCNN2020-12-05T04:10:55Z<p>Wtjung: /* Critiques */</p>
<hr />
<div>== Presented by == <br />
Qing Guo, Xueguang Ma, James Ni, Yuanxin Wang<br />
<br />
== Introduction == <br />
Mask RCNN [1] is a deep neural network architecture that aims to solve instance segmentation problems in computer vision which is important when attempting to identify different objects within the same image. RCNN base architectures first extract a regional proposal (a region of the image where the object of interest is proposed to lie) and then attempts to classify the object within it. <br />
Mask R-CNN extends Faster R-CNN [2] by adding a branch for predicting an object mask in parallel with the existing branch for bounding box recognition. This is done by using a Fully Convolutional Network as each mask branch in a pixel-by-pixel way. Mask R-CNN is simple to train and adds only a small overhead to Faster R-CNN, running at 5 fps. Moreover, Mask R-CNN is easy to generalize to other tasks, e.g., allowing us to estimate human poses in the same framework. Mask R-CNN achieved top results in all three tracks of the COCO suite of challenges [3], including instance segmentation, bounding-box object detection, and person keypoint detection.<br />
<br />
== Visual Perception Tasks == <br />
<br />
Figure 1 shows a visual representation of different types of visual perception tasks:<br />
<br />
- Image Classification: Predict a set of labels to characterize the contents of an input image<br />
<br />
- Object Detection: Build on image classification but localize each object in an image by placing bounding boxes around the objects<br />
<br />
- Semantic Segmentation: Associate every pixel in an input image with a class label<br />
<br />
- Instance Segmentation: Associate every pixel in an input image to a specific object. Instance segmentation combines image classification, object detection and semantic segmentation making it a complex task [1].<br />
<br />
[[File:instance segmentation.png | center]]<br />
<div align="center">Figure 1: Visual Perception tasks</div><br />
<br />
<br />
Mask RCNN is a deep neural network architecture for Instance Segmentation.<br />
<br />
== Related Work == <br />
Region Proposal Network: A Region Proposal Network (RPN) takes an image (of any size) as input and outputs a set of rectangular object proposals, each with an objectness score.<br />
<br />
ROI Pooling: The main use of ROI (Region of Interest) Pooling is to adjust the proposal to a uniform size. It’s better for the subsequent network to process. It maps the proposal to the corresponding position of the feature map, divide the mapped area into sections of the same size, and performs max pooling or average pooling operations on each section.<br />
<br />
Faster R-CNN: Faster R-CNN consists of two stages: Region Proposal Network and ROI Pooling. Region Proposal Network proposes candidate object bounding boxes. ROI Pooling, which is in essence Fast R-CNN, extracts features using RoIPool from each candidate box and performs classification and bounding-box regression. The features used by both stages can be shared for faster inference.<br />
<br />
[[File:FasterRCNN.png | center]]<br />
<div align="center">Figure 2: Faster RCNN architecture</div><br />
<br />
<br />
ResNet-FPN: FPN uses a top-down architecture with lateral connections to build an in-network feature pyramid from a single-scale input. FPN is a general architecture that can be used in conjunction with various networks, such as VGG, ResNet, etc. Faster R-CNN with an FPN backbone extracts RoI features from different levels of the feature pyramid according to their scale. Other than FPN, the rest of the approach is similar to vanilla ResNet. Using a ResNet-FPN backbone for feature extraction with Mask RCNN gives excellent gains in both accuracy and speed.<br />
<br />
[[File:ResNetFPN.png | center]]<br />
<div align="center">Figure 3: ResNetFPN architecture</div><br />
<br />
== Model Architecture == <br />
The structure of mask R-CNN is quite similar to the structure of faster R-CNN. <br />
Faster R-CNN has two stages, the RPN(Region Proposal Network) first proposes candidate object bounding boxes. Then RoIPool extracts the features from these boxes. After the features are extracted, these features data can be analyzed using classification and bounding-box regression. Mask R-CNN shares the identical first stage. But the second stage is adjusted to tackle the issue of simplifying the stages pipeline. Instead of only performing classification and bounding-box regression, it also outputs a binary mask for each RoI as <math>L=L_{cls}+L_{box}+L_{mask}</math>, where <math>L_{cls}</math>, <math>L_{box}</math>, <math>L_{mask}</math> represent the classification loss, bounding box loss and the average binary cross-entropy loss respectively.<br />
<br />
The important concept here is that, for most recent network systems, there's a certain order to follow when performing classification and regression, because classification depends on mask predictions. Mask R-CNN, on the other hand, applies bounding-box classification and regression in parallel, which effectively simplifies the multi-stage pipeline of the original R-CNN. And just for comparison, complete R-CNN pipeline stages involve 1. Make region proposals; 2. Feature extraction from region proposals; 3. SVM for object classification; 4. Bounding box regression. In conclusion, stages 3 and 4 are adjusted to simplify the network procedures.<br />
<br />
The system follows the multi-task loss, which by formula equals classification loss plus bounding-box loss plus the average binary cross-entropy loss.<br />
One thing worth noticing is that for other network systems, those masks across classes compete with each other, but in this particular case, with a <br />
per-pixel sigmoid and a binary loss the masks across classes no longer compete, which makes this formula the key for good instance segmentation results.<br />
<br />
'' RoIAlign''<br />
<br />
This concept is useful in stage 2 where the RoIPool extracts features from bounding-boxes. For each RoI as input, there will be a mask and a feature map as output. The mask is obtained using the FCN(Fully Convolutional Network) and the feature map is obtained using the RoIPool. The mask helps with spatial layout, which is crucial to the pixel-to-pixel correspondence. <br />
<br />
The two things we desire along the procedure are: pixel-to-pixel correspondence; no quantization is performed on any coordinates involved in the RoI, its bins, or the sampling points. Pixel-to-pixel correspondence makes sure that the input and output match in size. If there is a size difference, there will be information loss, and coordinates cannot be matched. <br />
<br />
RoIPool is standard for extracting a small feature map from each RoI. However, it performs quantization before subdividing into spatial bins which are further quantized. Quantization produces misalignments when it comes to predicting pixel accurate masks. Therefore, instead of quantization, the coordinates are computed using bilinear interpolation They use bilinear interpolation to get the exact values of the inputs features at the 4 RoI bins and aggregate the result (using max or average). These results are robust to the sampling location and number of points and to guarantee spatial correspondence.<br />
<br />
The network architectures utilized are called ResNet and ResNeXt. The depth can be either 50 or 101. ResNet-FPN(Feature Pyramid Network) is used for feature extraction. <br />
<br />
Some implementation details should be mentioned: first, an RoI is considered positive if it has IoU with a ground-truth box of at least 0.5 and negative otherwise. It is important because the mask loss Lmask is defined only on positive RoIs. Second, image-centric training is used to rescale images so that pixel correspondence is achieved. An example complete structure is, the proposal number is 1000 for FPN, and then run the box prediction branch on these proposals. The mask branch is then applied to the highest scoring 100 detection boxes. The mask branch can predict K masks per RoI, but only the kth mask will be used, where k is the predicted class by the classification branch. The m-by-m floating-number mask output is then resized to the RoI size and binarized at a threshold of 0.5.<br />
<br />
== Results ==<br />
[[File:ExpInstanceSeg.png | center]]<br />
<div align="center">Figure 4: Instance Segmentation Experiments</div><br />
<br />
Instance Segmentation: Based on COCO dataset, Mask R-CNN outperforms all categories comparing to MNC and FCIS which are state of the art model <br />
<br />
[[File:BoundingBoxExp.png | center]]<br />
<div align="center">Figure 5: Bounding Box Detection Experiments</div><br />
<br />
Bounding Box Detection: Mask R-CNN outperforms the base variants of all previous state-of-the-art models, including the winner of the COCO 2016 Detection Challenge.<br />
<br />
== Ablation Experiments ==<br />
[[File:BackboneExp.png | center]]<br />
<div align="center">Figure 6: Backbone Architecture Experiments</div><br />
<br />
(a) Backbone Architecture: Better backbones bring expected gains: deeper networks do better, FPN outperforms C4 features, and ResNeXt improves on ResNet. <br />
<br />
[[File:MultiVSInde.png | center]]<br />
<div align="center">Figure 7: Multinomial vs. Independent Masks Experiments</div><br />
<br />
(b) Multinomial vs. Independent Masks (ResNet-50-C4): Decoupling via perclass binary masks (sigmoid) gives large gains over multinomial masks (softmax).<br />
<br />
[[File: RoIAlign.png | center]]<br />
<div align="center">Figure 8: RoIAlign Experiments 1</div><br />
<br />
(c) RoIAlign (ResNet-50-C4): Mask results with various RoI layers. Our RoIAlign layer improves AP by ∼3 points and AP75 by ∼5 points. Using proper alignment is the only factor that contributes to the large gap between RoI layers. <br />
<br />
[[File: RoIAlignExp.png | center]]<br />
<div align="center">Figure 9: RoIAlign Experiments w Experiments</div><br />
<br />
(d) RoIAlign (ResNet-50-C5, stride 32): Mask-level and box-level AP using large-stride features. Misalignments are more severe than with stride-16 features, resulting in big accuracy gaps.<br />
<br />
[[File:MaskBranchExp.png | center]]<br />
<div align="center">Figure 10: Mask Branch Experiments</div><br />
<br />
(e) Mask Branch (ResNet-50-FPN): Fully convolutional networks (FCN) vs. multi-layer perceptrons (MLP, fully-connected) for mask prediction. FCNs improve results as they take advantage of explicitly encoding spatial layout.<br />
<br />
== Human Pose Estimation ==<br />
Mask RCNN can be extended to human pose estimation.<br />
<br />
The simple approach the paper presents is to model a keypoint’s location as a one-hot mask, and adopt Mask R-CNN to predict K masks, one for each of K keypoint types such as left shoulder, right elbow. The model has minimal knowledge of human pose and this example illustrates the generality of the model.<br />
<br />
[[File:HumanPose.png | center]]<br />
<div align="center">Figure 11: Keypoint Detection Results</div><br />
<br />
== Experiments on Cityscapes ==<br />
The model was also tested on Cityscapes dataset. From this dataset the authors used 2975 annotated images for training, 500 for validation, and 1525 for testing. The instance segmentation task involved eight categories: person, rider, car, truck, bus, train, motorcycle and bicycle. When the Mask R-CNN model was applied to the data it achieved 26.2 AP on the testing data which was an over 30% improvement on the previous best entry. <br />
<br />
<center><br />
[[ File:cityscapeDataset.png ]]<br />
<br />
<br />
Figure 12. Cityscapes Results<br />
</center><br />
<br />
== Conclusion ==<br />
Mask RCNN is a deep neural network aimed to solve the instance segmentation problems in machine learning or computer vision. Mask R-CNN is a conceptually simple, flexible, and general framework for object instance segmentation. It can efficiently detect objects in an image while simultaneously generating a high-quality segmentation mask for each instance. It does object detection and instance segmentation, and can also be extended to human pose estimation.<br />
It extends Faster R-CNN by adding a branch for predicting an object mask in parallel with the existing branch for bounding box recognition. Mask R-CNN is simple to train and adds only a small overhead to Faster R-CNN, running at 5 fps.<br />
<br />
== Critiques ==<br />
In Faster RCNN, the ROI boundary is quantized. However, mask RCNN avoids quantization and used the bilinear interpolation to compute exact values of features. By solving the misalignments due to quantization, the number and location of sampling points have no impact on the result.<br />
<br />
It may be better to compare the proposed model with other NN models or even non-NN methods like spectral clustering. Also, the applications can be further discussed like geometric mesh processing and motion analysis.<br />
<br />
The paper lacks the comparisons of different methods and Mask RNN on unlabeled data, as the paper only briefly mentioned that the authors found out that Mask R_CNN can benefit from extra data, even if the data is unlabelled.<br />
<br />
The Mask RCNN has many practical applications as well. A particular example, where Mask RCNNs are applied would be in autonomous vehicles. Namely, it would be able to help with isolating pedestrians, other vehicles, lights, etc.<br />
<br />
The Mask RCNN could be a candidate model to do short-term predictions on the physical behaviors of a person, which could be very useful at crime scenes.<br />
<br />
An interesting application of Mask RCNN would be on face recognition from CCTVs. Flurry pictures of crowded people could be obtained from CCTV, so that mask RCNN can be applied to distinguish each person.<br />
<br />
The main problem for CNN architectures like Mask RCNN is the running time. Due to slow running times, Single Shot Detector algorithms are preferred for applications like video or live stream detections, where a faster running time would mean a better response to changes in frames. It would be beneficial to have a graphical representation of the Mask RCNN running times against single shot detector algorithms such as YOLOv3.<br />
<br />
It is interesting to investigate a solution of embedding instance segmentation with semantic segmentation to improve time performance. Because in many situations, knowing the exact boundary of an object is not necessary.<br />
<br />
It will be better if we can have more comparisons with other models. It will also be nice if we can have more details about why Mask RCNN can perform better, and how about the efficiency of it?<br />
The authors mentioned that Mask R-CNN is a deep neural network architecture for Instance Segmentation. It's better to include more background information about this task. For example, challenges of this task (e.g. the model will need to take into account the overlapping of objects) and limitations of existing methods.<br />
<br />
It would be interesting to see how a postprocessing step with conditional random fields (CRF) might improve (or not?) segmentation. It would also have been interesting to see the performance of the method with lighter backbones since the backbones used to have a very large inference time which makes them unsuitable for many applications.<br />
<br />
An extension of the application of Mask RCNN in medical AI is to highlight areas of an MRI scan that correlate to certain behavioral/psychological patterns.<br />
<br />
The use of these in medical imaging systems seems rather useful, but it can also be extended to more general CCTV camera systems which can also detect physiological patterns.<br />
<br />
In the Human Pose Estimation section, we assume that Mask RCNN does not have any knowledge of human poses, and all the predictions are based on keypoints on human bodies, for example, left shoulder and right elbow. While in fact we may be able to achieve better performances here because currently this approach is strongly dependent on correct classifications of human body parts. That is, if the model messed up the position of left shoulder, the position estimation will be awful. It is important to remove the dependency on preceding predictions, so that even when previous steps fail, we may still expect a fair performance.<br />
<br />
It will be interesting to see if applying dropout can boost this Mask RCNN architecture's performance.<br />
<br />
== References ==<br />
[1] Kaiming He, Georgia Gkioxari, Piotr Dollár, Ross Girshick. Mask R-CNN. arXiv:1703.06870, 2017.<br />
<br />
[2] Shaoqing Ren, Kaiming He, Ross Girshick, Jian Sun. Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks, arXiv:1506.01497, 2015.<br />
<br />
[3] Tsung-Yi Lin, Michael Maire, Serge Belongie, Lubomir Bourdev, Ross Girshick, James Hays, Pietro Perona, Deva Ramanan, C. Lawrence Zitnick, Piotr Dollár. Microsoft COCO: Common Objects in Context. arXiv:1405.0312, 2015</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:J46hou&diff=49180User:J46hou2020-12-05T04:08:11Z<p>Wtjung: /* Popular Dataset Benchmark Result */</p>
<hr />
<div>DROCC: Deep Robust One-Class Classification<br />
== Presented by == <br />
Jinjiang Lian, Yisheng Zhu, Jiawen Hou, Mingzhe Huang<br />
== Introduction ==<br />
In this paper, the “one-class” classification, whose goal is to obtain accurate discriminators for a special class, has been studied. Popular uses of this technique include anomaly detection, which is widely used to detect unusual patterns in data. Anomaly detection is a well-studied area of research that aims to learn a model that accurately describes "normality". It has many applications, such as risk assessment for security purposes in many fields, health, and medical risk. However, the conventional approach of modeling with typical data using a simple function falls short when it comes to complex domains such as vision or speech. Another case where this would be useful is when recognizing a “wake-word” while waking up AI systems such as Alexa. <br />
<br />
Deep learning based on anomaly detection methods attempts to learn features automatically but has some limitations. One approach is based on extending the classical data modeling techniques over the learned representations, but in this case, all the points may be mapped to a single point, making the layer look "perfect". The second approach is based on learning the salient geometric structure of data and training the discriminator to predict the applied transformation. The result could be considered anomalous if the discriminator fails to predict the transformation accurately. Appropriate structures or transformations are necessary for these works in general, but are hard to find in practice, especially for domains like time-series or speech, for image data from several orientations, or when generative models are used for deep anomaly detection.<br />
<br />
Thus, in this paper, a new approach called Deep Robust One-Class Classification (DROCC) was presented to solve the above concerns. DROCC is based on the assumption that the points from the class of interest lie on a well-sampled, locally linear low-dimensional manifold. More specifically, we are presenting DROCC-LF which is an outlier-exposure style extension of DROCC. This extension combines the DROCC's anomaly detection loss with standard classification loss over the negative data and exploits the negative examples to learn a Mahalanobis distance.<br />
<br />
== Previous Work ==<br />
Traditional approaches for one-class problems include one-class SVM (Scholkopf et al., 1999) and Isolation Forest (Liu et al., 2008)[9]. One drawback of these approaches is that they involve careful feature engineering when applied to structured domains like images. The current state-of-the-art methodologies to tackle these kinds of problems are: <br />
<br />
1. Approach based on prediction transformations (Golan & El-Yaniv, 2018; Hendricks et al.,2019a) [1]. This work is based on learning the salient geometric structure of typical data by applying specific transformations to the input data and training the discriminator to predict the applied transformation. This approach has some shortcomings in the sense that it depends heavily on an appropriate domain-specific set of transformations that are in general hard to obtain. <br />
<br />
2. Approach of minimizing a classical one-class loss on the learned final layer representations such as DeepSVDD. (Ruff et al.,2018)[2]. Such work has proposed some heuristics to mitigate issues like setting the bias to zero but it is often insufficient in practice. This method suffers from the fundamental drawback of representation collapse, where the learned transformation might map all the points to a single point (like the origin), leading to a degenerate solution and poor discrimination between normal points and the anomalous points.<br />
<br />
3. Approach based on balancing unbalanced training datasets using methods such as SMOTE to synthetically create outlier data to train models on.<br />
<br />
== Motivation ==<br />
Anomaly detection is a well-studied problem with a large body of research (Aggarwal, 2016; Chandola et al., 2009) [3]. The goal is to identify the outliers: points which are not following a typical distribution. The following image provides a visual representation of an outlier/anomaly. <br />
[[File:abnormal.jpeg | thumb | center | 1000px | Abnormal Data (Data Driven Investor, 2020)]]<br />
Classical approaches for anomaly detection are based on modeling the typical data using simple functions over the low-dimensional subspace or a tree-structured partition of the input space to detect anomalies (Schölkopf et al., 1999; Liu et al., 2008; Lakhina et al., 2004) [4], such as constructing a minimum-enclosing ball around the typical data points (Tax & Duin, 2004) [5]. They broadly fall into three categories: AD via generative modeling, Deep Once Class SVM, Transformations based methods, and Side-information based AD. While these techniques are well-suited when the input is featured appropriately, they struggle on complex domains like vision and speech, where hand-designing features are difficult.<br />
<br />
'''AD via Generative Modeling:''' involves deep autoencoders and GAN based methods and have been deeply studied. But, this method solves a much harder problem than required and reconstructs the entire input during the decoding step.<br />
<br />
'''Deep Once Class SVM:''' Deep SVDD attempts to learn a neural network which maps data into a hypersphere. Mappings which fall within the hypersphere are considered "normal". It was the first method to introduce deep one-class classification for the purpose of anomaly detection, but is impeded by representation collapse. <br />
<br />
'''Transformations based methods:''' Are more recent methods that are based on self-supervised training. The training process of these methods applies transformations to the regular points and training the classifier to identify the transformations used. The model relies on the assumption that a point is normal iff the transformations applied to the point can be identified. Some proposed transformations are as simple as rotations and flips, or can be handcrafted and much more complicated. The various transformations that have been proposed are heavily domain dependent and are hard to design.<br />
<br />
'''Side-information based AD:''' incorporate labelled anomalous data or out-of-distribution samples. DROCC makes no assumptions regarding access to side-information.<br />
<br />
Another related problem is the one-class classification under limited negatives (OCLN). In this case, only a few negative samples are available. The goal is to find a classifier that would not misfire close negatives so that the false positive rate will be low. <br />
<br />
DROCC is robust to representation collapse by involving a discriminative component that is general and empirically accurate on most standard domains like tabular, time-series and vision without requiring any additional side information. DROCC is motivated by the key observation that generally, the typical data lies on a low-dimensional manifold, which is well-sampled in the training data and thus tends to be more accurate in practical problems. This is believed to be true even in complex domains such as vision, speech, and natural language (Pless & Souvenir, 2009). [6]<br />
<br />
== Model Explanation ==<br />
[[File:drocc_f1.jpg | center]]<br />
<div align="center">'''Figure 1'''</div><br />
<br />
(a): A normal data manifold with red dots representing generated anomalous points in Ni(r). <br />
<br />
(b): Decision boundary learned by DROCC when applied to the data from (a). Blue represents points classified as normal and red points are classified as abnormal. We observe from here that DROCC is able to capture the manifold accurately; whereas the classical methods, OC-SVM and DeepSVDD perform poorly as they both try to learn a minimum enclosing ball for the whole set of positive data points. <br />
<br />
(c), (d): First two dimensions of the decision boundary of DROCC and DROCC–LF, when applied to noisy data (Section 5.2). DROCC–LF is nearly optimal while DROCC’s decision boundary is inaccurate. Yellow color sine wave depicts the train data.<br />
<br />
== DROCC ==<br />
The model is based on the assumption that the true data lies on a manifold. As manifolds resemble Euclidean space locally, our discriminative component is based on classifying a point as anomalous if it is outside the union of small L2 norm balls around the training typical points (See Figure 1a, 1b for an illustration). Importantly, the above definition allows us to synthetically generate anomalous points, and we adaptively generate the most effective anomalous points while training via a gradient ascent phase reminiscent of adversarial training. In other words, DROCC has a gradient ascent phase to adaptively add anomalous points to our training set and a gradient descent phase to minimize the classification loss by learning a representation and a classifier on top of the representations to separate typical points from the generated anomalous points. In this way, DROCC automatically learns an appropriate representation (like DeepSVDD) but is robust to a representation collapse as mapping all points to the same value would lead to poor discrimination between normal points and the generated anomalous points.<br />
<br />
The algorithm that was used to train the model is laid out below in pseudocode.<br />
<center><br />
[[File:DROCCtrain.png]]<br />
</center><br />
<br />
For a DNN <math>f_\theta: \mathbb{R}^d \to \mathbb{R}</math> that is parameterized by a set of parameters <math>\theta</math>, DROCC estimates <math>\theta^{dr} = \min_\theta\ell^{dr}(\theta)</math> where <br />
$$\ell^{dr}(\theta) = \lambda\|\theta\|^2 + \sum_{i=1}^n[\ell(f_\theta(x_i),1)+\mu\max_{\tilde{x}_i \in N_i(r)}\ell(f_\theta(\tilde{x}_i),-1)]$$<br />
Here, <math>N_i(r) = \{\|\tilde{x}_i-x_i\|_2\leq\gamma\cdot r; r \leq \|\tilde{x}_i - x_j\|, \forall j=1,2,...n\}</math> contains all the points that are at least distance <math>r</math> from the training points. The <math>\gamma \geq 1</math> is a regularization term, and <math>\ell:\mathbb{R} \times \mathbb{R} \to \mathbb{R}</math> is a loss function. The <math>x_i</math> are normal points that should be classified as positive and the <math>\tilde{x}_i</math> are anomalous points that should be classified as negative. This formulation is a saddle point problem.<br />
<br />
== DROCC-LF ==<br />
To especially tackle problems such as anomaly detection and outlier exposure (Hendrycks et al., 2019a) [7], DROCC–LF, an outlier-exposure style extension of DROCC was proposed. Intuitively, DROCC–LF combines DROCC’s anomaly detection loss (that is over only the positive data points) with standard classification loss over the negative data. In addition, DROCC–LF exploits the negative examples to learn a Mahalanobis distance to compare points over the manifold instead of using the standard Euclidean distance, which can be inaccurate for high-dimensional data with relatively fewer samples. (See Figure 1c, 1d for illustration)<br />
<br />
== Popular Dataset Benchmark Result ==<br />
<br />
[[File:drocc_auc.jpg | center]]<br />
<div align="center">'''Figure 2: AUC result'''</div><br />
<br />
The CIFAR-10 dataset consists of 60000 32x32 color images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. The dataset is divided into five training batches and one test batch, each with 10000 images. The test batch contains exactly 1000 randomly selected images from each class. The training batches contain the remaining images in random order, but some training batches may contain more images from one class than another. Between them, the training batches contain exactly 5000 images from each class. The average AUC (with standard deviation) for one-vs-all anomaly detection on CIFAR-10 is shown in table 1. DROCC outperforms baselines on most classes, with gains as high as 20%, and notably, nearest neighbors (NN) beats all the baselines on 2 classes.<br />
<br />
[[File:drocc_f1score.jpg | center]]<br />
<div align="center">'''Figure 3: F1-Score'''</div><br />
<br />
Figure 3 shows F1-Score (with standard deviation) for one-vs-all anomaly detection on Thyroid, Arrhythmia, and Abalone datasets from the UCI Machine Learning Repository. DROCC outperforms the baselines on all three datasets by a minimum of 0.07 which is about an 11.5% performance increase.<br />
Results on One-class Classification with Limited Negatives (OCLN): <br />
[[File:ocln.jpg | center]]<br />
<div align="center">'''Figure 4: Sample positives, negatives and close negatives for MNIST digit 0 vs 1 experiment (OCLN).'''</div><br />
MNIST 0 vs. 1 Classification: <br />
We consider an experimental setup on the MNIST dataset, where the training data consists of Digit 0, the normal class, and Digit 1 as the anomaly. During the evaluation, in addition to samples from training distribution, we also have half zeros, which act as challenging OOD points (close negatives). These half zeros are generated by randomly masking 50% of the pixels (Figure 2). BCE performs poorly, with a recall of 54% only at a fixed FPR of 3%. DROCC–OE gives a recall value of 98:16% outperforming DeepSAD by a margin of 7%, which gives a recall value of 90:91%. DROCC–LF provides further improvement with a recall of 99:4% at 3% FPR. <br />
<br />
[[File:ocln_2.jpg | center]]<br />
<div align="center">'''Figure 5: OCLN on Audio Commands.'''</div><br />
Wake word Detection: <br />
Finally, we evaluate DROCC–LF on the practical problem of wake word detection with low FPR against arbitrary OOD negatives. To this end, we identify a keyword, say “Marvin” from the audio commands dataset (Warden, 2018) [8] as the positive class, and the remaining 34 keywords are labeled as the negative class. For training, we sample points uniformly at random from the above-mentioned dataset. However, for evaluation, we sample positives from the train distribution, but negatives contain a few challenging OOD points as well. Sampling challenging negatives itself is a hard task and is the key motivating reason for studying the problem. So, we manually list close-by keywords to Marvin such as Mar, Vin, Marvelous, etc. We then generate audio snippets for these keywords via a speech synthesis tool 2 with a variety of accents.<br />
Figure 5 shows that for 3% and 5% FPR settings, DROCC–LF is significantly more accurate than the baselines. For example, with FPR=3%, DROCC–LF is 10% more accurate than the baselines. We repeated the same experiment with the keyword "Seven" and observed a similar trend. In summary, DROCC–LF is able to generalize well against negatives that are “close” to the true positives even when such negatives were not supplied with the training data.<br />
<br />
== Conclusion and Future Work ==<br />
We introduced DROCC method for deep anomaly detection. It models normal data points using a low-dimensional sub-manifold inside the feature space, and the anomalous points are characterized via their Euclidean distance from the sub-manifold. Based on this intuition, DROCC’s optimization is formulated as a saddle point problem which is solved via a standard gradient descent-ascent algorithm. We then extended DROCC to OCLN problem where the goal is to generalize well against arbitrary negatives, assuming the positive class is well sampled and a small number of negative points are also available. Both the methods perform significantly better than strong baselines, in their respective problem settings. <br />
<br />
For computational efficiency, we simplified the projection set of both methods which can perhaps slow down the convergence of the two methods. Designing optimization algorithms that can work with the stricter set is an exciting research direction. Further, we would also like to rigorously analyze DROCC, assuming enough samples from a low-curvature manifold. Finally, as OCLN is an exciting problem that routinely comes up in a variety of real-world applications, we would like to apply DROCC–LF to a few high impact scenarios. Possible applications of this work are financial fraud detection, medical anomalies, or key words in audio processing.<br />
<br />
The results of this study showed that DROCC is comparatively better for anomaly detection across many different areas, such as tabular data, images, audio, and time series, when compared to existing state-of-the-art techniques.<br />
<br />
== References ==<br />
[1]: Golan, I. and El-Yaniv, R. Deep anomaly detection using geometric transformations. In Advances in Neural Information Processing Systems (NeurIPS), 2018.<br />
<br />
[2]: Ruff, L., Vandermeulen, R., Goernitz, N., Deecke, L., Siddiqui, S. A., Binder, A., M¨uller, E., and Kloft, M. Deep one-class classification. In International Conference on Machine Learning (ICML), 2018.<br />
<br />
[3]: Aggarwal, C. C. Outlier Analysis. Springer Publishing Company, Incorporated, 2nd edition, 2016. ISBN 3319475770.<br />
<br />
[4]: Sch¨olkopf, B., Williamson, R., Smola, A., Shawe-Taylor, J., and Platt, J. Support vector method for novelty detection. In Proceedings of the 12th International Conference on Neural Information Processing Systems, 1999.<br />
<br />
[5]: Tax, D. M. and Duin, R. P. Support vector data description. Machine Learning, 54(1), 2004.<br />
<br />
[6]: Pless, R. and Souvenir, R. A survey of manifold learning for images. IPSJ Transactions on Computer Vision and Applications, 1, 2009.<br />
<br />
[7]: Hendrycks, D., Mazeika, M., and Dietterich, T. Deep anomaly detection with outlier exposure. In International Conference on Learning Representations (ICLR), 2019a.<br />
<br />
[8]: Warden, P. Speech commands: A dataset for limited vocabulary speech recognition, 2018. URL https: //arxiv.org/abs/1804.03209.<br />
<br />
[9]: Liu, F. T., Ting, K. M., and Zhou, Z.-H. Isolation forest. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining, 2008.<br />
<br />
== Critiques/Insights ==<br />
<br />
1. It would be interesting to see this implemented in self-driving cars, for instance, to detect unusual road conditions.<br />
<br />
2. Figure 1 shows a good representation on how this model works. However, how can we know that this model is not prone to overfitting? There are many situations where there are valid points that lie outside of the line, especially new data that the model has never see before. An explanation on how this is avoided would be good.<br />
<br />
3.In the introduction part, it should first explain what is "one class", and then make a detailed application. Moreover, special definition words are used in many places in the text. No detailed explanation was given. In the end, the future application fields of DROCC and the research direction of the group can be explained.<br />
<br />
4. It will also be interesting to see if one change from using <math>\ell_{2}</math> Euclidean distance to other distances. When the low-dimensional manifold is highly non-linear, using the local linear distance to characterize anomalous points might fail.<br />
<br />
5. This is a nice summary and the authors introduce clearly on the performance of DROCC. It is nice to use Alexa as an example to catch readers' attention. I think it will be nice to include the algorithm of the DROCC or the architecture of DROCC in this summary to help us know the whole view of this method. Maybe it will be interesting to apply DROCC in biomedical studies? since one-class classification is often used in biomedical studies.<br />
<br />
6. For the second sentence in the motivation section, it's better to change "The goal is to identify the outliers: points which are not following a typical distribution" to "The goal is to identify the outliers: points that are not following a typical distribution". In addition, it should be noted that there is an important assumption which assumes the points from the class of interest lie on a well-sampled, locally linear low dimensional manifold when someone wants to use DROCC.<br />
<br />
7. The training method resembles adversarial learning with gradient ascent, however, there is no evaluation of this method on adversarial examples. This is quite unusual considering the paper proposed a method for robust one-class classification, and can be a security threat in real life in critical applications.<br />
<br />
8. The underlying idea behind OCLN is very similar to how neural networks are implemented in recommender systems and trained over positive/negative triplet models. In that case as well, due to the nature of implicit and explicit feedback, positive data tends to dominate the system. It would be interesting to see if insights from that area could be used to further boost the model presented in this paper.<br />
<br />
9. The paper shows the performance of DROCC being evaluated for time series data. It is interesting to see high AUC scores for DROCC against baselines like nearest neighbours and REBMs.Because detecting abnormal data in time series datasets is not common to practice.<br />
<br />
10. Figure1 presented results on a simple 2-D sine wave dataset to visualize the kind of classifiers learnt by DROCC. And the 1a is the positive data lies on a 1-D manifold. We can see from 1b that DROCC is able to capture the manifold accurately.<br />
<br />
11. In the MNIST 0 vs. 1 Classification dataset, why is 1 the only digit that is considered an anomoly? Couldn't all of the non-0 digits be left in the dataset to serve as "anomolies"?<br />
<br />
12. For future work the authors suggest considering DROCC for a low curvature manifold but do not motivate the benefits of such a direction.<br />
<br />
13. One of the problems is that in this model we might need to map all the points to one point to make the layer looks "perfect". However, this might not be a good choice since each point is distinct and if we map them together to one point, then this point cannot tell everything. If authors can specify more details on this it would be better.<br />
<br />
14. This project introduced DROCC for “one-class” classification. It will be interesting if such kind of classification can be compared with any other classification such as binary classification, etc. If “one-class” classification would be more speedy than the others.<br />
<br />
15. The dimensions and feature values must be so different across datasets in different domains. I would love to see how this algorithm is performing so well applied on different domains as it is mentioned that it could be used on datasets including images, audio, time-series, etc.<br />
<br />
16. It would be interesting to show the performance of DROCC against popular models used for outlier prediction such as PCA, EVA, etc. Perhaps show their accuracy scores so we can better compare.<br />
<br />
17. It would be greater if an visualization of how much performance DROCC improved compare to traditional binary classifier like SVM, isolation Forest.<br />
<br />
19. The paper is well organized and informative. It would be great if it included more details about the datasets. For example, some detailed information about CIFAR-10 can be found in this paper: [https://arxiv.org/pdf/1207.0580.pdf]</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:J46hou&diff=49179User:J46hou2020-12-05T04:05:50Z<p>Wtjung: /* Critiques/Insights */</p>
<hr />
<div>DROCC: Deep Robust One-Class Classification<br />
== Presented by == <br />
Jinjiang Lian, Yisheng Zhu, Jiawen Hou, Mingzhe Huang<br />
== Introduction ==<br />
In this paper, the “one-class” classification, whose goal is to obtain accurate discriminators for a special class, has been studied. Popular uses of this technique include anomaly detection, which is widely used to detect unusual patterns in data. Anomaly detection is a well-studied area of research that aims to learn a model that accurately describes "normality". It has many applications, such as risk assessment for security purposes in many fields, health, and medical risk. However, the conventional approach of modeling with typical data using a simple function falls short when it comes to complex domains such as vision or speech. Another case where this would be useful is when recognizing a “wake-word” while waking up AI systems such as Alexa. <br />
<br />
Deep learning based on anomaly detection methods attempts to learn features automatically but has some limitations. One approach is based on extending the classical data modeling techniques over the learned representations, but in this case, all the points may be mapped to a single point, making the layer look "perfect". The second approach is based on learning the salient geometric structure of data and training the discriminator to predict the applied transformation. The result could be considered anomalous if the discriminator fails to predict the transformation accurately. Appropriate structures or transformations are necessary for these works in general, but are hard to find in practice, especially for domains like time-series or speech, for image data from several orientations, or when generative models are used for deep anomaly detection.<br />
<br />
Thus, in this paper, a new approach called Deep Robust One-Class Classification (DROCC) was presented to solve the above concerns. DROCC is based on the assumption that the points from the class of interest lie on a well-sampled, locally linear low-dimensional manifold. More specifically, we are presenting DROCC-LF which is an outlier-exposure style extension of DROCC. This extension combines the DROCC's anomaly detection loss with standard classification loss over the negative data and exploits the negative examples to learn a Mahalanobis distance.<br />
<br />
== Previous Work ==<br />
Traditional approaches for one-class problems include one-class SVM (Scholkopf et al., 1999) and Isolation Forest (Liu et al., 2008)[9]. One drawback of these approaches is that they involve careful feature engineering when applied to structured domains like images. The current state-of-the-art methodologies to tackle these kinds of problems are: <br />
<br />
1. Approach based on prediction transformations (Golan & El-Yaniv, 2018; Hendricks et al.,2019a) [1]. This work is based on learning the salient geometric structure of typical data by applying specific transformations to the input data and training the discriminator to predict the applied transformation. This approach has some shortcomings in the sense that it depends heavily on an appropriate domain-specific set of transformations that are in general hard to obtain. <br />
<br />
2. Approach of minimizing a classical one-class loss on the learned final layer representations such as DeepSVDD. (Ruff et al.,2018)[2]. Such work has proposed some heuristics to mitigate issues like setting the bias to zero but it is often insufficient in practice. This method suffers from the fundamental drawback of representation collapse, where the learned transformation might map all the points to a single point (like the origin), leading to a degenerate solution and poor discrimination between normal points and the anomalous points.<br />
<br />
3. Approach based on balancing unbalanced training datasets using methods such as SMOTE to synthetically create outlier data to train models on.<br />
<br />
== Motivation ==<br />
Anomaly detection is a well-studied problem with a large body of research (Aggarwal, 2016; Chandola et al., 2009) [3]. The goal is to identify the outliers: points which are not following a typical distribution. The following image provides a visual representation of an outlier/anomaly. <br />
[[File:abnormal.jpeg | thumb | center | 1000px | Abnormal Data (Data Driven Investor, 2020)]]<br />
Classical approaches for anomaly detection are based on modeling the typical data using simple functions over the low-dimensional subspace or a tree-structured partition of the input space to detect anomalies (Schölkopf et al., 1999; Liu et al., 2008; Lakhina et al., 2004) [4], such as constructing a minimum-enclosing ball around the typical data points (Tax & Duin, 2004) [5]. They broadly fall into three categories: AD via generative modeling, Deep Once Class SVM, Transformations based methods, and Side-information based AD. While these techniques are well-suited when the input is featured appropriately, they struggle on complex domains like vision and speech, where hand-designing features are difficult.<br />
<br />
'''AD via Generative Modeling:''' involves deep autoencoders and GAN based methods and have been deeply studied. But, this method solves a much harder problem than required and reconstructs the entire input during the decoding step.<br />
<br />
'''Deep Once Class SVM:''' Deep SVDD attempts to learn a neural network which maps data into a hypersphere. Mappings which fall within the hypersphere are considered "normal". It was the first method to introduce deep one-class classification for the purpose of anomaly detection, but is impeded by representation collapse. <br />
<br />
'''Transformations based methods:''' Are more recent methods that are based on self-supervised training. The training process of these methods applies transformations to the regular points and training the classifier to identify the transformations used. The model relies on the assumption that a point is normal iff the transformations applied to the point can be identified. Some proposed transformations are as simple as rotations and flips, or can be handcrafted and much more complicated. The various transformations that have been proposed are heavily domain dependent and are hard to design.<br />
<br />
'''Side-information based AD:''' incorporate labelled anomalous data or out-of-distribution samples. DROCC makes no assumptions regarding access to side-information.<br />
<br />
Another related problem is the one-class classification under limited negatives (OCLN). In this case, only a few negative samples are available. The goal is to find a classifier that would not misfire close negatives so that the false positive rate will be low. <br />
<br />
DROCC is robust to representation collapse by involving a discriminative component that is general and empirically accurate on most standard domains like tabular, time-series and vision without requiring any additional side information. DROCC is motivated by the key observation that generally, the typical data lies on a low-dimensional manifold, which is well-sampled in the training data and thus tends to be more accurate in practical problems. This is believed to be true even in complex domains such as vision, speech, and natural language (Pless & Souvenir, 2009). [6]<br />
<br />
== Model Explanation ==<br />
[[File:drocc_f1.jpg | center]]<br />
<div align="center">'''Figure 1'''</div><br />
<br />
(a): A normal data manifold with red dots representing generated anomalous points in Ni(r). <br />
<br />
(b): Decision boundary learned by DROCC when applied to the data from (a). Blue represents points classified as normal and red points are classified as abnormal. We observe from here that DROCC is able to capture the manifold accurately; whereas the classical methods, OC-SVM and DeepSVDD perform poorly as they both try to learn a minimum enclosing ball for the whole set of positive data points. <br />
<br />
(c), (d): First two dimensions of the decision boundary of DROCC and DROCC–LF, when applied to noisy data (Section 5.2). DROCC–LF is nearly optimal while DROCC’s decision boundary is inaccurate. Yellow color sine wave depicts the train data.<br />
<br />
== DROCC ==<br />
The model is based on the assumption that the true data lies on a manifold. As manifolds resemble Euclidean space locally, our discriminative component is based on classifying a point as anomalous if it is outside the union of small L2 norm balls around the training typical points (See Figure 1a, 1b for an illustration). Importantly, the above definition allows us to synthetically generate anomalous points, and we adaptively generate the most effective anomalous points while training via a gradient ascent phase reminiscent of adversarial training. In other words, DROCC has a gradient ascent phase to adaptively add anomalous points to our training set and a gradient descent phase to minimize the classification loss by learning a representation and a classifier on top of the representations to separate typical points from the generated anomalous points. In this way, DROCC automatically learns an appropriate representation (like DeepSVDD) but is robust to a representation collapse as mapping all points to the same value would lead to poor discrimination between normal points and the generated anomalous points.<br />
<br />
The algorithm that was used to train the model is laid out below in pseudocode.<br />
<center><br />
[[File:DROCCtrain.png]]<br />
</center><br />
<br />
For a DNN <math>f_\theta: \mathbb{R}^d \to \mathbb{R}</math> that is parameterized by a set of parameters <math>\theta</math>, DROCC estimates <math>\theta^{dr} = \min_\theta\ell^{dr}(\theta)</math> where <br />
$$\ell^{dr}(\theta) = \lambda\|\theta\|^2 + \sum_{i=1}^n[\ell(f_\theta(x_i),1)+\mu\max_{\tilde{x}_i \in N_i(r)}\ell(f_\theta(\tilde{x}_i),-1)]$$<br />
Here, <math>N_i(r) = \{\|\tilde{x}_i-x_i\|_2\leq\gamma\cdot r; r \leq \|\tilde{x}_i - x_j\|, \forall j=1,2,...n\}</math> contains all the points that are at least distance <math>r</math> from the training points. The <math>\gamma \geq 1</math> is a regularization term, and <math>\ell:\mathbb{R} \times \mathbb{R} \to \mathbb{R}</math> is a loss function. The <math>x_i</math> are normal points that should be classified as positive and the <math>\tilde{x}_i</math> are anomalous points that should be classified as negative. This formulation is a saddle point problem.<br />
<br />
== DROCC-LF ==<br />
To especially tackle problems such as anomaly detection and outlier exposure (Hendrycks et al., 2019a) [7], DROCC–LF, an outlier-exposure style extension of DROCC was proposed. Intuitively, DROCC–LF combines DROCC’s anomaly detection loss (that is over only the positive data points) with standard classification loss over the negative data. In addition, DROCC–LF exploits the negative examples to learn a Mahalanobis distance to compare points over the manifold instead of using the standard Euclidean distance, which can be inaccurate for high-dimensional data with relatively fewer samples. (See Figure 1c, 1d for illustration)<br />
<br />
== Popular Dataset Benchmark Result ==<br />
<br />
[[File:drocc_auc.jpg | center]]<br />
<div align="center">'''Figure 2: AUC result'''</div><br />
<br />
The CIFAR-10 dataset consists of 60000 32x32 color images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. The dataset is divided into five training batches and one test batch, each with 10000 images. The test batch contains exactly 1000 randomly selected images from each class. The training batches contain the remaining images in random order, but some training batches may contain more images from one class than another. Between them, the training batches contain exactly 5000 images from each class. The average AUC (with standard deviation) for one-vs-all anomaly detection on CIFAR-10 is shown in table 1. DROCC outperforms baselines on most classes, with gains as high as 20%, and notably, nearest neighbors (NN) beats all the baselines on 2 classes.<br />
<br />
[[File:drocc_f1score.jpg | center]]<br />
<div align="center">'''Figure 3: F1-Score'''</div><br />
<br />
Figure 3 shows F1-Score (with standard deviation) for one-vs-all anomaly detection on Thyroid, Arrhythmia, and Abalone datasets from the UCI Machine Learning Repository. DROCC outperforms the baselines on all three datasets by a minimum of 0.07 which is about an 11.5% performance increase.<br />
Results on One-class Classification with Limited Negatives (OCLN): <br />
[[File:ocln.jpg | center]]<br />
<div align="center">'''Figure 4: Sample positives, negatives and close negatives for MNIST digit 0 vs 1 experiment (OCLN).'''</div><br />
MNIST 0 vs. 1 Classification: <br />
We consider an experimental setup on the MNIST dataset, where the training data consists of Digit 0, the normal class, and Digit 1 as the anomaly. During the evaluation, in addition to samples from training distribution, we also have half zeros, which act as challenging OOD points (close negatives). These half zeros are generated by randomly masking 50% of the pixels (Figure 2). BCE performs poorly, with a recall of 54% only at a fixed FPR of 3%. DROCC–OE gives a recall value of 98:16% outperforming DeepSAD by a margin of 7%, which gives a recall value of 90:91%. DROCC–LF provides further improvement with a recall of 99:4% at 3% FPR. <br />
<br />
[[File:ocln_2.jpg | center]]<br />
<div align="center">'''Figure 5: OCLN on Audio Commands.'''</div><br />
Wake word Detection: <br />
Finally, we evaluate DROCC–LF on the practical problem of wake word detection with low FPR against arbitrary OOD negatives. To this end, we identify a keyword, say “Marvin” from the audio commands dataset (Warden, 2018) [8] as the positive class, and the remaining 34 keywords are labeled as the negative class. For training, we sample points uniformly at random from the above-mentioned dataset. However, for evaluation, we sample positives from the train distribution, but negatives contain a few challenging OOD points as well. Sampling challenging negatives itself is a hard task and is the key motivating reason for studying the problem. So, we manually list close-by keywords to Marvin such as Mar, Vin, Marvelous, etc. We then generate audio snippets for these keywords via a speech synthesis tool 2 with a variety of accents.<br />
Figure 5 shows that for 3% and 5% FPR settings, DROCC–LF is significantly more accurate than the baselines. For example, with FPR=3%, DROCC–LF is 10% more accurate than the baselines. We repeated the same experiment with the keyword: Seven, and observed a similar trend. In summary, DROCC–LF is able to generalize well against negatives that are “close” to the true positives even when such negatives were not supplied with the training data.<br />
<br />
== Conclusion and Future Work ==<br />
We introduced DROCC method for deep anomaly detection. It models normal data points using a low-dimensional sub-manifold inside the feature space, and the anomalous points are characterized via their Euclidean distance from the sub-manifold. Based on this intuition, DROCC’s optimization is formulated as a saddle point problem which is solved via a standard gradient descent-ascent algorithm. We then extended DROCC to OCLN problem where the goal is to generalize well against arbitrary negatives, assuming the positive class is well sampled and a small number of negative points are also available. Both the methods perform significantly better than strong baselines, in their respective problem settings. <br />
<br />
For computational efficiency, we simplified the projection set of both methods which can perhaps slow down the convergence of the two methods. Designing optimization algorithms that can work with the stricter set is an exciting research direction. Further, we would also like to rigorously analyze DROCC, assuming enough samples from a low-curvature manifold. Finally, as OCLN is an exciting problem that routinely comes up in a variety of real-world applications, we would like to apply DROCC–LF to a few high impact scenarios. Possible applications of this work are financial fraud detection, medical anomalies, or key words in audio processing.<br />
<br />
The results of this study showed that DROCC is comparatively better for anomaly detection across many different areas, such as tabular data, images, audio, and time series, when compared to existing state-of-the-art techniques.<br />
<br />
== References ==<br />
[1]: Golan, I. and El-Yaniv, R. Deep anomaly detection using geometric transformations. In Advances in Neural Information Processing Systems (NeurIPS), 2018.<br />
<br />
[2]: Ruff, L., Vandermeulen, R., Goernitz, N., Deecke, L., Siddiqui, S. A., Binder, A., M¨uller, E., and Kloft, M. Deep one-class classification. In International Conference on Machine Learning (ICML), 2018.<br />
<br />
[3]: Aggarwal, C. C. Outlier Analysis. Springer Publishing Company, Incorporated, 2nd edition, 2016. ISBN 3319475770.<br />
<br />
[4]: Sch¨olkopf, B., Williamson, R., Smola, A., Shawe-Taylor, J., and Platt, J. Support vector method for novelty detection. In Proceedings of the 12th International Conference on Neural Information Processing Systems, 1999.<br />
<br />
[5]: Tax, D. M. and Duin, R. P. Support vector data description. Machine Learning, 54(1), 2004.<br />
<br />
[6]: Pless, R. and Souvenir, R. A survey of manifold learning for images. IPSJ Transactions on Computer Vision and Applications, 1, 2009.<br />
<br />
[7]: Hendrycks, D., Mazeika, M., and Dietterich, T. Deep anomaly detection with outlier exposure. In International Conference on Learning Representations (ICLR), 2019a.<br />
<br />
[8]: Warden, P. Speech commands: A dataset for limited vocabulary speech recognition, 2018. URL https: //arxiv.org/abs/1804.03209.<br />
<br />
[9]: Liu, F. T., Ting, K. M., and Zhou, Z.-H. Isolation forest. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining, 2008.<br />
<br />
== Critiques/Insights ==<br />
<br />
1. It would be interesting to see this implemented in self-driving cars, for instance, to detect unusual road conditions.<br />
<br />
2. Figure 1 shows a good representation on how this model works. However, how can we know that this model is not prone to overfitting? There are many situations where there are valid points that lie outside of the line, especially new data that the model has never see before. An explanation on how this is avoided would be good.<br />
<br />
3.In the introduction part, it should first explain what is "one class", and then make a detailed application. Moreover, special definition words are used in many places in the text. No detailed explanation was given. In the end, the future application fields of DROCC and the research direction of the group can be explained.<br />
<br />
4. It will also be interesting to see if one change from using <math>\ell_{2}</math> Euclidean distance to other distances. When the low-dimensional manifold is highly non-linear, using the local linear distance to characterize anomalous points might fail.<br />
<br />
5. This is a nice summary and the authors introduce clearly on the performance of DROCC. It is nice to use Alexa as an example to catch readers' attention. I think it will be nice to include the algorithm of the DROCC or the architecture of DROCC in this summary to help us know the whole view of this method. Maybe it will be interesting to apply DROCC in biomedical studies? since one-class classification is often used in biomedical studies.<br />
<br />
6. For the second sentence in the motivation section, it's better to change "The goal is to identify the outliers: points which are not following a typical distribution" to "The goal is to identify the outliers: points that are not following a typical distribution". In addition, it should be noted that there is an important assumption which assumes the points from the class of interest lie on a well-sampled, locally linear low dimensional manifold when someone wants to use DROCC.<br />
<br />
7. The training method resembles adversarial learning with gradient ascent, however, there is no evaluation of this method on adversarial examples. This is quite unusual considering the paper proposed a method for robust one-class classification, and can be a security threat in real life in critical applications.<br />
<br />
8. The underlying idea behind OCLN is very similar to how neural networks are implemented in recommender systems and trained over positive/negative triplet models. In that case as well, due to the nature of implicit and explicit feedback, positive data tends to dominate the system. It would be interesting to see if insights from that area could be used to further boost the model presented in this paper.<br />
<br />
9. The paper shows the performance of DROCC being evaluated for time series data. It is interesting to see high AUC scores for DROCC against baselines like nearest neighbours and REBMs.Because detecting abnormal data in time series datasets is not common to practice.<br />
<br />
10. Figure1 presented results on a simple 2-D sine wave dataset to visualize the kind of classifiers learnt by DROCC. And the 1a is the positive data lies on a 1-D manifold. We can see from 1b that DROCC is able to capture the manifold accurately.<br />
<br />
11. In the MNIST 0 vs. 1 Classification dataset, why is 1 the only digit that is considered an anomoly? Couldn't all of the non-0 digits be left in the dataset to serve as "anomolies"?<br />
<br />
12. For future work the authors suggest considering DROCC for a low curvature manifold but do not motivate the benefits of such a direction.<br />
<br />
13. One of the problems is that in this model we might need to map all the points to one point to make the layer looks "perfect". However, this might not be a good choice since each point is distinct and if we map them together to one point, then this point cannot tell everything. If authors can specify more details on this it would be better.<br />
<br />
14. This project introduced DROCC for “one-class” classification. It will be interesting if such kind of classification can be compared with any other classification such as binary classification, etc. If “one-class” classification would be more speedy than the others.<br />
<br />
15. The dimensions and feature values must be so different across datasets in different domains. I would love to see how this algorithm is performing so well applied on different domains as it is mentioned that it could be used on datasets including images, audio, time-series, etc.<br />
<br />
16. It would be interesting to show the performance of DROCC against popular models used for outlier prediction such as PCA, EVA, etc. Perhaps show their accuracy scores so we can better compare.<br />
<br />
17. It would be greater if an visualization of how much performance DROCC improved compare to traditional binary classifier like SVM, isolation Forest.<br />
<br />
19. The paper is well organized and informative. It would be great if it included more details about the datasets. For example, some detailed information about CIFAR-10 can be found in this paper: [https://arxiv.org/pdf/1207.0580.pdf]</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:J46hou&diff=49178User:J46hou2020-12-05T04:04:36Z<p>Wtjung: /* Critiques/Insights */</p>
<hr />
<div>DROCC: Deep Robust One-Class Classification<br />
== Presented by == <br />
Jinjiang Lian, Yisheng Zhu, Jiawen Hou, Mingzhe Huang<br />
== Introduction ==<br />
In this paper, the “one-class” classification, whose goal is to obtain accurate discriminators for a special class, has been studied. Popular uses of this technique include anomaly detection, which is widely used to detect unusual patterns in data. Anomaly detection is a well-studied area of research that aims to learn a model that accurately describes "normality". It has many applications, such as risk assessment for security purposes in many fields, health, and medical risk. However, the conventional approach of modeling with typical data using a simple function falls short when it comes to complex domains such as vision or speech. Another case where this would be useful is when recognizing a “wake-word” while waking up AI systems such as Alexa. <br />
<br />
Deep learning based on anomaly detection methods attempts to learn features automatically but has some limitations. One approach is based on extending the classical data modeling techniques over the learned representations, but in this case, all the points may be mapped to a single point, making the layer look "perfect". The second approach is based on learning the salient geometric structure of data and training the discriminator to predict the applied transformation. The result could be considered anomalous if the discriminator fails to predict the transformation accurately. Appropriate structures or transformations are necessary for these works in general, but are hard to find in practice, especially for domains like time-series or speech, for image data from several orientations, or when generative models are used for deep anomaly detection.<br />
<br />
Thus, in this paper, a new approach called Deep Robust One-Class Classification (DROCC) was presented to solve the above concerns. DROCC is based on the assumption that the points from the class of interest lie on a well-sampled, locally linear low-dimensional manifold. More specifically, we are presenting DROCC-LF which is an outlier-exposure style extension of DROCC. This extension combines the DROCC's anomaly detection loss with standard classification loss over the negative data and exploits the negative examples to learn a Mahalanobis distance.<br />
<br />
== Previous Work ==<br />
Traditional approaches for one-class problems include one-class SVM (Scholkopf et al., 1999) and Isolation Forest (Liu et al., 2008)[9]. One drawback of these approaches is that they involve careful feature engineering when applied to structured domains like images. The current state-of-the-art methodologies to tackle these kinds of problems are: <br />
<br />
1. Approach based on prediction transformations (Golan & El-Yaniv, 2018; Hendricks et al.,2019a) [1]. This work is based on learning the salient geometric structure of typical data by applying specific transformations to the input data and training the discriminator to predict the applied transformation. This approach has some shortcomings in the sense that it depends heavily on an appropriate domain-specific set of transformations that are in general hard to obtain. <br />
<br />
2. Approach of minimizing a classical one-class loss on the learned final layer representations such as DeepSVDD. (Ruff et al.,2018)[2]. Such work has proposed some heuristics to mitigate issues like setting the bias to zero but it is often insufficient in practice. This method suffers from the fundamental drawback of representation collapse, where the learned transformation might map all the points to a single point (like the origin), leading to a degenerate solution and poor discrimination between normal points and the anomalous points.<br />
<br />
3. Approach based on balancing unbalanced training datasets using methods such as SMOTE to synthetically create outlier data to train models on.<br />
<br />
== Motivation ==<br />
Anomaly detection is a well-studied problem with a large body of research (Aggarwal, 2016; Chandola et al., 2009) [3]. The goal is to identify the outliers: points which are not following a typical distribution. The following image provides a visual representation of an outlier/anomaly. <br />
[[File:abnormal.jpeg | thumb | center | 1000px | Abnormal Data (Data Driven Investor, 2020)]]<br />
Classical approaches for anomaly detection are based on modeling the typical data using simple functions over the low-dimensional subspace or a tree-structured partition of the input space to detect anomalies (Schölkopf et al., 1999; Liu et al., 2008; Lakhina et al., 2004) [4], such as constructing a minimum-enclosing ball around the typical data points (Tax & Duin, 2004) [5]. They broadly fall into three categories: AD via generative modeling, Deep Once Class SVM, Transformations based methods, and Side-information based AD. While these techniques are well-suited when the input is featured appropriately, they struggle on complex domains like vision and speech, where hand-designing features are difficult.<br />
<br />
'''AD via Generative Modeling:''' involves deep autoencoders and GAN based methods and have been deeply studied. But, this method solves a much harder problem than required and reconstructs the entire input during the decoding step.<br />
<br />
'''Deep Once Class SVM:''' Deep SVDD attempts to learn a neural network which maps data into a hypersphere. Mappings which fall within the hypersphere are considered "normal". It was the first method to introduce deep one-class classification for the purpose of anomaly detection, but is impeded by representation collapse. <br />
<br />
'''Transformations based methods:''' Are more recent methods that are based on self-supervised training. The training process of these methods applies transformations to the regular points and training the classifier to identify the transformations used. The model relies on the assumption that a point is normal iff the transformations applied to the point can be identified. Some proposed transformations are as simple as rotations and flips, or can be handcrafted and much more complicated. The various transformations that have been proposed are heavily domain dependent and are hard to design.<br />
<br />
'''Side-information based AD:''' incorporate labelled anomalous data or out-of-distribution samples. DROCC makes no assumptions regarding access to side-information.<br />
<br />
Another related problem is the one-class classification under limited negatives (OCLN). In this case, only a few negative samples are available. The goal is to find a classifier that would not misfire close negatives so that the false positive rate will be low. <br />
<br />
DROCC is robust to representation collapse by involving a discriminative component that is general and empirically accurate on most standard domains like tabular, time-series and vision without requiring any additional side information. DROCC is motivated by the key observation that generally, the typical data lies on a low-dimensional manifold, which is well-sampled in the training data and thus tends to be more accurate in practical problems. This is believed to be true even in complex domains such as vision, speech, and natural language (Pless & Souvenir, 2009). [6]<br />
<br />
== Model Explanation ==<br />
[[File:drocc_f1.jpg | center]]<br />
<div align="center">'''Figure 1'''</div><br />
<br />
(a): A normal data manifold with red dots representing generated anomalous points in Ni(r). <br />
<br />
(b): Decision boundary learned by DROCC when applied to the data from (a). Blue represents points classified as normal and red points are classified as abnormal. We observe from here that DROCC is able to capture the manifold accurately; whereas the classical methods, OC-SVM and DeepSVDD perform poorly as they both try to learn a minimum enclosing ball for the whole set of positive data points. <br />
<br />
(c), (d): First two dimensions of the decision boundary of DROCC and DROCC–LF, when applied to noisy data (Section 5.2). DROCC–LF is nearly optimal while DROCC’s decision boundary is inaccurate. Yellow color sine wave depicts the train data.<br />
<br />
== DROCC ==<br />
The model is based on the assumption that the true data lies on a manifold. As manifolds resemble Euclidean space locally, our discriminative component is based on classifying a point as anomalous if it is outside the union of small L2 norm balls around the training typical points (See Figure 1a, 1b for an illustration). Importantly, the above definition allows us to synthetically generate anomalous points, and we adaptively generate the most effective anomalous points while training via a gradient ascent phase reminiscent of adversarial training. In other words, DROCC has a gradient ascent phase to adaptively add anomalous points to our training set and a gradient descent phase to minimize the classification loss by learning a representation and a classifier on top of the representations to separate typical points from the generated anomalous points. In this way, DROCC automatically learns an appropriate representation (like DeepSVDD) but is robust to a representation collapse as mapping all points to the same value would lead to poor discrimination between normal points and the generated anomalous points.<br />
<br />
The algorithm that was used to train the model is laid out below in pseudocode.<br />
<center><br />
[[File:DROCCtrain.png]]<br />
</center><br />
<br />
For a DNN <math>f_\theta: \mathbb{R}^d \to \mathbb{R}</math> that is parameterized by a set of parameters <math>\theta</math>, DROCC estimates <math>\theta^{dr} = \min_\theta\ell^{dr}(\theta)</math> where <br />
$$\ell^{dr}(\theta) = \lambda\|\theta\|^2 + \sum_{i=1}^n[\ell(f_\theta(x_i),1)+\mu\max_{\tilde{x}_i \in N_i(r)}\ell(f_\theta(\tilde{x}_i),-1)]$$<br />
Here, <math>N_i(r) = \{\|\tilde{x}_i-x_i\|_2\leq\gamma\cdot r; r \leq \|\tilde{x}_i - x_j\|, \forall j=1,2,...n\}</math> contains all the points that are at least distance <math>r</math> from the training points. The <math>\gamma \geq 1</math> is a regularization term, and <math>\ell:\mathbb{R} \times \mathbb{R} \to \mathbb{R}</math> is a loss function. The <math>x_i</math> are normal points that should be classified as positive and the <math>\tilde{x}_i</math> are anomalous points that should be classified as negative. This formulation is a saddle point problem.<br />
<br />
== DROCC-LF ==<br />
To especially tackle problems such as anomaly detection and outlier exposure (Hendrycks et al., 2019a) [7], DROCC–LF, an outlier-exposure style extension of DROCC was proposed. Intuitively, DROCC–LF combines DROCC’s anomaly detection loss (that is over only the positive data points) with standard classification loss over the negative data. In addition, DROCC–LF exploits the negative examples to learn a Mahalanobis distance to compare points over the manifold instead of using the standard Euclidean distance, which can be inaccurate for high-dimensional data with relatively fewer samples. (See Figure 1c, 1d for illustration)<br />
<br />
== Popular Dataset Benchmark Result ==<br />
<br />
[[File:drocc_auc.jpg | center]]<br />
<div align="center">'''Figure 2: AUC result'''</div><br />
<br />
The CIFAR-10 dataset consists of 60000 32x32 color images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. The dataset is divided into five training batches and one test batch, each with 10000 images. The test batch contains exactly 1000 randomly selected images from each class. The training batches contain the remaining images in random order, but some training batches may contain more images from one class than another. Between them, the training batches contain exactly 5000 images from each class. The average AUC (with standard deviation) for one-vs-all anomaly detection on CIFAR-10 is shown in table 1. DROCC outperforms baselines on most classes, with gains as high as 20%, and notably, nearest neighbors (NN) beats all the baselines on 2 classes.<br />
<br />
[[File:drocc_f1score.jpg | center]]<br />
<div align="center">'''Figure 3: F1-Score'''</div><br />
<br />
Figure 3 shows F1-Score (with standard deviation) for one-vs-all anomaly detection on Thyroid, Arrhythmia, and Abalone datasets from the UCI Machine Learning Repository. DROCC outperforms the baselines on all three datasets by a minimum of 0.07 which is about an 11.5% performance increase.<br />
Results on One-class Classification with Limited Negatives (OCLN): <br />
[[File:ocln.jpg | center]]<br />
<div align="center">'''Figure 4: Sample positives, negatives and close negatives for MNIST digit 0 vs 1 experiment (OCLN).'''</div><br />
MNIST 0 vs. 1 Classification: <br />
We consider an experimental setup on the MNIST dataset, where the training data consists of Digit 0, the normal class, and Digit 1 as the anomaly. During the evaluation, in addition to samples from training distribution, we also have half zeros, which act as challenging OOD points (close negatives). These half zeros are generated by randomly masking 50% of the pixels (Figure 2). BCE performs poorly, with a recall of 54% only at a fixed FPR of 3%. DROCC–OE gives a recall value of 98:16% outperforming DeepSAD by a margin of 7%, which gives a recall value of 90:91%. DROCC–LF provides further improvement with a recall of 99:4% at 3% FPR. <br />
<br />
[[File:ocln_2.jpg | center]]<br />
<div align="center">'''Figure 5: OCLN on Audio Commands.'''</div><br />
Wake word Detection: <br />
Finally, we evaluate DROCC–LF on the practical problem of wake word detection with low FPR against arbitrary OOD negatives. To this end, we identify a keyword, say “Marvin” from the audio commands dataset (Warden, 2018) [8] as the positive class, and the remaining 34 keywords are labeled as the negative class. For training, we sample points uniformly at random from the above-mentioned dataset. However, for evaluation, we sample positives from the train distribution, but negatives contain a few challenging OOD points as well. Sampling challenging negatives itself is a hard task and is the key motivating reason for studying the problem. So, we manually list close-by keywords to Marvin such as Mar, Vin, Marvelous, etc. We then generate audio snippets for these keywords via a speech synthesis tool 2 with a variety of accents.<br />
Figure 5 shows that for 3% and 5% FPR settings, DROCC–LF is significantly more accurate than the baselines. For example, with FPR=3%, DROCC–LF is 10% more accurate than the baselines. We repeated the same experiment with the keyword: Seven, and observed a similar trend. In summary, DROCC–LF is able to generalize well against negatives that are “close” to the true positives even when such negatives were not supplied with the training data.<br />
<br />
== Conclusion and Future Work ==<br />
We introduced DROCC method for deep anomaly detection. It models normal data points using a low-dimensional sub-manifold inside the feature space, and the anomalous points are characterized via their Euclidean distance from the sub-manifold. Based on this intuition, DROCC’s optimization is formulated as a saddle point problem which is solved via a standard gradient descent-ascent algorithm. We then extended DROCC to OCLN problem where the goal is to generalize well against arbitrary negatives, assuming the positive class is well sampled and a small number of negative points are also available. Both the methods perform significantly better than strong baselines, in their respective problem settings. <br />
<br />
For computational efficiency, we simplified the projection set of both methods which can perhaps slow down the convergence of the two methods. Designing optimization algorithms that can work with the stricter set is an exciting research direction. Further, we would also like to rigorously analyze DROCC, assuming enough samples from a low-curvature manifold. Finally, as OCLN is an exciting problem that routinely comes up in a variety of real-world applications, we would like to apply DROCC–LF to a few high impact scenarios. Possible applications of this work are financial fraud detection, medical anomalies, or key words in audio processing.<br />
<br />
The results of this study showed that DROCC is comparatively better for anomaly detection across many different areas, such as tabular data, images, audio, and time series, when compared to existing state-of-the-art techniques.<br />
<br />
== References ==<br />
[1]: Golan, I. and El-Yaniv, R. Deep anomaly detection using geometric transformations. In Advances in Neural Information Processing Systems (NeurIPS), 2018.<br />
<br />
[2]: Ruff, L., Vandermeulen, R., Goernitz, N., Deecke, L., Siddiqui, S. A., Binder, A., M¨uller, E., and Kloft, M. Deep one-class classification. In International Conference on Machine Learning (ICML), 2018.<br />
<br />
[3]: Aggarwal, C. C. Outlier Analysis. Springer Publishing Company, Incorporated, 2nd edition, 2016. ISBN 3319475770.<br />
<br />
[4]: Sch¨olkopf, B., Williamson, R., Smola, A., Shawe-Taylor, J., and Platt, J. Support vector method for novelty detection. In Proceedings of the 12th International Conference on Neural Information Processing Systems, 1999.<br />
<br />
[5]: Tax, D. M. and Duin, R. P. Support vector data description. Machine Learning, 54(1), 2004.<br />
<br />
[6]: Pless, R. and Souvenir, R. A survey of manifold learning for images. IPSJ Transactions on Computer Vision and Applications, 1, 2009.<br />
<br />
[7]: Hendrycks, D., Mazeika, M., and Dietterich, T. Deep anomaly detection with outlier exposure. In International Conference on Learning Representations (ICLR), 2019a.<br />
<br />
[8]: Warden, P. Speech commands: A dataset for limited vocabulary speech recognition, 2018. URL https: //arxiv.org/abs/1804.03209.<br />
<br />
[9]: Liu, F. T., Ting, K. M., and Zhou, Z.-H. Isolation forest. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining, 2008.<br />
<br />
== Critiques/Insights ==<br />
<br />
1. It would be interesting to see this implemented in self-driving cars, for instance, to detect unusual road conditions.<br />
<br />
2. Figure 1 shows a good representation on how this model works. However, how can we know that this model is not prone to overfitting? There are many situations where there are valid points that lie outside of the line, especially new data that the model has never see before. An explanation on how this is avoided would be good.<br />
<br />
3.In the introduction part, it should first explain what is "one class", and then make a detailed application. Moreover, special definition words are used in many places in the text. No detailed explanation was given. In the end, the future application fields of DROCC and the research direction of the group can be explained.<br />
<br />
4. It will also be interesting to see if one change from using <math>\ell_{2}</math> Euclidean distance to other distances. When the low-dimensional manifold is highly non-linear, using the local linear distance to characterize anomalous points might fail.<br />
<br />
5. This is a nice summary and the authors introduce clearly on the performance of DROCC. It is nice to use Alexa as an example to catch readers' attention. I think it will be nice to include the algorithm of the DROCC or the architecture of DROCC in this summary to help us know the whole view of this method. Maybe it will be interesting to apply DROCC in biomedical studies? since one-class classification is often used in biomedical studies.<br />
<br />
6. For the second sentence in the motivation section, it's better to change "The goal is to identify the outliers: points which are not following a typical distribution" to "The goal is to identify the outliers: points that are not following a typical distribution". In addition, it should be noted that there is an important assumption which assumes the points from the class of interest lie on a well-sampled, locally linear low dimensional manifold when someone wants to use DROCC.<br />
<br />
7. The training method resembles adversarial learning with gradient ascent, however, there is no evaluation of this method on adversarial examples. This is quite unusual considering the paper proposed a method for robust one-class classification, and can be a security threat in real life in critical applications.<br />
<br />
8. The underlying idea behind OCLN is very similar to how neural networks are implemented in recommender systems and trained over positive/negative triplet models. In that case as well, due to the nature of implicit and explicit feedback, positive data tends to dominate the system. It would be interesting to see if insights from that area could be used to further boost the model presented in this paper.<br />
<br />
9. The paper shows the performance of DROCC being evaluated for time series data. It is interesting to see high AUC scores for DROCC against baselines like nearest neighbours and REBMs.Because detecting abnormal data in time series datasets is not common to practice.<br />
<br />
10. Figure1 presented results on a simple 2-D sine wave dataset to visualize the kind of classifiers learnt by DROCC. And the 1a is the positive data lies on a 1-D manifold. We can see from 1b that DROCC is able to capture the manifold accurately.<br />
<br />
11. In the MNIST 0 vs. 1 Classification dataset, why is 1 the only digit that is considered an anomoly? Couldn't all of the non-0 digits be left in the dataset to serve as "anomolies"?<br />
<br />
12. For future work the authors suggest considering DROCC for a low curvature manifold but do not motivate the benefits of such a direction.<br />
<br />
13. One of the problems is that in this model we might need to map all the points to one point to make the layer looks "perfect". However, this might not be a good choice since each point is distinct and if we map them together to one point, then this point cannot tell everything. If authors can specify more details on this it would be better.<br />
<br />
14. This project introduced DROCC for “one-class” classification. It will be interesting if such kind of classification can be compared with any other classification such as binary classification, etc. If “one-class” classification would be more speedy than the others.<br />
<br />
15. The dimensions and feature values must be so different across datasets in different domains. I would love to see how this algorithm is performing so well applied on different domains as it is mentioned that it could be used on datasets including images, audio, time-series, etc.<br />
<br />
16. It would be interesting to show the performance of DROCC against popular models used for outlier prediction such as PCA, EVA, etc. Perhaps show their accuracy scores so we can better compare.<br />
<br />
17. It would be greater if an visualization of how much performance DROCC improved compare to traditional binary classifier like SVM, isolation Forest.<br />
<br />
19. The paper is well organized and informative. It would be great if it included more details about the datasets. For example, some detailed information about CIFAR-10 can be found in this paper: https://arxiv.org/pdf/1207.0580.pdf</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Summary_for_survey_of_neural_networked-based_cancer_prediction_models_from_microarray_data&diff=48849Summary for survey of neural networked-based cancer prediction models from microarray data2020-12-02T05:56:05Z<p>Wtjung: /* Background */</p>
<hr />
<div>== Presented by == <br />
Rao Fu, Siqi Li, Yuqin Fang, Zeping Zhou<br />
<br />
== Introduction == <br />
Microarray technology can help researchers quickly detect genetic information and is widely used to analyze genetic diseases. Researchers use this technology to compare normal and abnormal cancerous tissues to gain insights into the pathology of cancer. <br />
<br />
However, due to the high dimensionality of the gene expressions, the accuracy, and computation time of the model might be affected. The following two approaches are adopted to cope with this problem: the feature selection method and the feature creating method. The formal one, similar to the well-known principal component analysis method, aims to focus on key features and ignore minor noises. On the other hand, the latter one, similar to scale-invariant feature transformations, aims to combine existing features or map them to a new low-dimensional space.<br />
<br />
Compared to neural network models presented by others, this paper is specifically designed for predicting cancer using gene expression data. Thus, we will review the latest neural network-based cancer prediction models by presenting the methodology of preprocessing, filtering, prediction, and clustering gene expressions.<br />
<br />
== Background == <br />
<br />
'''Neural Network''' <br><br />
Neural networks are often used to solve non-linear complex problems. It is an operational model consisting of a large number of neurons connected to each other by different weights. In this network structure, each neuron is related to an activation function such as sigmoid or rectified linear activation functions. To train the network, the inputs are fed forward and the activation function value is calculated at every neuron. The difference between the output of the neural network and the actual value is what we call an error.<br />
The backpropagation mechanism is one of the most commonly used algorithms in solving neural network problems. By using this algorithm, we optimize the objective function by propagating back the generated error through the network to adjust the weights.<br />
In the next sections, we will use the above algorithm but with different network architectures and a different number of neurons to review the neural network-based cancer prediction models for learning the gene expression features.<br />
<br />
'''Cancer prediction models'''<br><br />
Cancer prediction models often contain more than 1 method to achieve high prediction accuracy with a more accurate prognosis and it also aims to reduce the cost of patients.<br />
<br />
High dimensionality and spatial structure are the two main factors that can affect the accuracy of the cancer prediction models. They add irrelevant noisy features to our selected models. We have 3 ways to determine the accuracy of a model.<br />
<br />
The first way is called the ROC curve. ROC curves, receiver operating characteristic curves, are graphs that show the true positive rate against the false-positive rate [4]. It reflects the sensitivity of the response to the same signal stimulus under different criteria. To test its validity, we need to consider it with the confidence interval. Usually, a model is considered acceptable when its ROC is greater than 0.7. <br />
<br />
A different machine learning problem is predicting the survival time. The performance of a model that predicts survival time can be measured using two metrics. The problem of predicting survival time can be seen as a ranking problem, where survival times of different subjects are ranked against each other. CI (Concordance Index) is a measure of how good a model ranks survival times and explains the concordance probability of the predicted and observed survival. The closer its value to 0.7, the better the model is. We can express the ordering of survival times in an order graph <math display="inline">G = (V, E)</math> where the vertices <math display="inline">V</math> are the individual survival times, and the edges <math display="inline">E_{ij}</math> from individual <math display="inline">i</math> to <math display="inline">j</math> indicate that <math display="inline">T_i < T_j</math>, where <math display="inline">T_i, T_j</math> are the survival times for individuals <math display="inline">i,j</math> respectively. Then we can write <math display="inline">CI = \frac{1}{|E|}\sum_{i,j}I(f(i) < f(j))</math> where <math display="inline">|E|</math> is the number of edges in the order graph (ie. the total number of comparable pairs), and <math display="inline">I(f(i) < f(j)) = 1</math> if the predictor <math display="inline">f</math> correctly ranks <math display="inline">T_i < T_j</math> [3].<br />
<br />
Another metric is the Brier score, which measures the average difference between the observed and the estimated survival rate in a given period of time. It ranges from 0 to 1, and a lower score indicates higher accuracy. It is defined as <math display="inline">\frac{1}{n}\sum_{i=1}^n(f_i - o_i)^2</math> where <math display="inline">f_i</math> is the predicted survival rate, and <math display="inline">o_i</math> is the observed survival rate [2].<br />
<br />
== Neural network-based cancer prediction models ==<br />
An extensive search relevant to neural network-based cancer prediction was performed using Google scholar and other electronic databases namely PubMed and Scopus with keywords such as “Neural Networks AND Cancer Prediction” and “gene expression clustering”, and only articles between 2013 and 2018 with available accessibility were considered. The chosen papers covered cancer classification, discovery, survivability prediction, and statistical analysis models. The following figure 1 shows a graph representing the number of citations including filtering, predictive, and clustering for chosen papers. <br />
<br />
[[File:f1.png]]<br />
<br />
'''Datasets and preprocessing''' <br><br />
Most studies investigating automatic cancer prediction and clustering used datasets such as the TCGA, UCI, NCBI Gene Expression Omnibus and Kentridge biomedical databases. There are a few techniques used in processing dataset including removing the genes that have zero expression across all samples, Normalization, filtering with p-value > <math>10^{-5}</math> to remove some unwanted technical variation and <math>\log_2</math> transformations. Statistical methods, neural networks, were applied to reduce the dimensionality of the gene expressions by selecting a subset of genes. Principle Component Analysis (PCA) can also be used as an initial preprocessing step to extract the dataset's features. The PCA method linearly transforms the dataset features into lower dimensional space without capturing the complex relationships between the features. However, simply removing the genes that were not measured by the other datasets could not overcome the class imbalance problem. In that case, one research used Synthetic Minority Class Over Sampling method to generate synthetic minority class samples, which may lead to a sparse matrix problem. Clustering was also applied in some studies for labeling data by grouping the samples into high-risk, low-risk groups, and so on. <br />
<br />
The following table presents the dataset used by considered reference, the applied normalization technique, the cancer type and the dimensionality of the datasets.<br />
<br />
[[File:Datasets and preprocessing.png]]<br />
<br />
'''Neural network architecture''' <br><br />
Most recent studies reveal that neural network methods are used for filtering, predicting, and clustering in cancer prediction. <br />
<br />
*''filtering'': Filter the gene expressions to eliminate noise or reduce dimensionality. Then use the resulted features with statistical methods or machine learning classification and clustering tools as figure 2 indicates.<br />
<br />
*''predicting'': Extract features and improve the accuracy of prediction (classification).<br />
<br />
*''clustering'': Divide the gene expressions or samples based on similarity.<br />
<br />
[[File:filtering gane.png]]<br />
<br />
All the neurons in the neural network work together as feature detectors to learn the features from the input. In order to categorize a neural network as filtering, predicting, or clustering method, we looked at the overall role that network provided within the framework of cancer prediction. Filtering methods are trained to remove the input’s noise and to extract the most representative features that best describe the unlabeled gene expressions. Predicting methods are trained to extract the features that are significant to prediction, therefore its objective functions measure how accurately the network is able to predict the class of input. Clustering methods are trained to divide unlabeled samples into groups based on their similarities.<br />
<br />
'''Building neural networks-based approaches for gene expression prediction''' <br><br />
According to our survey, the representative codes are generated by filtering methods with dimensionality M smaller or equal to N, where N is the dimensionality of the input. Some other machine learning algorithms such as naïve Bayes or k-means can be used together with the filtering.<br />
Predictive neural networks are supervised, which find the best classification accuracy; meanwhile, clustering methods are unsupervised, which group similar samples or genes together. <br />
The goal of training prediction is to enhance the classification capability, and the goal of training classification is to find the optimal group for a new test set with unknown labels.<br />
<br />
'''Neural network filters for cancer prediction''' <br><br />
In the preprocessing step to classification, clustering, and statistical analysis, the autoencoders are more and more commonly-used, to extract generic genomic features. An autoencoder is composed of the encoder part and the decoder part. The encoder part is to learn the mapping between high-dimensional unlabeled input I(x) and the low-dimensional representations in the middle layer(s), and the decoder part is to learn the mapping from the middle layer’s representation to the high-dimensional output O(x). The reconstruction of the input can take the Root Mean Squared Error (RMSE) or the Logloss function as the objective function. <br />
<br />
$$ RMSE = \sqrt{ \frac{\sum{(I(x)-O(x))^2}}{n} } $$<br />
<br />
$$ Logloss = \sum{(I(x)\log(O(x)) + (1 - I(x))\log(1 - O(x)))} $$<br />
<br />
There are several types of autoencoders, such as stacked denoising autoencoders, contractive autoencoders, sparse autoencoders, regularized autoencoders, and variational autoencoders. The architecture of the networks varies in many parameters, such as depth and loss function. Each example of an autoencoder mentioned above has a different number of hidden layers, different activation functions (e.g. sigmoid function, exponential linear unit function), and different optimization algorithms (e.g. stochastic gradient descent optimization, Adam optimizer).<br />
<br />
Overfitting is a major problem that most autoencoders need to deal with to achieve high efficiency of the extracted features. Regularization, dropout, and sparsity are common solutions.<br />
<br />
The neural network filtering methods were used by different statistical methods and classifiers. The conventional methods include Cox regression model analysis, Support Vector Machine (SVM), K-means clustering, t-SNE and so on. The classifiers could be SVM or AdaBoost or others.<br />
<br />
By using neural network filtering methods, the model can be trained to learn low-dimensional representations, remove noises from the input, and gain better generalization performance by re-training the classifier with the newest output layer.<br />
<br />
'''Neural network prediction methods for cancer''' <br><br />
The prediction based on neural networks can build a network that maps the input features to an output with a number of neurons, which could be one or two for binary classification or more for multi-class classification. It can also build several independent binary neural networks for the multi-class classification, where the technique called “one-hot encoding” is applied.<br />
<br />
The codeword is a binary string <math>C'k</math> of length k whose j’th position is set to 1 for the j’th class, while other positions remain 0. The process of the neural networks is to map the input to the codeword iteratively, whose objective function is minimized in each iteration.<br />
<br />
Such cancer classifiers were applied to identify cancerous/non-cancerous samples, a specific cancer type, or the survivability risk. MLP models were used to predict the survival risk of lung cancer patients with several gene expressions as input. The deep generative model DeepCancer, the RBM-SVM and RBM-logistic regression models, the convolutional feedforward model DeepGene, Extreme Learning Machines (ELM), the one-dimensional convolutional framework model SE1DCNN, and GA-ANN model are all used for solving cancer issues mentioned above. This paper indicates that the performance of neural networks with MLP architecture as a classifier is better than those of SVM, logistic regression, naïve Bayes, classification trees, and KNN.<br />
<br />
'''Neural network clustering methods in cancer prediction''' <br><br />
Neural network clustering belongs to unsupervised learning. The input data are divided into different groups according to their feature similarity.<br />
The single-layered neural network SOM, which is unsupervised and without a backpropagation mechanism, is one of the traditional model-based techniques to be applied to gene expression data. The measurement of its accuracy could be Rand Index (RI), which can be improved to Adjusted Random Index (ARI) and Normalized Mutation Information (NMI).<br />
<br />
$$ RI=\frac{TP+TN}{TP+TN+FP+FN}$$<br />
<br />
In general, gene expression clustering considers either the relevance of samples-to-cluster assignment or that of gene-to-cluster assignment or both. The high dimensionality of gene expression samples poses a problem for traditional clustering algorithms such as k-means clustering, which uses a distance function to separate samples. Such an approach is not viable for high dimensional datasets. To solve the high dimensionality problem, there are two methods: clustering ensembles by running a single clustering algorithm several times, each of which has different initialization or number of parameters; and projective clustering by only considering a subset of the original features.<br />
<br />
SOM was applied on discriminating future tumor behavior using molecular alterations, whose results were not easy to be obtained by classic statistical models. Then this paper introduces two ensemble clustering frameworks: Random Double Clustering-based Cluster Ensembles (RDCCE) and Random Double Clustering-based Fuzzy Cluster Ensembles (RDCFCE). Their accuracies are high, but they have not taken the gene-to-cluster assignment into consideration.<br />
<br />
Also, the paper provides a double SOM based Clustering Ensemble Approach (SOM2CE) and double NG-based Clustering Ensemble Approach (NG2CE), which are robust to noisy genes. Moreover, Projective Clustering Ensemble (PCE) combines the advantages of both projective clustering and ensemble clustering, which is better than SOM and RDCFCE when there are irrelevant genes.<br />
<br />
== Summary ==<br />
<br />
Cancer is a disease with a high mortality rate that kills millions of people every year, and it’s essential to analyze gene expression for discovering gene abnormalities and increasing survivability as a consequence. The previous analysis in the paper reveals that neural networks are essentially used for filtering the gene expressions, predicting their class, or clustering them.<br />
<br />
Neural network filtering methods are used to reduce the dimensionality of the gene expressions and remove their noise. In the article, the authors recommended deep architectures in comparison to shallow architectures for best practice, as they combine many nonlinearities. <br />
<br />
Neural network prediction methods can be used for both binary and multi-class problems. In the binary case, the network architecture has only one or two output neurons that diagnose a given sample as cancerous or non-cancerous. In comparison, the number of the output neurons in multi-class problems is equal to the number of classes. The authors suggested that the deep architecture with convolution layers which was the most recently used model proved efficient capability in predicting cancer subtypes, as it captures the spatial correlations between gene expressions.<br />
Clustering is another analysis tool that is used to divide the gene expressions into groups. The authors indicated that a hybrid approach combining both the assembling, clustering and projective clustering is more accurate than using single-point clustering algorithms, such as SOM, since those methods do not have the capability to distinguish the noisy genes.<br />
<br />
==Discussion==<br />
There are some technical problems that can be considered and improved for building new models. <br><br />
<br />
1. Overfitting: Since gene expression datasets are high dimensional and have a relatively small number of samples, it would be likely to properly fits the training data but not accurate for test samples due to the lack of generalization capability. The ways to avoid overfitting can be: (1). adding weight penalties using regularization; (2). using the average predictions from many models trained on different datasets; (3). dropout. (4) Augmentation of the dataset to produce more "observations".<br><br />
<br />
2. Model configuration and training: In order to reduce both the computational and memory expenses but also with high prediction accuracy, it’s crucial to properly set the network parameters. The possible ways can be: (1). proper initialization; (2). pruning the unimportant connections by removing the zero-valued neurons; (3). using ensemble learning framework by training different models using different parameter settings or using different parts of the dataset for each base model; (4). Using SMOTE for dealing with class imbalance on the high dimensional level. <br><br />
<br />
3. Model evaluation: In Braga-Neto and Dougherty's research, they have investigated several model evaluation methods: cross-validation, substitution and bootstrap methods. The cross-validation was found to be unreliable for small size data since it displayed excessive variance. The bootstrap method proved more accurate predictability.<br><br />
<br />
4. Study producibility: A study needs to be reproducible to enhance research reliability so that others can replicate the results using the same algorithms, data and methodology. Hence, the query used for getting the dataset should be stated.<br />
<br />
==Conclusion==<br />
This paper reviewed the most recent neural network-based cancer prediction models and gene expression analysis tools. The analysis indicates that the neural network methods are able to serve as filters, predictors, and clustering methods, and also showed that the role of the neural network determines its general architecture. The authors showed that Neural Network filtering methods are a way of reducing the dimensionality of the gene expressions, as well as removing their noise for better model fitting. To give suggestions for future neural network-based approaches, the authors highlighted some critical points that have to be considered such as overfitting and class imbalance, and suggest choosing different network parameters or combining two or more of the presented approaches. One of the biggest challenges for cancer prediction modelers is deciding on the network architecture (i.e. the number of hidden layers and neurons), as there are currently no guidelines to follow to obtain high prediction accuracy. The authors discovered that there is no algorithm available to concretely determine an optimal number of hidden layers or nodes and found that many papers simply implemented a trial and error method to reduce loss in the model.<br />
<br />
==Critiques==<br />
<br />
While results indicate that the functionality of the neural network determines its general architecture, the decision on the number of hidden layers, neurons, hypermeters, and learning algorithms is made using trial-and-error techniques. Therefore improvements in this area of the model might need to be explored in order to obtain better results and in order to make more convincing statements.<br />
<br />
An issue that one must be mindful of is the underlying distribution of data. Cancer is an extremely complex genetic disease and the predictions would depend on so many variables, a number of which will not even be present in the dataset as they might have been collected. So there is a need for extensive validation when it comes to applying deep learning methods to cancer-related data.<br />
<br />
In the field of medical sciences and molecular biology, interpretability of results is imperative as often experts seek not just to solve the issue at hand but to understand the causal relationships. Having a high ROC value may not necessarily convince other experts on the validity of the finding because the underlying details of cancer symptoms have been abstracted in a complex neural network as a black box. However, the neural network clustering method suggested in this paper does offer a good compromise because it enables humans to visual low-level features but still gives experts the control on making various predictions using well-studied traditional techniques.<br />
<br />
With high dimensionality features, kernel SVM is another option for cancer prediction. Jiang et. al. developed a Hadamard Kernel for predicting breast cancer using gene expression data, and it utilizes the Kernel trick to avoid high computational efforts (link: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5763304/). Compared against linear, quadratic, RBF and correlation kernels, Hadamard Kernel performs best with the highest averaged area under the ROC curve (AUC) value. It may be interesting to compare the performance and accuracy between the Hadamard Kernel and cancer prediction models with various number of hidden layers and neurons.<br />
<br />
Although the authors presented technical details about data processing, training approaches evaluation metric, and addressed many practical issues that can be considered for cancer prediction, no novel methods or models are proposed. It's more like a proof-of-concept about the feasibility of different models on cancer prediction.<br />
<br />
It would be interesting for the researchers to compare the performance between causal inference and neural network models on this data.<br />
<br />
As the authors indicate neural networks would be a useful tool for cancer prediction models, the article is lacking an example for implementing neural networks to provide persuasive support for their arguments.<br />
<br />
The inheritance of cancer is complex and changeable. The predicted variables are therefore very complicated, so for the model of the learning data set, a more adequate training set is needed to learn. And multi-party verification of the learned model.<br />
<br />
The authors mentioned many different neural network models and compared them. It would be better if more details of a commonly applied model with relatively high accuracy could be given such as how the model is built. An article named Convolutional neural network models for cancer type prediction based on gene expression gives explanations of CNN in detail.<br />
<br />
The authors briefly discussed methods and algorithms being used in the presented paper in their summary. However, a very little amount of technical details were provided to the readers. The summary itself is lacking specific examples for the aforementioned algorithms, and datasets which were used in the original analysis were only introduced in one or two sentences. As a result, the summary and conclusion appear to be unconvincing to the readers.<br />
<br />
PCA can still be used as an initial preprocessing step even if it is used in a neural network whose data dimension is reduced. By merging the PCA component with a random number of original functions, some good techniques can be adopted to enable the network to capture more useful relationships<br />
<br />
The key part of this model is to extract the features from the model. However, cancer may depends on many explanatory variables. Thus how do we know which feature should we extract in the data preprocessing. Since there are correlations between each variables. Authors did not specify this situation.<br />
<br />
From the biology side of view, gene expression is really complicated. Thus reducing dimension may or may not be the best way of predicting cancer, and this should be a controversial topic.<br />
<br />
==Reference==<br />
[1] Daoud, M., & Mayo, M. (2019). A survey of neural network-based cancer prediction models from microarray data. Artificial Intelligence in Medicine, 97, 204–214.<br />
<br />
[2] Brier GW. 1950. Verification of forecasts expressed in terms of probabilities. Monthly Weather Review 78: 1–3<br />
<br />
[3] Harald Steck, Balaji Krishnapuram, Cary Dehing-oberije, Philippe Lambin, Vikas C. Raykar. On ranking in survival analysis: Bounds on the concordance index. In Advances in Neural Information Processing Systems (2008), pp. 1209-1216<br />
<br />
[4] Google Developers. (2020 February 10). ''Classification: ROC Curve and AUC.'' Machine Learning Crash Course. https://developers.google.com/machine-learning/crash-course/classification/roc-and-auc<br />
<br />
[5] Mostavi, M., Chiu, YC., Huang, Y. et al. Convolutional neural network models for cancer type prediction based on gene expression. ''BMC Med Genomics'' '''13''', 44 (2020). https://doi.org/10.1186/s12920-020-0677-2</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Superhuman_AI_for_Multiplayer_Poker&diff=48848Superhuman AI for Multiplayer Poker2020-12-02T05:51:27Z<p>Wtjung: /* Experimental Results */</p>
<hr />
<div>== Presented by == <br />
Hansa Halim, Sanjana Rajendra Naik, Samka Marfua, Shawrupa Proshasty<br />
<br />
== Introduction ==<br />
<br />
A super-intelligence is a hypothetical agent that possesses intelligence surpassing that of the brightest and most gifted human minds. In the past two decades, most of the superhuman AI that has been built is only able to beat human players in two-player zero-sum games. Zero-sum games are games where a gain for one player results in an equivalent loss for the other player such that the sum of the gains and losses always equals zero. They almost dominated most of the board games in these twenty years. The most popular AI in the board games are the chess AI deep blue and the go chess AI Alpha-go. The most common strategy that the AI uses to beat those games is to find the most optimal Nash equilibrium. A Nash equilibrium is a pair of strategies such that either single-player switching to any ''other'' choice of strategy (while the other player's strategy remains unchanged) will result in a lower payout for the switching player. Intuitively this is similar to a locally optimal strategy for the players but is (i) not guaranteed to exist and (ii) may not be the truly optimal strategy. An example of this is the Prisoner's dilemma, where two individuals each have the option to testify against the other or to remain silent. Although the optimal choice is to remain silent, the individuals have an incentive to act in their own self-interest which results in a less than optimal outcome.<br />
<br />
More specifically, in the game of poker, we only have AI models that can beat human players in two-player settings. Poker is a great challenge in AI and game theory because it captures the challenges in hidden information so elegantly. This means that developing a superhuman AI in multiplayer poker is the remaining great milestone in this field, because there is no polynomial-time algorithm that can find a Nash equilibrium in two-player non-zero-sum games. The discovery of such an algorithm would have surprising and profound implications in computational complexity theory.<br />
<br />
In this paper, the AI which we call Pluribus is capable of defeating human professional poker players in Texas hold'em poker which is a six-player poker game and is the most commonly played format in the world. The algorithm that is used is not guaranteed to converge to a Nash algorithm outside of two-player zero-sum games. However, it uses a strong strategy that is capable of consistently defeating elite human professionals. This shows that despite not having strong theoretical guarantees on performance, they are capable of applying a wider class of superhuman strategies.<br />
<br />
Knowing the basics of Texas hold'em poker may help to understand how the AI can play. Texas hold'em is played using a standard deck of 52 cards. For each hand, players are each dealt two cards, which only they can see. Once players receive their two cards, and prior to turning any other cards, players (in clockwise order) have the option to call, raise, fold, or check. Call means to match the highest bet so far, raise means to increase the bet, fold means the player wishes to leave the current hand, and check means to continue without betting any further. A player can only check if no one has bet yet this round or if they have already matched the highest bet. Once this round of betting is over three cards, called the Flop, are turned over, and the betting happens in a clockwise manner again. This is followed by revealing another card, the Turn, and then proceeding with another round of betting. Now, the final card, the River, is turned. There is one more round of betting before the players reveal their cards and determine the winner. The player with the best five-card hand (out of the 7 cards) wins the hand and the money in the pot. The following is a list of the Texas hold'em hands from best to worst:<br />
* Royal Flush (A,K,Q,J,10 all of the same suit)<br />
* Straight Flush(5 in a row from the same suit)<br />
* 4 of a kind<br />
* Full house (3 of a kind + a pair)<br />
* Flush (5 cards of the same suit)<br />
* Straight (5 cards in a row)<br />
* 3 of a kind<br />
* 2 different pairs<br />
* A pair<br />
* Highest card<br />
<br />
== Nash Equilibrium in Multiplayer Games ==<br />
<br />
Many AI has reached superhuman performance in games like checkers, chess, two-player limit poker, Go, and two-player no-limit poker. Nash equilibrium has been proven to exist in all finite games and numerous infinite games but the challenge is to find the equilibrium. It is the best possible strategy and is unbeatable in two-player zero-sum games since it guarantees to not lose in expectation regardless of what the opponent is doing.<br />
<br />
To have a deeper understanding of Nash Equilibria, we must first define some basic game theory concepts. The first one being a strategic game, in-game theory a strategic game consists of a set of players, for each player a set of actions and for each player preferences (or payoffs) over the set of action profiles (set of combination of actions). With these three elements, we can model a wide variety of situations. Now a Nash Equilibrium is an action profile, with the property that no player can do better by changing their action, given that all other players' actions remain the same. A common illustration of Nash equilibria is the Prisoner's Dilemma. We also have mixed strategies and mixed strategy Nash equilibria. A mixed strategy is when instead of a player choosing an action they apply a probability distribution to their set of actions and pick randomly. Note that with mixed strategies we must look at the expected payoff of the player given the other players' strategies. Therefore a mixed strategy Nash Equilibria involves at least one player playing with a mixed strategy where no player can increase their expected payoff by changing their action, given that all other players' actions remain the same. Then we can define a pure Nash Equilibria to where no one is playing a mixed strategy. We also must be aware that a single game can have multiple pure Nash equilibria and mixed Nash equilibria. Also, Nash Equilibria are purely theoretical and depend on players acting optimally and being rational, this is not always the case with humans and we can act very irrationally. Therefore empirically we will see that games can have very unexpected outcomes and you may be able to get a better payoff if you move away from a strictly theoretical strategy and take advantage of your opponent's irrational behavior. <br />
<br />
The insufficiency with current AI systems is that they only try to achieve Nash equilibriums instead of trying to actively detect and exploit weaknesses in opponents. At the Nash equilibrium, there is no incentive for any player to change their initial strategy, so it is a stable state of the system. For example, let's consider the game of Rock-Paper-Scissors, the Nash equilibrium is to randomly pick any option with equal probability. However, we can see that this means the best strategy that the opponent can have will result in a tie. Therefore, in this example, our player cannot win in expectation. Now let's try to combine the Nash equilibrium strategy and opponent exploitation. We can initially use the Nash equilibrium strategy and then change our strategy over time to exploit the observed weaknesses of our opponent. For example, we switch to always play Rock against our opponent who always plays Scissors. However, shifting away from the Nash equilibrium strategy opens up the possibility for our opponent to use our strategy against ourselves. For example, they notice we always play Rock and thus they will now always play Paper.<br />
<br />
Trying to approximate a Nash equilibrium is hard in theory, and in games with more than two players, it can only find a handful of possible strategies per player. Currently, existing techniques to find ways to exploit an opponent require way too many samples and are not competitive enough outside of small games. Finding a Nash equilibrium in three or more players is a great challenge. Even we can efficiently compute a Nash equilibrium in games with more than two players, it is still highly questionable if playing the Nash equilibrium strategy is a good choice. Additionally, if each player tries to find their own version of a Nash equilibrium, we could have infinitely many strategies and each player’s version of the equilibrium might not even be a Nash equilibrium.<br />
<br />
Consider the Lemonade Stand example from Figure 1 Below. We have 4 players and the goal for each player is to find a spot in the ring that is furthest away from every other player. This way, each lemonade stand can cover as much selling region as possible and generate maximum revenue. In the left circle, we have three different Nash equilibria distinguished by different colors which would benefit everyone. The right circle is an illustration of what would happen if each player decides to calculate their own Nash equilibrium.<br />
<br />
[[File:Lemonade_Example.png| 600px |center ]]<br />
<br />
<div align="center">Figure 1: Lemonade Stand Example</div><br />
<br />
From the right circle in Figure 1, we can see that when each player tries to calculate their own Nash equilibria independently, the joint strategy can hardly lead to equally-spaced players along the ring, which is not a Nash equilibrium. This shows that attempting to find a Nash equilibrium is not the best strategy outside of two-player zero-sum games, and our goal should not be focused on finding a specific game-theoretic solution. Instead, we need to focus on observations and empirical results that consistently defeat human opponents.<br />
<br />
== Theoretical Analysis ==<br />
Pluribus uses forms of abstraction to make computations scalable. To simplify the complexity due to too many decision points, some actions are eliminated from consideration and similar decision points are grouped together and treated as identical. This process is called abstraction. Pluribus uses two kinds of abstraction: '''Action abstraction and information abstraction'''. Action abstraction reduces the number of different actions the AI needs to consider. For instance, it does not consider all bet sizes (the exact number of bets it considers varies between 1 and 14 depending on the situation). Information abstraction groups together decision points that reveal similar information. For instance, the player’s cards and revealed board cards. This is only used to reason about situations on future betting rounds, never the current betting round.<br />
<br />
Pluribus uses a built-in strategy - '''“Blueprint strategy”''', which it gradually improves by searching in real-time in situations it finds itself in during the course of the game. In the first betting round, pluribus uses the initial blueprint strategy when the number of decision points is small. The blueprint strategy is computed using Monte Carlo Counterfactual Regret Minimization (MCCFR) algorithm. CFR is commonly used in imperfect information games AI which is trained by repeatedly playing against copies of itself, without any data of human or prior AI play used as input. For ease of computation of CFR in this context, poker is represented as a game tree. A game tree is a tree structure where each node represents either a player’s decision, a chance event, or a terminal outcome and edges represent actions taken. <br />
<br />
[[File:Screen_Shot_2020-11-17_at_11.57.00_PM.png| 600px |center ]]<br />
<br />
<div align="center">Figure 1: Kuhn Poker (Simpler form of Poker) </div><br />
<br />
At the start of each iteration, MCCFR stimulates a hand of poker randomly (Cards held by a player at a given time) and designates one player as the traverser of the game tree. Once that is completed, the AI reviews the decision made by the traverser at a decision point in the game and investigates whether the decision was profitable. The AI compares its decision with other actions available to the traverser at that point and also with the future hypothetical decisions that would have been made following the other available actions. To evaluate a decision, the Counterfactual Regret factor is used. This is the difference between what the traverser would have expected to receive for choosing an action and actually received on the iteration. Thus regret is a numeric value, where a positive regret indicates you regret your decision, a negative regret indicates you are happy with your decision and zero regret indicates that you are indifferent.<br />
<br />
The value of counterfactual regret for a decision is adjusted over the iterations as more scenarios or decision points are encountered. This means at the end of each iteration, the traverser’s strategy is updated so actions with higher counterfactual regret are chosen with higher probability. CFR minimizes regret over many iterations until the average strategy overall iterations converge and the average strategy is the approximated Nash equilibrium. CFR guarantees in all finite games that all counterfactual regrets grow sublinearly in the number of iterations. Pluribus uses Linear CFR in early iterations to reduce the influence of initial bad iterations i.e it assigns a weight of T to regret contributions at iteration T. This causes the influence of the first iteration to decay at a rate of <math>\frac{1}{\sum_{t=1}^Tt} = \frac{2}{T(T+1)}</math>, compared to a rate of <math>\frac{1}{T}</math> in the original CFR algorithm. This leads to the strategy of improving more quickly in practice.<br />
<br />
An additional feature of Pluribus is that in the sub games, instead of assuming that all players play according to a single strategy, Pluribus considers that each player may choose between k different strategies specialized to each player when a decision point is reached. This results in the searcher choosing a more balanced strategy. For instance, if a player never bluffs while holding the best possible hand, then the opponents would learn that fact and always fold in that scenario. To fold in that scenario is a more balanced strategy than to bet.<br />
Therefore, the blueprint strategy is produced offline for the entire game and it is gradually improved while making real-time decisions during the game.<br />
<br />
== Experimental Results ==<br />
To test how well Pluribus functions, it was tested against human players in 2 formats. The first format included 5 human players and one copy of Pluribus (5H+1AI). The 13 human participants were poker players who have won more than $1M playing professionally and were provided with cash incentives to play their best. 10,000 hands of poker were played over 12 days with the 5H+1AI format. Players were anonymized with aliases that remained consistent throughout all their games. The aliases helped the players keep track of the tendencies and types of games played by each player over the 10,000 hands. <br />
<br />
The second format included one human player and 5 copies of Pluribus (1H+5AI). There were 2 more professional players who split another 10,000 hands of poker by playing 5000 hands each and followed the same aliasing process as the first format.<br />
The performance was measured using milli big blinds per game, mbb/game, (i.e. the initial amount of money the second player has to put in the pot) which is the standard measure in the AI field. Additionally, AIVAT was used as the variance reduction technique to control for luck in the games. Significance tests were run at a 95% significance level with one-tailed t-tests to check the profitability of Pluribus's performance.<br />
<br />
Applying AIVAT the following were the results:<br />
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"<br />
! scope="col" | Format !! scope="col" | Average mbb/game !! scope="col" | Standard Error in mbb/game !! scope="col" | P-value of being profitable <br />
|-<br />
! scope="row" | 5H+1AI <br />
| 48 || 25 || 0.028 <br />
|-<br />
! scope="row" | 1H+5AI <br />
| 32 || 15 || 0.014<br />
|}<br />
[[File:top.PNG| 950px | x450px |left]]<br />
<br />
<br />
<div align="center">"Figure 3. Performance of Pluribus in the 5 humans + 1 AI experiment. The dots show Pluribus's performance at the end of each day of play. (Top) The lines show the win rate (solid line) plus or minus the standard error (dashed lines). (Bottom) The lines show the cumulative number of mbbs won (solid line) plus or minus the standard error (dashed lines). The relatively steady performance of Pluribus over the course of the 10,000-hand experiment also suggests that the humans were unable to find exploitable weaknesses in the bot."</div> <br />
<br />
Optimal play in Pluribus looks different from well-known poker conventions: A standard convention of “limping” in poker (calling the 'big blind' rather than folding or raising) is confirmed to be not optimal by Pluribus since it initially experimented with it but eliminated this from its strategy over its games of self-play. On the other hand, another convention of “donk betting” (starting a round by betting when someone else ended the previous round with a call) that is dismissed by players was adopted by Pluribus much more often than played by humans and is proven to be profitable.<br />
<br />
== Discussion and Critiques ==<br />
<br />
Pluribus' Blueprint strategy and Abstraction methods effectively reduce the computational power required. Hence it was computed in 8 days and required less than 512 GB of memory, and costs about $144 to produce. This is in sharp contrast to all the other recent superhuman AI milestones for games. This is a great way the researchers have condensed down the problem to fit the current computational powers. <br />
<br />
Pluribus definitely shows that we can capture observational data and empirical results to construct a superhuman AI without requiring theoretical guarantees, this can be a baseline for future AI inventions and help in the research of AI. It would be interesting to use Pluribus's way of using a non-theoretical approach in more real-life problems such as autonomous driving or stock market trading.<br />
<br />
Extending this idea beyond two-player zero-sum games will have many applications in real life. For example, the emergence of strategies used by the superhuman AI that were seen as "non-optimal" to humans, poses interesting questions about applying such an AI to fields such as business, economics and financial markets.<br />
<br />
It should be noted that there's one interesting fact that even though a player knows all possible outcomes in a Nash Equilibrium game, he/she would still not be beneficial for changing strategies, which means that the player will still choose to keep his/her original strategy.<br />
<br />
The summary for Superhuman AI for Multiplayer Poker is very well written, with a detailed explanation of the concept, steps, and result and with a combination of visual images. However, it seems that the experiment of the study is not well designed. For example, sample selection is not strict and well defined, this could cause selection bias introduced into the result and thus making it not generalizable.<br />
<br />
Superhuman AI, while sounding superior, is actually not uncommon. There have been many endeavours on mastering poker such as the Recursive Belief-based Learning (ReBeL) by Facebook Research. They pursued a method of reinforcement learning on a partially observable Markov decision process which was inspired by the recent successes of AlphaZero. For Pluribus to demonstrate how effective it is compared to the state-of-the-art, it should run some experiments against ReBeL.<br />
<br />
This is a very interesting topic, and this summary is clear enough for readers to understand. I think this application not only can apply in poker, maybe thinking of more applications in other areas? There are many famous AI that really changing our life. For example, AlphaGo and AlphaStar, which are developed by Google DeepMind, defeated professional gamers. Discussing more this will be interesting.<br />
<br />
One of the biggest issues when applying AI to games against humans (when not all information is known, ie, opponents' cards) is the assumption is generally made that the human players are rational players which follow a certain set of "rules" based on the information that they know. This could be an issue with the fact that Pluribus has trained itself by playing itself instead of humans. While the results clearly show that Pluribus has found some kind of 'optimal' method to play, it would be interesting to see if it could actually maximize its profits by learning the trends of its human opponents over time (learning on the fly with information gained each hand while it's playing). In addition to that, the paper may discuss how human action could be changed in the game when they play with Superhuman AI. We can see that playing card games require various strategy and different people can have a different set of actions in the same game and in the same situation.<br />
<br />
One interesting software called Piosolver leverages a similar tree-based algorithm presented in the paper to recommend the move that is deemed game theory optimal (GTO). In the poker world, GTO is a play-style that is based on mathematics and is considered a "defensive" strategy. Following the rock, paper, scissors analogy from the paper, a GTO play-style is synonymous with choosing randomly between the three options, whereas an exploitative strategy involves reading a human player's tendencies and adjusting the strategy accordingly. Piosolver is used by many professional poker players to enhance their game and gain intuition on what the best move is in certain situations.<br />
<br />
Another way to train the proposed model can be a poker game with two or more AI players. That method was used by AlphaGo to train a better model. <br />
<br />
Games with various AI players would be an interesting topic, and through comparing different AI, their shortcomes could be observed and improved. More discussions on this would be of interests.<br />
<br />
Similar to Pluribis, another [https://science.sciencemag.org/content/356/6337/508 paper] discussed a different AI program, called DeepStack, which also has defeated professional poker players at a 2-player Texas hold'em variant. However, instead of finding the Nash Equilibria, DeepStack uses recursive reasoning, decomposition, and a form of intuition that is automatically learned from self-play.<br />
<br />
AI can calculate the exact odds of a specific card or set of cards dropped on the flop. The difficulty is that in poker, the player cannot see the opponent's hand. This kind of hidden information also brings other complications (such as fraud), which is more challenging for AI at the game. At the same time, using AI to train each other is also an interesting topic.<br />
<br />
In inspiration from the critique about fraud, could this AI be used to detect cheating, such as illicitly obtaining information from an opponents hand, in games between human players? For instance by comparing the decision of the humans versus that of the AI's given the same information. Or are the strategies between human players and this AI too different to make conclusions about suspicious humans plays?<br />
<br />
I think AI can possibly calculate the probability of winning each round based on the hand it gets, and modify that probability while replicating the experiments. This may help building the model more accuratly and efficiently.<br />
<br />
The summary is overall well-defined, but authors should explain more on the algorithm parts such that specify how this model perform in each step to achieve the optimal solution. Also, there is an issue such that we must have an assumption before applying this model. The assumption is human need to follow the "rules". What if this model is against with human which does not follow the "rules" then how this model will learn from it? This is a question to contemplate.<br />
<br />
This is an interesting topic discussing AI players in the game. It would be attractive if the summary can provide a brief comparison about the advantages and shortages of the AI players nowadays.<br />
<br />
It is interesting to see how AI has some strategy playing against people in Texas poker. However, this game also relies on how people behave in their body language and the way they talk. This paper focuses more on theoretical side but not 100% true in real life.<br />
<br />
== Conclusion ==<br />
<br />
As Pluribus’s strategy was not developed with any human data and was trained only by self-play, it is an unbiased and a different perspective on how optimal play can be attained.<br />
Developing a superhuman AI for multiplayer poker is a widely recognized milestone in this area and the major remaining milestone in computer poker.<br />
Pluribus’s success shows that despite the lack of known strong theoretical guarantees on performance in multiplayer games, there are large-scale, complex multiplayer imperfect information settings in which a carefully constructed self-play-with-search algorithm can produce superhuman strategies.<br />
<br />
== References ==<br />
<br />
Noam Brown and Tuomas Sandholm (July 11, 2019). Superhuman AI for multiplayer poker. Science 365.<br />
<br />
Osborne, Martin J.; Rubinstein, Ariel (July 12, 1994). A Course in Game Theory. Cambridge, MA: MIT. p. 14.<br />
<br />
Justin Sermeno. (November 17, 2020). Vanilla Counterfactual Regret Minimization for Engineers. https://justinsermeno.com/posts/cfr/#:~:text=Counterfactual%20regret%20minimization%20%28CFR%29%20is%20an%20algorithm%20that,decision.%20It%20can%20be%20positive%2C%20negative%2C%20or%20zero<br />
<br />
Brown, N., Bakhtin, A., Lerer, A., & Gong, Q. (2020). Combining deep reinforcement learning and search for imperfect-information games. Advances in Neural Information Processing Systems, 33.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Gtompkin&diff=48847User:Gtompkin2020-12-02T05:49:36Z<p>Wtjung: /* Related Work */</p>
<hr />
<div>== Presented by == <br />
Grace Tompkins, Tatiana Krikella, Swaleh Hussain<br />
<br />
== Introduction ==<br />
<br />
One of the fundamental challenges in machine learning and data science is dealing with missing and incomplete data. This paper proposes a theoretically justified methodology for using incomplete data in neural networks, eliminating the need for direct completion of data by imputation or other commonly used methods in the existing literature. The authors propose identifying missing data points with a parametric density and then training it together with the rest of the network's parameters. The neuron's response at the first hidden layer is generalized by taking its expected value to process this probabilistic representation. This process is essentially calculating the average activation of the neuron over imputations drawn from the missing data's density. The proposed approach is advantageous as it has the ability to train neural networks using incomplete observations from datasets, which are ubiquitous in practice. This approach also requires minimal adjustments and modifications to existing architectures. Theoretical results of this study show that this process does not lead to a loss of information, while experimental results showed the practical uses of this methodology on several different types of networks.<br />
<br />
== Related Work ==<br />
<br />
Currently, dealing with incomplete inputs in machine learning requires filling absent attributes based on complete, observed data. Two commonly used methods are mean imputation and <math>k</math>-nearest neighbors (k-NN) imputation. In the former (mean imputation), the missing value is replaced by the mean of all available values of that feature in the dataset. In the latter (k-NN imputation), the missing value is also replaced by the mean, however, it is now computed using only the k "closest" samples in the dataset. For example, if the dataset is numerical, Euclidean distance could be used as a measure of "closeness". Other methods for dealing with missing data involve training separate neural networks and extreme learning machines. Probabilistic models of incomplete data can also be built depending on the mechanism missingness (i.e. whether the data is Missing At Random (MAR), Missing Completely At Random (MCAR), or Missing Not At Random (MNAR)), which can be fed into a particular learning model. Further, the decision function can also be trained using available/visible inputs alone. Previous work using neural networks for missing data includes a paper by Bengio and Gringras [1] where the authors used recurrent neural networks with feedback into the input units to fill absent attributes solely to minimize the learning criterion. Goodfellow et. al. [2] also used neural networks by introducing a multi-prediction deep Boltzmann machine that could perform classification on data with missingness in the inputs.<br />
<br />
== Layer for Processing Missing Data ==<br />
<br />
In this approach, the adaptation of a given neural network to incomplete data relies on two steps: the estimation of the missing data and the generalization of the neuron's activation. <br />
<br />
Let <math>(x,J)</math> represent a missing data point, where <math>x \in \mathbb{R}^D </math>, and <math>J \subset \{1,...,D\} </math> is a set of attributes with missing data. <math>(x,J)</math> therefore represents an "incomplete" data point for which <math>|J|</math>-many entries are unknown - examples of this could be a list of daily temperature readings over a week where temperature was not recorded on the third day (<math>x\in \mathbb{R}^7, J= \{3\}</math>), an audio transcript that goes silent for certain timespans, or images that are partially masked out (as discussed in the examples).<br />
<br />
For each missing point <math>(x,J)</math>, define an affine subspace consisting of all points which coincide with <math>x</math> on known coordinates <math>J'=\{1,…,N\}/J</math>: <br />
<br />
<center><math>S=Aff[x,J]=span(e_J) </math></center> <br />
where <math>e_J=[e_j]_{j\in J}</math> and <math>e_j</math> is the <math> j^{th}</math> canonical vector in <math>\mathbb{R}^D </math>.<br />
<br />
Assume that the missing data points come from the D-dimensional probability distribution, <math>F</math>. In their approach, the authors assume that the data points follow a mixture of Gaussians (GMM) with diagonal covariance matrices. By choosing diagonal covariance matrices, the number of model parameters is reduced. To model the missing points <math>(x,J)</math>, the density <math>F</math> is restricted to the affine subspace <math>S</math>. Thus, possible values of <math>(x,J)</math> are modelled using the conditional density <math>F_S: S \to \mathbb{R} </math>, <br />
<br />
<center><math>F_S(x) = \begin{cases}<br />
\frac{1}{\int_{S} F(s) \,ds}F(x) & \text{if $x \in S$,} \\<br />
0 & \text{otherwise.}<br />
\end{cases} </math></center><br />
<br />
To process the missing data by a neural network, the authors propose that only the first hidden layer needs modification. Specifically, they generalize the activation functions of all the neurons in the first hidden layer of the network to process the probability density functions representing the missing data points. For the conditional density function <math>F_S</math>, the authors define the generalized activation of a neuron <math>n: \mathbb{R}^D \to \mathbb{R}</math> on <math>F_S </math> as: <br />
<br />
<center><math>n(F_S)=E[n(x)|x \sim F_S]=\int n(x)F_S(x) \,dx</math>,</center> <br />
provided that the expectation exists. <br />
<br />
The following two theorems describe how to apply the above generalizations to both the ReLU and the RBF neurons, respectively. <br />
<br />
'''Theorem 3.1''' Let <math>F = \sum_i{p_iN(m_i, \Sigma_i)}</math> be the mixture of (possibly degenerate) Gaussians. Given weights <math>w=(w_1, ..., w_D) \in \mathbb{R}^D,</math><math> b \in \mathbb{R} </math>, we have<br />
<br />
<center><math>\text{ReLU}_{w,b}(F)=\sum_i{p_iNR\big(\frac{w^{\top}m_i+b}{\sqrt{w^{\top}\Sigma_iw}}}\big)</math></center> <br />
<br />
where <math>NR(x)=\text{ReLU}[N(x,1)]</math> and <math>\text{ReLU}_{w,b}(x)=\text{max}(w^{\top}+b, 0)</math>, <math>w \in \mathbb{R}^D </math> and <math> b \in \mathbb{R}</math> is the bias.<br />
<br />
'''Theorem 3.2''' Let <math>F = \sum_i{p_iN(m_i, \Sigma_i)}</math> be the mixture of (possibly degenerate) Gaussians and let the RBF unit be parametrized by <math>N(c, \Gamma) </math>. We have: <br />
<br />
<center><math>\text{RBF}_{c, \Gamma}(F) = \sum_{i=1}^k{p_iN(m_i-c, \Gamma+\Sigma_i)}(0)</math>.</center> <br />
<br />
In the case where the data set contains no missing values, the generalized neurons reduce to classical ones, since the distribution <math>F</math> is only used to estimate possible values at missing attributes. However, if one wishes to use an incomplete data set in the testing stage, then an incomplete data set must be used to train the model.<br />
<br />
<math> </math><br />
<br />
== Theoretical Analysis ==<br />
<br />
The main theoretical results, which are summarized below, show that using generalized neuron's activation at the first layer does not lead to the loss of information. <br />
<br />
Let the generalized response of a neuron <math>n: \mathbb{R}^D \rightarrow \mathbb{R}</math> evaluated on a probability measure <math>\mu</math> which is given by <br />
<center><math>n(\mu) := \int n(x)d\mu(x)</math></center><br />
<br />
'''Theorem 4.1.''' Let <math>\mu</math>, <math>v</math> be probabilistic measures satisfying <math>\int ||x|| d \mu(x) < \infty</math>. If <br />
<center><math>ReLU_{w,b}(\mu) = ReLU_{w,b}(\nu) \text{ for } w \in \mathbb{R}^D, b \in \mathbb{R}</math></center> then <math>\nu = \mu</math>.<br />
<br />
Theorem 4.1 shows that a neural network with generalized ReLU units is able to identify any two probability measures. The proof presented by the authors uses the Universal Approximation Property (UAP), and is summarized as follows. <br />
<br />
''Sketch of Proof'' Let <math>w \in \mathbb{R}^D</math> be fixed and define the set <center><math>F_w = \{p: \mathbb{R} \rightarrow \mathbb{R}: \int p(w^Tx)d\mu(x) = \int p(w^Tx)d\nu(x)\}.</math></center> The first step of the proof involves showing that <math>F_w</math> contains all continuous and bounded functions. The authors show this by showing that a piecewise continuous function that is affine linear on specific intervals, <math>Q</math>, is in the set <math>F_w</math>. This involves re-writing <math>Q</math> as a sum of tent-like piecewise linear functions, <math>T</math> and showing that <math>T \in F_w</math> (since it is sufficient to only show <math>T \in F_w</math>). <br />
<br />
Next, the authors show that an arbitrary bounded continuous function <math>G</math> is in <math>F_w</math> by the Lebesgue dominated convergence theorem. <br />
<br />
Then, as <math>cos(\cdot), sin(\cdot) \in F_w</math>, the function <center><math>exp(ir) = cos(r) + i sin(r) \in F_w</math></center> and we have the equality <center><math>\int exp(iw^Tx)d\mu(x) = \int exp(iw^Tx)d\nu(x).</math></center> Since <math>w</math> was arbitrarily chosen, we can conclude that <math>\mu = \nu</math> as the characteristic functions of the two measures coincide. <br />
<br />
A result analogous to Theorem 4.1 for RBF can also be obtained.<br />
<br />
'''Theorem 2.1''' Let <math>\mu, \nu</math> be probabilistic measures. If<br />
$$ RBF_{m,\alpha}(\mu) = RBF_{m,\alpha}(\nu) \text{ for every } m \in \mathbb{R}^D, \alpha > 0,$$<br />
then <math>\nu = \mu</math>.<br />
<br />
More general results can be obtained making stronger assumptions on the probability measures. For example, if a given family of neurons satisfies UAP, then their generalization can identify any probability measure with compact support.<br />
<br />
'''Theorem 2.2''' Let <math>\mu, \nu</math> be probabilistic measures with compact support. Let <math>\mathcal{N}</math> be a family of functions having UAP. If<br />
$$n(\mu) = n(\nu) \text{ for every } n \in \mathcal{N},$$<br />
then <math>\nu = \mu</math>.<br />
<br />
A detailed proof Theorems 2.1 and 2.2 can be found in section 2 of the Supplementary Materials, which can be downloaded [https://proceedings.neurips.cc/paper/2018/hash/411ae1bf081d1674ca6091f8c59a266f-Abstract.html here].<br />
<br />
== Experimental Results ==<br />
The model was applied to three types of algorithms: an Autoencoder (AE), a multilayer perceptron, and a radial basis function network.<br />
<br />
'''Autoencoder'''<br />
<br />
Corrupted images were restored as a part of this experiment. Grayscale handwritten digits were obtained from the MNIST database. A 13 by 13 (169 pixels) square was removed from each 28 by 28 (784 pixels) image. The location of the square was uniformly sampled for each image. The autoencoder used included 5 hidden layers. The first layer used ReLU activation functions while the subsequent layers utilized sigmoids. The loss function was computed using pixels from outside the mask. <br />
<br />
Popular imputation techniques were compared against the conducted experiment:<br />
<br />
''k-nn:'' Replaced missing features with the mean of respective features calculated using K nearest training samples. Here, K=5. <br />
<br />
''mean:'' Replaced missing features with the mean of respective features calculated using all incomplete training samples.<br />
<br />
''dropout:'' Dropped input neutrons with missing values. <br />
<br />
Moreover, a context encoder (CE) was trained by replacing missing features with their means. Unlike mean imputation, the complete data was used in the training phase. The method under study performed better than the imputation methods inside and outside the mask. Additionally, the method under study outperformed CE based on the whole area and area outside the mask. <br />
<br />
The mean square error of reconstruction is used to test each method. The errors calculated over the whole area, inside and outside the mask are shown in Table 1, which indicates the method introduced in this paper is the most competitive.<br />
<br />
[[File:Group3_Table1.png |center]]<br />
<div align="center">Table 1: Mean square error of reconstruction on MNIST incomplete images scaled to [0, 1]</div><br />
<br />
'''Multilayer Perceptron'''<br />
<br />
A multilayer perceptron with 3 ReLU hidden layers was applied to a multi-class classification problem on the Epileptic Seizure Recognition (ESR) data set taken from [3]. Each 178-dimensional vector (out of 11500 samples) is the EEG recording of a given person for 1 second, categorized into one of 5 classes. To generate missing attributes, 25%, 50%, 75%, and 90% of observations were randomly removed. The aforementioned imputation methods were used in addition to Multiple Imputation by Chained Equation (mice) and a mixture of Gaussians (GMM). The former utilizes the conditional distribution of data by Markov chain Monte Carlo techniques to draw imputations. The latter replaces missing features with values sampled from GMM estimated from incomplete data using the EM algorithm. <br />
<br />
Double 5-fold cross-validation was used to report classification results. The classical accuracy measure is usually being used for accessing the results. The model under study outperformed classical imputation methods, which give reasonable results only for a low number of missing values. The method under study performs nearly as well as CE, even though CE had access to complete training data. <br />
<br />
'''Radial Basis Function Network'''<br />
<br />
RBFN can be considered as a minimal architecture implementing our model, which contains only one hidden layer. A cross-entropy function was applied to a softmax in the output layer. Two-class data sets retrieved from the UCI repository [4] with internally missing attributes were used. Since the classification is binary, two additional SVM kernel models which work directly with incomplete data without performing any imputations were included in the experiment:<br />
<br />
''geom:'' The objective function is based on the geometric interpretation of the margin and aims to maximize the margin of each sample in its own subspace [5].<br />
<br />
''karma:'' This algorithm iteratively tunes kernel classifier under low-rank assumptions [6].<br />
<br />
The above SVM methods were combined with RBF kernel function. The number of RBF units was selected in the inner cross-validation from the range {25, 50, 75, 100}. Initial centers of RBFNs were randomly selected from training data while variances were samples from <math>N(0,1)</math> distribution. For SVM methods, the margin parameter <math>C</math> and kernel radius <math>\gamma</math> were selected from <math>\{2^k :k=−5,−3,...,9\}</math> for both parameters. For karma, additional parameter <math>\gamma_{karma}</math> was selected from the set <math>\{1, 2\}</math>.<br />
<br />
The model under study outperformed imputation techniques in almost all cases. It partially confirms that the use of raw incomplete data in neural networks is a usually better approach than filling missing attributes before the learning process. Moreover, it obtained more accurate results than modified kernel methods, which directly work on incomplete data. The performance of the model was once again comparable to, and in some cases better than CE, which had access to the complete data.<br />
<br />
== Conclusion ==<br />
<br />
The results with these experiments along with the theoretical results conclude that this novel approach for dealing with missing data through a modification of a neural network is beneficial and outperforms many existing methods. This approach, which utilizes representing missing data with a probability density function, allows a neural network to determine a more generalized and accurate response of the neuron.<br />
<br />
== Critiques ==<br />
<br />
- A simulation study where the mechanism of missingness is known will be interesting to examine. Doing this will allow us to see when the proposed method is better than existing methods, and under what conditions.<br />
<br />
- This method of imputing incomplete data has many limitations. In most cases, we have a missing data point. Consequently, we are facing a relatively small amount of data that does not require training of a neural network. For a large dataset, missing records do not seem to be very crucial because obtaining data will be relatively easier. Thus, using an empirical way of imputing data such as a majority vote will be sufficient.<br />
<br />
- An interesting application of this problem is in NLP. In NLP, especially Question Answering, there is a problem where a query is given and an answer must be retrieved, but the knowledge base is incomplete. There is therefore a requirement for the model to be able to infer information from the existing knowledge base in order to answer the question. Although this problem is a little more contrived than the one mentioned here, it is nevertheless similar in nature because it requires the ability to probabilistically determine some value which can then be used as a response.<br />
<br />
- For the first sentence in the introduction section, it might be better to use "to deal with missing and incomplete data" rather than use "dealing with missing and incomplete data"<br />
<br />
- Based on my R experience, there is one useful function "knnImputation" under the package DMwR, which is very helpful when dealing with missing and incomplete data. This function uses the k-nearest neighbors to fill in the unknown (NA) values in a data set. The users can modify the number of neighbors required in the calculation of each missing value.<br />
<br />
- The experiments in this paper evaluate this method against low amounts of missing data. It would be interesting to see the properties of this imputation when a majority of the data is missing, and see if this method can outperform dropout training in this setting (dropout is known to be surprisingly robust even at high drop levels).<br />
<br />
- This problem can possibly be applied to face recognition were given a blurry image of a person's face, the neural network can make the image clearer such that the face of the person would be visible for humans to see and also possible for the software to identify who the person is.<br />
<br />
- This novel approach can also be applied to restoring damaged handwritten historical documents. By feeding in a damaged document with portions of unreadable texts, the neural network can add missing information utilizing a trained context encoder<br />
<br />
- It will be interesting to see how this method performs with audio data, i.e. say if there are parts of an audio file that are missing, whether the neural network will be able to learn the underlying distribution and impute the missing sections of speech.<br />
<br />
- In general, data are usually missing in a specific part of the content. For example, old books usually have first couple page or last couple pages that are missing. It would be interesting to see that how the distribution of "missing data" will be applied in those cases.<br />
<br />
- In this paper, the researchers were able to outperform existing imputation methods using neural networks. It would be really nice to see how does the usage of neural networks impacts the need for amount of data, and how much more training is required in comparison to the other algorithms provided in this paper. <br />
<br />
- It might be an interesting approach to investigate how the size of missing data may influence the training. For example, in the MNIST AutoEncoder, some algorithms involve masks to generate more general images and avoid overfitting. The approach could compare the result by changing the size of the "missing" part to illustrate to what degree can we ignore the missing data and view them as assistance.<br />
<br />
- It would be nice to see how the method under study outperforms both the Multilayer Perceptron and Radial Basis Function Network methods mentioned in the summary. Since these two methods are placed under the Experimental Results section, it is expected that they consist of more justifications and supporting evidence (analytical results, tables, graphs, etc.) than what have been presented, so that it is more convincing for the readers.<br />
<br />
- Both KNN imputation and mean imputation are valid techniques to solve the problem, one possible future study on this topic is to explore which one of the two methods above will perform better given the sparsity of the dataset. Another possible study is that if there are methods that can make better inferences on original input, an example is described in the paper (https://cs.uwaterloo.ca/~ilyas/papers/WuMLSys2020.pdf), where imputation is performed based on learning structural properties of data distributions.<br />
<br />
- This project has many applications in the real world. One of the examples is to forecast the average height in Canada. We know that people are born and die every second. Also, many people are not accessible which means we cannot obtain the exactly true data. Thus, we can just feed the observed data to the neural network then we will get the predicted result. This can also be applied to the investment world since the data are changing every second.<br />
<br />
-If the performance and efficiency of training neural network models with the original dataset directly are generally better than traditional algorithms, this would be a practical method to use. <br />
<br />
- It is interesting to see how we can fill missing data with machine learning. The standard way to fill missing numeric data is to use mean, and using KNN is really a talented way of doing this.<br />
<br />
- This project focus on dealing with incomplete or missing data which has a wide application on many aspects such as image recognition and damaged documents, etc. This is a novel application for machine learning and inspires us for the usage of the missing data.<br />
<br />
== References ==<br />
[1] Yoshua Bengio and Francois Gingras. Recurrent neural networks for missing or asynchronous<br />
data. In Advances in neural information processing systems, pages 395–401, 1996.<br />
<br />
[2] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016.<br />
<br />
[3] Ralph G Andrzejak, Klaus Lehnertz, Florian Mormann, Christoph Rieke, Peter David, and Christian E Elger. Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state. Physical Review E, 64(6):061907, 2001.<br />
<br />
[4] Arthur Asuncion and David J. Newman. UCI Machine Learning Repository, 2007.<br />
<br />
[5] Gal Chechik, Geremy Heitz, Gal Elidan, Pieter Abbeel, and Daphne Koller. Max-margin classification of data with absent features. Journal of Machine Learning Research, 9:1–21, 2008.<br />
<br />
[6] Elad Hazan, Roi Livni, and Yishay Mansour. Classification with low rank and missing data. In Proceedings of The 32nd International Conference on Machine Learning, pages 257–266, 2015.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=48557Improving neural networks by preventing co-adaption of feature detectors2020-12-01T01:19:02Z<p>Wtjung: </p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by the decaying factor <math>f</math> = 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors. This was reduced to about 130 errors with 0.5 dropout and different L2 constraints for each hidden unit input weight. By omitting a random 20% of the input pixels in addition to the aforementioned changes, the number of errors was further reduced to 110. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
TIMIT dataset includes voice samples of 630 American English speakers varying across 8 different dialects. It is often used to evaluate the performance of automatic speech recognition systems. Using Kaldi, the dataset was pre-processed to extract input features in the form of log filter bank responses.<br />
<br />
=== Pre-training and Training ===<br />
<br />
For pretraining, they pretrained their neural network with a deep belief network and the first layer was built using Restricted Boltzmann Machine (RBM). Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged.<br />
<br />
Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
<br />
=== Classification Test and Performance ===<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input.<br />
<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Thus, neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias a.k.a weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. The figure above gives a visual representation of a Convolutional Neural Network. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. When looking at the image example, green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + \epsilon <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90 minutes and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
Removing incorrect labels, The CIFAR-10 dataset is a subset of the Tiny Images dataset with 10 classes. It contains 5000 training images and 1000 testing images for each class. The dataset has 32 x 32 color images searched from the web and the images are labeled with the noun used to search the image.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
Two models, one with dropout and one without dropout, were built to test the performance of dropout on CIFAR-10. All models have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The model with dropout enables us to use more parameters because dropout forces a strong regularization on the network. Thus, a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected, but not convolutional, and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. The model with one additional locally-connected layer and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution images, and they are labeled among 1000 different categories. The data were collected from the web and manually labeled using MTerk tool, which is a crowd-sourcing tool provided by Amazon.<br />
Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error rate using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
They mostly focused on the model with dropout because the one without dropout had a similar approach, but there was a serious issue with overfitting. They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. This could reduce the network’s capacity to overfit the training data and helped generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for testing their model patched at the center, four corners, and their horizontal reflections. To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
The network contains seven weight layers. The first five are convolutional, and the last two are globally-connected. Max-pooling layers follow the layer number 1,2, and 5. And then, the output of the last globally-connected layer was fed to a 1000-way softmax output layers. Using this architecture, the authors achieved the error rate of 48.6%. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
Like the previous datasets, such as the MNIST, TIMIT, Reuters, and CIFAR-10, we also see a significant improvement for the ImageNet dataset. Including complicated architectures like this one, introducing dropout generalizes models better and gives lower test error rates.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.<br />
<br />
= Critiques =<br />
It is a very brilliant idea to dropout half of the neurons to reduce co-adaptations. It is mentioned that for fully connected layers, dropout in all hidden layers works better than dropout in only one hidden layer. There is another paper Dropout: A Simple Way to Prevent Neural Networks from<br />
Overfitting[https://www.cs.toronto.edu/~hinton/absps/JMLRdropout.pdf] gives a more detailed explanation.<br />
<br />
It will be interesting to see how this paper could be used to prevent overfitting of LSTMs.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=48556Improving neural networks by preventing co-adaption of feature detectors2020-12-01T01:14:40Z<p>Wtjung: /* Models for ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by the decaying factor <math>f</math> = 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors. This was reduced to about 130 errors with 0.5 dropout and different L2 constraints for each hidden unit input weight. By omitting a random 20% of the input pixels in addition to the aforementioned changes, the number of errors was further reduced to 110. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
TIMIT dataset includes voice samples of 630 American English speakers varying across 8 different dialects. It is often used to evaluate the performance of automatic speech recognition systems. Using Kaldi, the dataset was pre-processed to extract input features in the form of log filter bank responses.<br />
<br />
=== Pre-training and Training ===<br />
<br />
For pretraining, they pretrained their neural network with a deep belief network and the first layer was built using Restricted Boltzmann Machine (RBM). Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged.<br />
<br />
Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
<br />
=== Classification Test and Performance ===<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input.<br />
<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Thus, neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias a.k.a weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. The figure above gives a visual representation of a Convolutional Neural Network. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. When looking at the image example, green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + \epsilon <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90 minutes and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
Removing incorrect labels, The CIFAR-10 dataset is a subset of the Tiny Images dataset with 10 classes. It contains 5000 training images and 1000 testing images for each class. The dataset has 32 x 32 color images searched from the web and the images are labeled with the noun used to search the image.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
Two models, one with dropout and one without dropout, were built to test the performance of dropout on CIFAR-10. All models have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The model with dropout enables us to use more parameters because dropout forces a strong regularization on the network. Thus, a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected, but not convolutional, and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. The model with one additional locally-connected layer and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution images, and they are labeled among 1000 different categories. The data were collected from the web and manually labeled using MTerk tool, which is a crowd-sourcing tool provided by Amazon.<br />
Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error rate using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
They mostly focused on the model with dropout because the one without dropout had a similar approach, but there was a serious issue with overfitting. They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. This could reduce the network’s capacity to overfit the training data and helped generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for testing their model patched at the center, four corners, and their horizontal reflections. To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
The network contains seven weight layers. The first five are convolutional, and the last two are globally-connected. Max-pooling layers follow the layer number 1,2, and 5. And then, the output of the last globally-connected layer was fed to a 1000-way softmax output layers. Using this architecture, the authors achieved the error rate of 48.6%. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
Like the previous datasets, such as the MNIST, TIMIT, Reuters, and CIFAR-10, we also see a significant improvement for the ImageNet dataset. Including complicated architectures like this one, introducing dropout generalizes models better and gives lower test error rates.<br />
<br />
= Critiques =<br />
It is a very brilliant idea to dropout half of the neurons to reduce co-adaptations. It is mentioned that for fully connected layers, dropout in all hidden layers works better than dropout in only one hidden layer. There is another paper Dropout: A Simple Way to Prevent Neural Networks from<br />
Overfitting[https://www.cs.toronto.edu/~hinton/absps/JMLRdropout.pdf] gives a more detailed explanation.<br />
<br />
It will be interesting to see how this paper could be used to prevent overfitting of LSTMs.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=48555Improving neural networks by preventing co-adaption of feature detectors2020-12-01T01:13:00Z<p>Wtjung: /* ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by the decaying factor <math>f</math> = 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors. This was reduced to about 130 errors with 0.5 dropout and different L2 constraints for each hidden unit input weight. By omitting a random 20% of the input pixels in addition to the aforementioned changes, the number of errors was further reduced to 110. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
TIMIT dataset includes voice samples of 630 American English speakers varying across 8 different dialects. It is often used to evaluate the performance of automatic speech recognition systems. Using Kaldi, the dataset was pre-processed to extract input features in the form of log filter bank responses.<br />
<br />
=== Pre-training and Training ===<br />
<br />
For pretraining, they pretrained their neural network with a deep belief network and the first layer was built using Restricted Boltzmann Machine (RBM). Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged.<br />
<br />
Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
<br />
=== Classification Test and Performance ===<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input.<br />
<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Thus, neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias a.k.a weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. The figure above gives a visual representation of a Convolutional Neural Network. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. When looking at the image example, green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + \epsilon <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90 minutes and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
Removing incorrect labels, The CIFAR-10 dataset is a subset of the Tiny Images dataset with 10 classes. It contains 5000 training images and 1000 testing images for each class. The dataset has 32 x 32 color images searched from the web and the images are labeled with the noun used to search the image.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
Two models, one with dropout and one without dropout, were built to test the performance of dropout on CIFAR-10. All models have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The model with dropout enables us to use more parameters because dropout forces a strong regularization on the network. Thus, a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected, but not convolutional, and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. The model with one additional locally-connected layer and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution images, and they are labeled among 1000 different categories. The data were collected from the web and manually labeled using MTerk tool, which is a crowd-sourcing tool provided by Amazon.<br />
Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error rate using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
They mostly focused on the model with dropout because the one without dropout had a similar approach, but there was a serious issue with overfitting. They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. This could reduce the network’s capacity to overfit the training data and helped generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for testing their model patched at the center, four corners, and their horizontal reflections.<br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
The network contains seven weight layers. The first five are convolutional, and the last two are globally-connected. Max-pooling layers follow the layer number 1,2, and 5. And then, the output of the last globally-connected layer was fed to a 1000-way softmax output layers. Using this architecture, the authors achieved the error rate of 48.6%. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
Like the previous datasets, such as the MNIST, TIMIT, Reuters, and CIFAR-10, we also see a significant improvement for the ImageNet dataset. Including complicated architectures like this one, introducing dropout generalizes models better and gives lower test error rates.<br />
<br />
= Critiques =<br />
It is a very brilliant idea to dropout half of the neurons to reduce co-adaptations. It is mentioned that for fully connected layers, dropout in all hidden layers works better than dropout in only one hidden layer. There is another paper Dropout: A Simple Way to Prevent Neural Networks from<br />
Overfitting[https://www.cs.toronto.edu/~hinton/absps/JMLRdropout.pdf] gives a more detailed explanation.<br />
<br />
It will be interesting to see how this paper could be used to prevent overfitting of LSTMs.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47845Improving neural networks by preventing co-adaption of feature detectors2020-11-29T22:08:15Z<p>Wtjung: /* Models for ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. <br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. When looking at the image example, green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + \epsilon <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90 minutes and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images are labeled with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. Two models both have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network. Thus, a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected, but not convolutional, and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error rate using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the model without dropout had a similar approach, but there was a serious issue with overfitting): <br />
<br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. This could reduce the network’s capacity to overfit the training data and helped generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, four corners, and their horizontal reflections).<br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
= Critiques =<br />
It is a very brilliant idea to dropout half of the neurons to reduce co-adaptations. It is mentioned that for fully connected layers, dropout in all hidden layers works better than dropout in only one hidden layer. There is another paper Dropout: A Simple Way to Prevent Neural Networks from<br />
Overfitting[https://www.cs.toronto.edu/~hinton/absps/JMLRdropout.pdf] gives a more detailed explanation.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47844Improving neural networks by preventing co-adaption of feature detectors2020-11-29T22:07:24Z<p>Wtjung: /* ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. <br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. When looking at the image example, green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + \epsilon <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90 minutes and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images are labeled with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. Two models both have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network. Thus, a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected, but not convolutional, and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error rate using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. When applying 50% dropout to the 6th layer, the error rate was brought down to 42.4%.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
<br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. This could reduce the network’s capacity to overfit the training data and helped generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, four corners, and their horizontal reflections).<br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
= Critiques =<br />
It is a very brilliant idea to dropout half of the neurons to reduce co-adaptations. It is mentioned that for fully connected layers, dropout in all hidden layers works better than dropout in only one hidden layer. There is another paper Dropout: A Simple Way to Prevent Neural Networks from<br />
Overfitting[https://www.cs.toronto.edu/~hinton/absps/JMLRdropout.pdf] gives a more detailed explanation.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47537Improving neural networks by preventing co-adaption of feature detectors2020-11-29T01:40:51Z<p>Wtjung: /* Models for ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images are labeled with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. Two models both have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected but not convolutional and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, the error rate of 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47535Improving neural networks by preventing co-adaption of feature detectors2020-11-29T01:40:34Z<p>Wtjung: /* ImageNet Dataset */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images are labeled with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. Two models both have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected but not convolutional and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, the error rate of 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47521Improving neural networks by preventing co-adaption of feature detectors2020-11-29T01:21:54Z<p>Wtjung: /* ImageNet Dataset */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden units being present during training. Hence there are fewer co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; Reuters Corpus Volume; CIFAR-10; and ImageNet.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with mini-batches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross-validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation of 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist2.png|center|400px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images are labeled with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. Two models both have CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers. A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
<br />
Additional changes were made with the model with dropout. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. This fourth weight layer is locally connected but not convolutional and contains 16 banks of filters of size 3 × 3 with 50% dropout. Lastly, the softmax layer takes its input from this fourth weight layer.<br />
<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset was the error rate of 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, the error rate of 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47266Improving neural networks by preventing co-adaption of feature detectors2020-11-28T07:13:44Z<p>Wtjung: /* Models for ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|600px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters Corpus Volume =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
<br />
= CIFAR-10 =<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
=== Models for CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
The authors have shown a consistent improvement by the models trained with dropout in classifying objects in the following datasets: MNIST; TIMIT; Reuters Corpus Volume I; CIFAR-10; and ImageNet.</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat441F21&diff=47252stat441F212020-11-28T06:21:33Z<p>Wtjung: /* Paper presentation */</p>
<hr />
<div><br />
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== [[F20-STAT 441/841 CM 763-Proposal| Project Proposal ]] ==<br />
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= Record your contributions here [https://docs.google.com/spreadsheets/d/10CHiJpAylR6kB9QLqN7lZHN79D9YEEW6CDTH27eAhbQ/edit?usp=sharing]=<br />
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Use the following notations:<br />
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P: You have written a summary/critique of the paper.<br />
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T: You had a technical contribution on a paper (excluding the paper that you present).<br />
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E: You had an editorial contribution on a paper (excluding the paper that you present).<br />
<br />
=Paper presentation=<br />
{| class="wikitable"<br />
<br />
{| border="1" cellpadding="3"<br />
|-<br />
|width="60pt"|Date<br />
|width="250pt"|Name <br />
|width="15pt"|Paper number <br />
|width="700pt"|Title<br />
|width="15pt"|Link to the paper<br />
|width="30pt"|Link to the summary<br />
|width="30pt"|Link to the video<br />
|-<br />
|Sep 15 (example)||Ri Wang || ||Sequence to sequence learning with neural networks.||[http://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Going_Deeper_with_Convolutions Summary] || [https://youtu.be/JWozRg_X-Vg?list=PLehuLRPyt1HzXDemu7K4ETcF0Ld_B5adG&t=539]<br />
|-<br />
|Week of Nov 16 ||Sharman Bharat, Li Dylan,Lu Leonie, Li Mingdao || 1|| Risk prediction in life insurance industry using supervised learning algorithms || [https://rdcu.be/b780J Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Bsharman Summary] ||<br />
[https://www.youtube.com/watch?v=TVLpSFYgF0c&feature=youtu.be]<br />
|-<br />
|Week of Nov 16 || Delaney Smith, Mohammad Assem Mahmoud || 2|| Influenza Forecasting Framework based on Gaussian Processes || [https://proceedings.icml.cc/static/paper_files/icml/2020/1239-Paper.pdf Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Influenza_Forecasting_Framework_based_on_Gaussian_Processes Summary]|| [https://www.youtube.com/watch?v=HZG9RAHhpXc&feature=youtu.be]<br />
|-<br />
|Week of Nov 16 || Tatianna Krikella, Swaleh Hussain, Grace Tompkins || 3|| Processing of Missing Data by Neural Networks || [http://papers.nips.cc/paper/7537-processing-of-missing-data-by-neural-networks.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Gtompkin Summary] || [https://learn.uwaterloo.ca/d2l/ext/rp/577051/lti/framedlaunch/6ec1ebea-5547-46a2-9e4f-e3dc9d79fd54]<br />
|-<br />
|Week of Nov 16 ||Jonathan Chow, Nyle Dharani, Ildar Nasirov ||4 ||Streaming Bayesian Inference for Crowdsourced Classification ||[https://papers.nips.cc/paper/9439-streaming-bayesian-inference-for-crowdsourced-classification.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Streaming_Bayesian_Inference_for_Crowdsourced_Classification Summary] || [https://www.youtube.com/watch?v=UgVRzi9Ubws]<br />
|-<br />
|Week of Nov 16 || Matthew Hall, Johnathan Chalaturnyk || 5|| Neural Ordinary Differential Equations || [https://papers.nips.cc/paper/7892-neural-ordinary-differential-equations.pdf] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_ODEs Summary]||<br />
|-<br />
|Week of Nov 16 || Luwen Chang, Qingyang Yu, Tao Kong, Tianrong Sun || 6|| Adversarial Attacks on Copyright Detection Systems || Paper [https://proceedings.icml.cc/static/paper_files/icml/2020/1894-Paper.pdf] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Adversarial_Attacks_on_Copyright_Detection_Systems Summary] || [https://www.youtube.com/watch?v=bQI9S6bCo8o]<br />
|-<br />
|Week of Nov 16 || Casey De Vera, Solaiman Jawad || 7|| IPBoost – Non-Convex Boosting via Integer Programming || [https://arxiv.org/pdf/2002.04679.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=IPBoost Summary] || [https://www.youtube.com/watch?v=4DhJDGC4pyI&feature=youtu.be]<br />
|-<br />
|Week of Nov 16 || Yuxin Wang, Evan Peters, Yifan Mou, Sangeeth Kalaichanthiran || 8|| What Game Are We Playing? End-to-end Learning in Normal and Extensive Form Games || [https://arxiv.org/pdf/1805.02777.pdf] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=what_game_are_we_playing Summary] || [https://www.youtube.com/watch?v=9qJoVxo3hnI&feature=youtu.be]<br />
|-<br />
|Week of Nov 16 || Yuchuan Wu || 9|| || || ||<br />
|-<br />
|Week of Nov 16 || Zhou Zeping, Siqi Li, Yuqin Fang, Fu Rao || 10|| A survey of neural network-based cancer prediction models from microarray data || [https://www.sciencedirect.com/science/article/pii/S0933365717305067 Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Y93fang Summary] || [https://youtu.be/B8pPUU8ypO0]<br />
|-<br />
|Week of Nov 23 ||Jinjiang Lian, Jiawen Hou, Yisheng Zhu, Mingzhe Huang || 11|| DROCC: Deep Robust One-Class Classification || [https://proceedings.icml.cc/static/paper_files/icml/2020/6556-Paper.pdf paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:J46hou Summary] || [https://www.youtube.com/watch?v=tvCEvvy54X8&ab_channel=JJLian]<br />
|-<br />
|Week of Nov 23 || Bushra Haque, Hayden Jones, Michael Leung, Cristian Mustatea || 12|| Combine Convolution with Recurrent Networks for Text Classification || [https://arxiv.org/pdf/2006.15795.pdf Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Cvmustat Summary] || [https://www.youtube.com/watch?v=or5RTxDnZDo]<br />
|-<br />
|Week of Nov 23 || Taohao Wang, Zeren Shen, Zihao Guo, Rui Chen || 13|| Large Scale Landmark Recognition via Deep Metric Learning || [https://arxiv.org/pdf/1908.10192.pdf paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:T358wang Summary] || [https://www.youtube.com/watch?v=K9NypDNPLJo Video]<br />
|-<br />
|Week of Nov 23 || Qianlin Song, William Loh, Junyue Bai, Phoebe Choi || 14|| Task Understanding from Confusing Multi-task Data || [https://proceedings.icml.cc/static/paper_files/icml/2020/578-Paper.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Task_Understanding_from_Confusing_Multi-task_Data Summary] || [https://youtu.be/i_5PQdfuH-Y]<br />
|-<br />
|Week of Nov 23 || Rui Gong, Xuetong Wang, Xinqi Ling, Di Ma || 15|| Semantic Relation Classification via Convolution Neural Network|| [https://www.aclweb.org/anthology/S18-1127.pdf paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Semantic_Relation_Classification——via_Convolution_Neural_Network Summary]|| [https://www.youtube.com/watch?v=m9o3NuMUKkU&ab_channel=DiMa video]<br />
|-<br />
|Week of Nov 23 || Xiaolan Xu, Robin Wen, Yue Weng, Beizhen Chang || 16|| Graph Structure of Neural Networks || [https://proceedings.icml.cc/paper/2020/file/757b505cfd34c64c85ca5b5690ee5293-Paper.pdf Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Graph_Structure_of_Neural_Networks Summary] || [https://youtu.be/x9eUgEwntcs Video]<br />
|-<br />
|Week of Nov 23 ||Hansa Halim, Sanjana Rajendra Naik, Samka Marfua, Shawrupa Proshasty || 17|| Superhuman AI for multiplayer poker || [https://www.cs.cmu.edu/~noamb/papers/19-Science-Superhuman.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Superhuman_AI_for_Multiplayer_Poker Summary]|| [https://www.youtube.com/watch?v=kazqcOwbtTI Video]<br />
|-<br />
|Week of Nov 23 ||Guanting Pan, Haocheng Chang, Zaiwei Zhang || 18|| Point-of-Interest Recommendation: Exploiting Self-Attentive Autoencoders with Neighbor-Aware Influence || [https://arxiv.org/pdf/1809.10770.pdf Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Point-of-Interest_Recommendation:_Exploiting_Self-Attentive_Autoencoders_with_Neighbor-Aware_Influence Summary] || [https://www.youtube.com/watch?v=aAwjaos_Hus Video]<br />
|-<br />
|Week of Nov 23 || Jerry Huang, Daniel Jiang, Minyan Dai || 19|| Neural Speed Reading Via Skim-RNN ||[https://arxiv.org/pdf/1711.02085.pdf?fbclid=IwAR3EeFsKM_b5p9Ox7X9mH-1oI3U3oOKPBy3xUOBN0XvJa7QW2ZeJJ9ypQVo Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_Speed_Reading_via_Skim-RNN Summary]|| [https://youtu.be/vOENmt9jgVE Video]<br />
|-<br />
|Week of Nov 23 ||Ruixian Chin, Yan Kai Tan, Jason Ong, Wen Cheen Chiew || 20|| DivideMix: Learning with Noisy Labels as Semi-supervised Learning || [https://openreview.net/pdf?id=HJgExaVtwr Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Yktan Summary]|| [https://www.youtube.com/watch?v=48xYZXifjS0&ab_channel=SeakraChin]<br />
|-<br />
|Week of Nov 30 || Banno Dion, Battista Joseph, Kahn Solomon || 21|| Music Recommender System Based on Genre using Convolutional Recurrent Neural Networks || [https://www.sciencedirect.com/science/article/pii/S1877050919310646] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Music_Recommender_System_Based_using_CRNN#Evaluation_of_Music_Recommendation_System: Summary] || [https://youtu.be/eGUV3zwLwqQ]<br />
|-<br />
|Week of Nov 30 || Sai Arvind Budaraju, Isaac Ellmen, Dorsa Mohammadrezaei, Emilee Carson || 22|| A universal SNP and small-indel variant caller using deep neural networks||[https://www.nature.com/articles/nbt.4235.epdf?author_access_token=q4ZmzqvvcGBqTuKyKgYrQ9RgN0jAjWel9jnR3ZoTv0NuM3saQzpZk8yexjfPUhdFj4zyaA4Yvq0LWBoCYQ4B9vqPuv8e2HHy4vShDgEs8YxI_hLs9ov6Y1f_4fyS7kGZ Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_universal_SNP_and_small-indel_variant_caller_using_deep_neural_networks Summary] ||<br />
|-<br />
|Week of Nov 30 || Daniel Fagan, Cooper Brooke, Maya Perelman || 23|| Efficient kNN Classification With Different Number of Nearest Neighbors || [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7898482 Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=User:Dfagan Summary]||<br />
|-<br />
|Week of Nov 30 || Karam Abuaisha, Evan Li, Jason Pu, Nicholas Vadivelu || 24|| Being Bayesian about Categorical Probability || [https://proceedings.icml.cc/static/paper_files/icml/2020/3560-Paper.pdf Paper] || ||<br />
|-<br />
|Week of Nov 30 || Anas Mahdi Will Thibault Jan Lau Jiwon Yang || 25|| Loss Function Search for Face Recognition<br />
|| [https://proceedings.icml.cc/static/paper_files/icml/2020/245-Paper.pdf] paper || Summary [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Loss_Function_Search_for_Face_Recognition] ||<br />
|-<br />
|Week of Nov 30 ||Zihui (Betty) Qin, Wenqi (Maggie) Zhao, Muyuan Yang, Amartya (Marty) Mukherjee || 26|| Deep Learning for Cardiologist-level Myocardial Infarction Detection in Electrocardiograms || [https://arxiv.org/pdf/1912.07618.pdf?fbclid=IwAR0RwATSn4CiT3qD9LuywYAbJVw8YB3nbex8Kl19OCExIa4jzWaUut3oVB0 Paper] || Summary [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Deep_Learning_for_Cardiologist-level_Myocardial_Infarction_Detection_in_Electrocardiograms&fbclid=IwAR1Tad2DAM7LT6NXXuSYDZtHHBvN0mjZtDdCOiUFFq_XwVcQxG3hU-3XcaE] || [https://www.youtube.com/watch?v=kiYbAmd_3IA]<br />
|-<br />
|Week of Nov 30 || Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim || 27|| Improving neural networks by preventing co-adaption of feature detectors || [https://arxiv.org/pdf/1207.0580.pdf paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors Summary] ||<br />
|-<br />
|Week of Nov 30 || Yawen Wang, Danmeng Cui, ZiJie Jiang, Mingkang Jiang, Haotian Ren, Haris Bin Zahid || 28|| A Brief Survey of Text Mining: Classification, Clustering and Extraction Techniques || [https://arxiv.org/pdf/1707.02919.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Describtion_of_Text_Mining Summary] ||<br />
|-<br />
|Week of Nov 30 || Qing Guo, XueGuang Ma, James Ni, Yuanxin Wang || 29|| Mask R-CNN || [https://arxiv.org/pdf/1703.06870.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Mask_RCNN Summary] ||<br />
|-<br />
|Week of Nov 30 || Junyi Yang, Jill Yu Chieh Wang, Yu Min Wu, Calvin Li || 30|| Research paper classifcation systems based on TF‑IDF and LDA schemes || [https://hcis-journal.springeropen.com/articles/10.1186/s13673-019-0192-7?fbclid=IwAR3swO-eFrEbj1BUQfmomJazxxeFR6SPgr6gKayhs38Y7aBG-zX1G3XWYRM Paper] || ||<br />
|-<br />
|Week of Nov 30 || Daniel Zhang, Jacky Yao, Scholar Sun, Russell Parco, Ian Cheung || 31 || Speech2Face: Learning the Face Behind a Voice || [https://arxiv.org/pdf/1905.09773.pdf?utm_source=thenewstack&utm_medium=website&utm_campaign=platform Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Speech2Face:_Learning_the_Face_Behind_a_Voice Summary] ||<br />
|-<br />
|Week of Nov 30 || Siyuan Xia, Jiaxiang Liu, Jiabao Dong, Yipeng Du || 32 || Evaluating Machine Accuracy on ImageNet || [https://proceedings.icml.cc/static/paper_files/icml/2020/6173-Paper.pdf] || ||<br />
|-<br />
|Week of Nov 30 || Mushi Wang, Siyuan Qiu, Yan Yu || 33 || Surround Vehicle Motion Prediction Using LSTM-RNN for Motion Planning of Autonomous Vehicles at Multi-Lane Turn Intersections || [https://ieeexplore.ieee.org/abstract/document/8957421 Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Surround_Vehicle_Motion_Prediction Summary] ||</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47251Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:20:53Z<p>Wtjung: /* Presented by */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Dae Hyun Kim<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
<br />
=== CIFAR-10 Dataset ===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
=== CIFAR-10 ===<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47249Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:19:46Z<p>Wtjung: /* Presented by */</p>
<hr />
<div>== Presented by ==<br />
Stan Lee, Seokho Lim, Kyle Jung, Daehyun Kim<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
=CIFAR-10=<br />
<br />
===Models for CIFAR-10:===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47248Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:18:46Z<p>Wtjung: /* CIFAR-10 */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
=CIFAR-10=<br />
<br />
===Models for CIFAR-10:===<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47247Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:18:14Z<p>Wtjung: </p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
=CIFAR-10=<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47246Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:17:27Z<p>Wtjung: /* CIFAR-10 */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
'''CIFAR-10'''<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47245Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:17:16Z<p>Wtjung: /* CIFAR-10 */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
=== '''CIFAR-10''' ===<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47244Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:16:55Z<p>Wtjung: /* CIFAR-10 */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=3|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation:<br />
[[File:local response norm.png|upright=2|center|]] where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
=== CIFAR-10 ===<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47241Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:15:44Z<p>Wtjung: /* ImageNet dataset */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=2|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation below, where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
local response norm.png<br />
<br />
[[File:local response norm.png|thumb|upright=2|center|alt=text|Figure 3: An RNN and it’s unfolded counterpart]]<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet Dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet Dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47239Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:14:27Z<p>Wtjung: /* ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=2|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation below, where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|600px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47238Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:14:09Z<p>Wtjung: /* ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=2|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation below, where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|800px|center]] <br />
<br />
[[File:layer2.png|500px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47237Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:12:59Z<p>Wtjung: /* ImageNet */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=2|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response normalization layers which are implemented in lateral form and used on neurons with unbounded activations and permits the detection of high-frequency features with a big neuron response. This regularizer encourages competition among neurons belonging to different banks. Normalization is done by dividing the activity of a neuron in bank <math>i</math> at position <math>(x,y)</math> by the equation below, where the sum runs over <math>N</math> ‘adjacent’ banks of neurons at the same position as in the topographic organization of neuron bank. The constants, <math>N</math>, <math>alpha</math> and <math>betas</math> are hyper-parameters whose values are determined using a validation set. This technique is replaced by better techniques such as the combination of dropout and regularization methods (<math>L1</math> and <math>L2</math>)<br />
<br />
=== Neuron nonlinearities ===<br />
<br />
All of the neurons for this model use the max-with-zero nonlinearity where output within a neuron is computed as <math> a^{i}_{x,y} = max(0, z^i_{x,y})</math> where <math> z^i_{x,y} </math> is the total input to the neuron. The reason they use nonlinearity is because it has several advantages over traditional saturating neuron models, such as significant reduction in training time required to reach a certain error rate. Another advantage is that nonlinearity reduces the need for contrast-normalization and data pre-processing since neurons do not saturate- meaning activities simply scale up little by little with usually large input values. For this model’s only pre-processing step, they subtract the mean activity from each pixel and the result is a centered data.<br />
<br />
=== Objective function ===<br />
<br />
The objective function of their network maximizes the multinomial logistic regression objective which is the same as minimizing the average cross-entropy across training cases between the true label and the model’s predicted label.<br />
<br />
=== Weight Initialization === <br />
<br />
It’s important to note that if a neuron always receives a negative value during training, it will not learn because its output is uniformly zero under the max-with-zero nonlinearity. Hence, the weights in their model were sampled from a zero-mean normal distribution with a high enough variance. High variance in weights will set a certain number of neurons with positive values for learning to happen, and in practice, it’s necessary to try out several candidates for variances until a working initialization is found. In their experiment, setting a positive constant, or 1, as biases of the neurons in the hidden layers was helpful in finding it.<br />
<br />
=== Training ===<br />
<br />
In this model, a batch size of 128 samples and momentum of 0.9, we train our model using stochastic gradient descent. The update rule for weight <math>w</math> is $$ v_{i+1} = 0.9v_i + <\frac{dE}{dw_i}> i$$ $$w_{i+1} = w_i + v_{i+1} $$ where <math>i</math> is the iteration index, <math>v</math> is a momentum variable, <math>\epsilon</math> is the learning rate and <math>\frac{dE}{dw}</math> is the average over the <math>i</math>th batch of the derivative of the objective with respect to <math>w_i</math>. The whole training process on CIFAR-10 takes roughly 90minuts and ImageNet takes 4 days with dropout and two days without.<br />
<br />
=== Learning ===<br />
To determine the learning rate for the network, it is a must to start with an equal learning rate for each layer which produces the largest reduction in the objective function with power of ten. Usually, it is in the order of <math>10^{-2}</math> or <math>10^{-3}</math>. In this case, they reduce the learning rate twice by a factor of ten before termination of training.<br />
<br />
= CIFAR-10 =<br />
<br />
Models for CIFAR-10:<br />
<br />
CIFAR-10 is a popular object recognition dataset with size 32 x 32 color images searched from the web. It contains 10 classes and the images were labels with the noun used to search the image. It has images of 6000 train images and 1000 test images of a single dominant object from the label name for each 10 classes.<br />
<br />
They implemented two different models for CIFAR-10, one with dropout and the other without. The one with dropout enables us to use more parameters because dropout forces a strong regularization on the network, and a fourth weight layer is added to take the input from the previous pooling layer. We add a fourth weight layer that is locally connected but not convolutional and this layer contains 16 banks of filters of size 3 × 3 (50% dropout). And then, the softmax layer takes its input from this fourth weight layer.<br />
<br />
The one without dropout is a CNN with three convolutional layers each with a pooling layer. The max-pooling method is performed by the pooling layer which follows the first convolutional layer, and the average-pooling method is performed by remaining 2 pooling layers. The first and second pooling layers with <math>N = 9, α = 0.001</math>, and <math>β = 0.75</math> are followed by response normalization layers.<br />
<br />
A ten-unit softmax layer, which is used to output a probability distribution over class labels, is connected with the upper-most pooling layer. Using filter size of 5×5, all convolutional layers have 64 filter banks.<br />
Thus, with a neural network with 3 convolutional hidden layers with 3 max-pooling layers, the classification error achieved 16.6% to beat 18.5% from the best published error rate without using transformed data. Then, adding one locally-connected layer after these 6 layers and dropout at the last hidden layer produced the error rate of 15.6%.<br />
<br />
[[File:CIFAR-10.png|thumb|upright=2|center|alt=text|Figure 4: CIFAR-10 Sample Dataset]]<br />
<br />
= ImageNet =<br />
<br />
===ImageNet dataset===<br />
<br />
ImageNet is a dataset of millions of high-resolution labeled images in thousands of categories which were collected from the web and labelled by human labellers using MTerk tool (Amazon’s Mechanical Turk crowd-sourcing tool). Because this dataset has millions of labeled images in thousands of categories, it is very difficult to have perfect accuracy on this dataset even for humans because the ImageNet images may contain multiple objects and there are a large number of object classes. ImageNet and CIFAR-10 are very similar, but the scale of ImageNet is about 20 times bigger (1,300,000 vs 60,000). The size of ImageNet is about 1.3 million training images, 50,000 validation images, and 150,000 testing images. They used resized images of 256 x 256 pixels for their experiments.<br />
<br />
'''An ambiguous example to classify:'''<br />
<br />
[[File:imagenet1.png|200px|center]]<br />
<br />
When this paper was written, the best score on this dataset is 45.7% by High-dimensional signature compression for large-scale image classification (J. Sanchez, F. Perronnin, CVPR11 (2011)). The authors of this paper could achieve a comparable performance of 48.6% error using a single neural network with five convolutional hidden layers with a max-pooling layer in between, followed by two globally connected layers and a final 1000-way softmax layer. Also, 42.4% could be achieved by using 50% dropout in the 6th hidden layer.<br />
<br />
'''ImageNet dataset:'''<br />
<br />
[[File:imagenet2.png|400px|center]]<br />
<br />
It was demonstrated that making a large number of decisions was important for the architecture of the net design for the speech recognition (TIMIT) and object recognition datasets (CIFAR-10 and ImageNet). A separate validation set which evaluated the performance of a large number of different architectures was used to make those decisions, and then they chose the best performance architecture with dropout on the validation set so that they could apply it to the real test set.<br />
<br />
===Models for ImageNet===<br />
<br />
The models for ImageNet with dropout (the one without dropout had a similar approach, but there was a serious issue with overfitting): <br />
They used a convolutional neural network trained by 224×224 patches randomly extracted from the 256 × 256 images. It can reduce the network’s capacity to overfit the training data and helps generalization as a form of data augmentation. The method of averaging the prediction of the net on ten 224 × 224 patches of the 256 × 256 input image was used for a testing (patched at the center, the four corner patches, and their horizontal reflections). <br />
<br />
To maximize the performance on the validation set, this complicated network architecture was used and it was found that dropout was very effective. Also, it was demonstrated that using non-convolutional higher layers with the number of parameters worked well with dropout, but it had a negative impact to the performance without dropout.<br />
<br />
[[File:modelh2.png|700px|center]] <br />
<br />
[[File:layer2.png|500px|center]]<br />
<br />
= Conclusion =<br />
<br />
Training with dropout improved the performance...</div>Wtjunghttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Improving_neural_networks_by_preventing_co-adaption_of_feature_detectors&diff=47235Improving neural networks by preventing co-adaption of feature detectors2020-11-28T06:11:25Z<p>Wtjung: /* ImageNet dataset */</p>
<hr />
<div>== Presented by ==<br />
Kyle Jung, Dae Hyun Kim, Seokho Lim, Stan Lee<br />
<br />
= Introduction to Dropout & Dataset =<br />
In this paper, Hinton et al. introduces a novel way to improve neural networks’ performance. By omitting neurons in hidden layers with a probability of 0.5, each hidden unit is prevented from relying on other hidden unit being present during training, hence there are less co-adaptations among them on the training data. Called “dropout,” this process is also an efficient alternative to training many separate networks and average their predictions on the test set.<br />
They used the standard, stochastic gradient descent algorithm and separated training data into mini-batches. An upper bound was set on the L2 norm of incoming weight vector for each hidden neuron, which was normalized if its size exceeds the bound. They found that using a constraint, instead of a penalty, forced model to do a more thorough search of the weight-space, when coupled with the very large learning rate that decays during training. <br />
Their dropout models included all of the hidden neurons, and their outgoing weights were halved to account for the chances of omission. The models were shown to result in lower test error rates on several datasets: MNIST; TIMIT; CIFAR-10; ImageNet; and Reuters Corpus Volume.<br />
<br />
= MNIST =<br />
The MNIST dataset contains 70,000 digit images of size 28 x 28. To see the impact of dropout, they used 4 different neural networks (784-800-800-10, 784-1200-1200-10, 784-2000-2000-10, 784-1200-1200-1200-10), using the same dropout rates as 50% for hidden neurons and 20% for visible neurons. Stochastic gradient descent was used with minibatches of size 100 and a cross-entropy objective function as the loss function. Weights were updated after each minibatch, and training was done for 3000 epochs. An exponentially decaying learning rate <math>\epsilon</math> was used, with the initial value set as 10.0, and it was multiplied by 0.998 at the end of each epoch. At each hidden layer, the incoming weight vector for each hidden neuron was set an upper bound of its length, <math>l</math>, and they found from cross validation that the results were the best when <math>l</math> = 15. Initial weights values were pooled from a normal distribution with mean 0 and standard deviation 0.01. To update weights, an additional variable, ''p'', called momentum, was used to accelerate learning. The initial value of <math>p</math> was 0.5, and it increased linearly to the final value 0.99 during the first 500 epochs, remaining unchanged after. Also, when updating weights, the learning rate was multiplied by <math>1 – p</math>. <math>L</math> denotes the gradient of loss function.<br />
<br />
[[File:weights_mnist.png|center|700px]]<br />
<br />
The best published result for a standard feedforward neural network was 160 errors, and it was reduced to about 130 errors with dropout. By omitting a random 20% of the input pixels, it was further reduced to 110 errors. The following figure visualizes the result.<br />
[[File:mnist_figure.png|center|500px]]<br />
A publicly available pre-trained deep belief net resulted in 118 errors, and it was reduced to 92 errors when the model was fine-tuned with dropout. Another publicly available model was a deep Boltzmann machine, and it resulted in 103, 97, 94, 93 and 88 when the model was fine-tuned using standard backpropagation and was unrolled. They were reduced to 83, 79, 78, 78, and 77 when the model was fine-tuned with dropout – the mean of 79 errors was a record for models that do not use prior knowledge or enhanced training sets.<br />
<br />
= TIMIT = <br />
<br />
Consisting of recordings of 630 speakers of 8 dialects of American English each reading 10 phonetically-rich sentences, the TIMIT is a standard dataset used for evaluation of automatic speech recognition systems. The objective is to convert a given speech signal into a transcription sequence of phones. Hidden Markov Models (HMMs) is an acoustic model that is typically used to deal with variance and determines a level of fit from coefficients of input to each state of HMMs. Recent results show that mapping feedforward neural networks with an acoustic input coupled with a probability distribution over HMM states perform better than the traditional Gaussian mixture models on speech recognition datasets including TIMIT.<br />
<br />
A Neural network was constructed to output the classification error rate on the test set of TIMIT dataset. They have built the neural network with four fully-connected hidden layers with 4000 neurons per layer. The output layer distinguishes distinct classes from one hundred 185 softmax output neurons that are merged into 39 classes. After constructing the neural network, 21 adjacent frames with an advance of 10ms per frame was given as an input. The results show that applying dropout with 50% of hidden units on various neural networks exceed classification performance from the neural networks without dropout. The decoder, a network that knows transition probabilities between HMM states, runs the Viterbi algorithm on class probabilities for each frame from the output of the neural network to predict the best single sequence of HMM states. The classification error achieved 19.7% with dropout and 22.7% without dropout.<br />
<br />
=== Pre-training ===<br />
<br />
Deep Belief Network was used to pretrain the neural network. Since the inputs are real-valued, Gaussian RBM was used for pretraining the first layer. Initializing visible biases with zero, weights were sampled from random numbers that followed normal distribution <math>N(0, 0.01)</math>. Each visible neuron’s variance was set to 1.0 and remained unchanged during training. Minimizing Contrastive Divergence (CD) was used to facilitate learning. Since momentum is used to speed up learning, it was initially set to 0.5 and increased linearly to 0.9 over 20 epochs. The average gradient had 0.001 of a learning rate which was then multiplied by <math>(1-momentum)</math> and L2 weight decay was set to 0.001. After setting up the hyperparameters, the model was done training after 100 epochs. Binary RBMs were used for training all subsequent layers with a learning rate of 0.01. Then, <math>p</math> was set as the mean activation of a neuron in the data set and the visible bias of each neuron was initialized to <math>log(p/(1 − p))</math>. Training each layer with 50 epochs, all remaining hyper-parameters were the same as those for the Gaussian RBM.<br />
<br />
=== Dropout tuning ===<br />
<br />
The initial weights were set in a neural network from the pretrained RBMs. To finetune the network with dropout-backpropagation, momentum was initially set to 0.5 and increased linearly up to 0.9 over 10 epochs. The model had a small constant learning rate of 1.0 and it was used to apply to the average gradient on a minibatch. The model also retained all other hyperparameters the same as the model from MNIST dropout finetuning. The model required approximately 200 epochs to converge. For comparison purpose, they also finetuned the same network with standard backpropagation with a learning rate of 0.1 with the same hyperparameters.<br />
Comparing the performance of dropout with standard backpropagation on several network architectures and input representations, dropout consistently achieved lower error and cross-entropy. Results showed that it significantly controls overfitting, making the method robust to choices of network architecture. It also allowed much larger nets to be trained and removed the need for early stopping. Neural network architectures with dropout are not very sensitive to the choice of learning rate and momentum.<br />
<br />
= Reuters =<br />
Reuters Corpus Volume I archives 804,414 news documents that belong to 103 topics. Under four major themes - corporate/industrial, economics, government/social, and markets – they belonged to 63 classes. After removing 11 classes with no data and one class with insufficient data, they are left with 50 classes and 402,738 documents. The documents were divided into training and test sets equally and randomly, with each document representing the 2000 most frequent words in the dataset, excluding stopwords.<br />
<br />
They trained two neural networks, with size 2000-2000-1000-50, one using dropout and backpropagation, and the other using standard backpropagation. The training hyperparameters are the same as that in MNIST, but training was done for 500 epochs.<br />
<br />
In the following figure, we see the significant improvements by the model with dropout in the test set error. On the right side, we see that the learning with dropout also proceeds smoother. <br />
<br />
[[File:reuters_figure.png|700px|center]]<br />
<br />
= CNN =<br />
<br />
Feed-forward neural networks consist of several layers of neurons where each neuron in a layer applies a linear filter to the input image data and is passed on to the neurons in the next layer. When calculating the neuron’s output, scalar bias aka weights is applied to the filter with nonlinear activation function as parameters of the network that are learned by training data. [[File:cnnbigpicture.jpeg|thumb|upright=2|center|alt=text|Figure: Overview of Convolutional Neural Network]] There are several differences between Convolutional Neural networks and ordinary neural networks. First, CNN’s neurons are organized topographically into a bank and laid out on a 2D grid, so it reflects the organization of dimensions of the input data. Secondly, neurons in CNN apply filters which are local, and which are centered at the neuron’s location in the topographic organization. Meaning that useful metrics or clues to identify the object in an input image which can be found by examining local neighborhoods of the image. Next, all neurons in a bank apply the same filter at different locations in the input image. By looking at the image example. Green is an input to one neuron bank, yellow is filter bank, and pink is the output of one neuron bank (convolved feature). A bank of neurons in a CNN applies a convolution operation, aka filters, to its input where a single layer in a CNN typically has multiple banks of neurons, each performing a convolution with a different filter. The resulting neuron banks become distinct input channels into the next layer. The whole process reduces the net’s representational capacity, but also reduces the capacity to overfit.<br />
[[File:bankofneurons.gif|thumb|upright=2|center|alt=text|Figure: Bank of neurons]]<br />
<br />
=== Pooling ===<br />
<br />
Pooling layer summarizes the activities of local patches of neurons in the convolutional layer by subsampling the output of a convolutional layer. Pooling is useful for extracting dominant features, to decrease the computational power required to process the data through dimensionality reduction. The procedure of pooling goes on like this; output from convolutional layers is divided into sections called pooling units and they are laid out topographically, connected to a local neighborhood of other pooling units from the same convolutional output. Then, each pooling unit is computed with some function which could be maximum and average. Maximum pooling returns the maximum value from the section of the image covered by the pooling unit while average pooling returns the average of all the values inside the pooling unit (see example). In result, there are fewer total pooling units than convolutional unit outputs from the previous layer, this is due to larger spacing between pixels on pooling layers. Using the max-pooling function reduces the effect of outliers and improves generalization.<br />
[[File:maxandavgpooling.jpeg|thumb|upright=2|center|alt=text|Figure: Max pooling and Average pooling]]<br />
<br />
=== Local Response Normalization === <br />
<br />
This network includes local response