http://wiki.math.uwaterloo.ca/statwiki/api.php?action=feedcontributions&user=S3wen&feedformat=atom statwiki - User contributions [US] 2023-02-01T13:09:46Z User contributions MediaWiki 1.28.3 http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Implicit_Causal_Models_for_Genome-wide_Association_Studies&diff=36289 stat946w18/Implicit Causal Models for Genome-wide Association Studies 2018-04-07T22:18:48Z <p>S3wen: /* Real-data Analysis */</p> <hr /> <div>==Introduction and Motivation==<br /> There is currently much progress in probabilistic models which could lead to the development of rich generative models. The models have been applied with neural networks, implicit densities, and with scalable algorithms to very large data for their Bayesian inference. However, most of the models are focused on capturing statistical relationships rather than causal relationships. Causal relationships are relationships where one event is a result of another event, i.e. a cause and effect. Causal models give us a sense of how manipulating the generative process could change the final results. <br /> <br /> Genome-wide association studies (GWAS) are examples of causal relationships. Genome is basically the sum of all DNAs in an organism and contain information about the organism's attributes. Specifically, GWAS is about figuring out how genetic factors cause disease among humans. Here the genetic factors we are referring to are single nucleotide polymorphisms (SNPs), and getting a particular disease is treated as a trait, i.e., the outcome. In order to know about the reason of developing a disease and to cure it, the causation between SNPs and diseases is investigated: first, predict which one or more SNPs cause the disease; second, target the selected SNPs to cure the disease. <br /> <br /> The figure below depicts an example Manhattan plot for a GWAS. Each dot represents an SNP. The x-axis is the chromosome location, and the y-axis is the negative log of the association p-value between the SNP and the disease, so points with the largest values represent strongly associated risk loci.<br /> <br /> [[File:gwas-example.jpg|500px|center]]<br /> <br /> This paper focuses on two challenges to combining modern probabilistic models and causality. The first one is how to build rich causal models with specific needs by GWAS. In general, probabilistic causal models involve a function &lt;math&gt;f&lt;/math&gt; and a noise &lt;math&gt;n&lt;/math&gt;. For working simplicity, we usually assume &lt;math&gt;f&lt;/math&gt; as a linear model with Gaussian noise. However problems like GWAS require models with nonlinear, learnable interactions among the inputs and the noise.<br /> <br /> The second challenge is how to address latent population-based confounders. Latent confounders are issues when we apply the causal models since we cannot observe them nor know the underlying structure. For example, in GWAS, both latent population structure, i.e., subgroups in the population with ancestry differences, and relatedness among sample individuals produce spurious correlations among SNPs to the trait of interest. The existing methods cannot easily accommodate the complex latent structure.<br /> <br /> For the first challenge, the authors develop implicit causal models, a class of causal models that leverages neural architectures with an implicit density. With GWAS, implicit causal models generalize previous methods to capture important nonlinearities, such as gene-gene and gene-population interaction. Building on this, for the second challenge, they describe an implicit causal model that adjusts for population-confounders by sharing strength across examples (genes).<br /> <br /> There has been an increasing number of works on causal models which focus on causal discovery and typically have strong assumptions such as Gaussian processes on noise variable or nonlinearities for the main function.<br /> <br /> ==Implicit Causal Models==<br /> Implicit causal models are an extension of probabilistic causal models. Probabilistic causal models will be introduced first.<br /> <br /> === Probabilistic Causal Models ===<br /> Probabilistic causal models have two parts: deterministic functions of noise and other variables. Consider background noise &lt;math&gt;\epsilon&lt;/math&gt;, representing unknown background quantities which are jointly independent and global variable &lt;math&gt;\beta&lt;/math&gt;, some function of this noise, where<br /> <br /> [[File: eq1.1.png|800px|center]]<br /> <br /> Each &lt;math&gt;\beta&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt; is a function of noise; &lt;math&gt;y&lt;/math&gt; is a function of noise and &lt;math&gt;x&lt;/math&gt;，<br /> <br /> [[File: eqt1.png|800px|center]]<br /> <br /> The target is the causal mechanism &lt;math&gt;f_y&lt;/math&gt; so that the causal effect &lt;math&gt;p(y|do(X=x),\beta)&lt;/math&gt; can be calculated. &lt;math&gt;do(X=x)&lt;/math&gt; means that we specify a value of &lt;math&gt;X&lt;/math&gt; under the fixed structure &lt;math&gt;\beta&lt;/math&gt;. By other paper’s work, it is assumed that &lt;math&gt;p(y|do(x),\beta) = p(y|x, \beta)&lt;/math&gt;.<br /> <br /> [[File: f_1.png|650px|center|]]<br /> <br /> <br /> An example of probabilistic causal models is additive noise model. <br /> <br /> [[File: eq2.1.png|800px|center]]<br /> <br /> &lt;math&gt;f(.)&lt;/math&gt; is usually a linear function or spline functions for nonlinearities. &lt;math&gt;\epsilon&lt;/math&gt; is assumed to be standard normal, as well as &lt;math&gt;y&lt;/math&gt;. Thus the posterior &lt;math&gt;p(\theta | x, y, \beta)&lt;/math&gt; can be represented as <br /> <br /> [[File: eqt2.png|800px|center]]<br /> <br /> where &lt;math&gt;p(\theta)&lt;/math&gt; is the prior which is known. Then, variational inference or MCMC can be applied to calculate the posterior distribution.<br /> <br /> ===Implicit Causal Models===<br /> The difference between implicit causal models and probabilistic causal models is the noise variable. Instead of an additive noise term, implicit causal models directly take noise &lt;math&gt;\epsilon&lt;/math&gt; into a neural network and output &lt;math&gt;x&lt;/math&gt;.<br /> <br /> The causal diagram has changed to:<br /> <br /> [[File: f_2.png|650px|center|]]<br /> <br /> <br /> They used fully connected neural network with a fair amount of hidden units to approximate each causal mechanism. Below is the formal description: <br /> <br /> [[File: theorem.png|650px|center|]]<br /> <br /> <br /> ==Implicit Causal Models with Latent Confounders==<br /> Previously, they assumed the global structure is observed. Next, the unobserved scenario is being considered.<br /> <br /> ===Causal Inference with a Latent Confounder===<br /> Same as before, the interest is the causal effect &lt;math&gt;p(y|do(x_m), x_{-m})&lt;/math&gt;. Here, the SNPs other than &lt;math&gt;x_m&lt;/math&gt; is also under consideration. However, it is confounded by the unobserved confounder &lt;math&gt;z_n&lt;/math&gt;. As a result, the standard inference method cannot be used in this case.<br /> <br /> The paper proposed a new method which include the latent confounders. For each subject &lt;math&gt;n=1,…,N&lt;/math&gt; and each SNP &lt;math&gt;m=1,…,M&lt;/math&gt;,<br /> <br /> [[File: eqt4.png|800px|center]]<br /> <br /> <br /> The mechanism for latent confounder &lt;math&gt;z_n&lt;/math&gt; is assumed to be known. SNPs depend on the confounders and the trait depends on all the SNPs and the confounders as well. <br /> <br /> The posterior of &lt;math&gt;\theta&lt;/math&gt; is needed to be calculate in order to estimate the mechanism &lt;math&gt;g_y&lt;/math&gt; as well as the causal effect &lt;math&gt;p(y|do(x_m), x_{-m})&lt;/math&gt;, so that it can be explained how changes to each SNP &lt;math&gt;X_m&lt;/math&gt; cause changes to the trait &lt;math&gt;Y&lt;/math&gt;.<br /> <br /> [[File: eqt5.png|800px|center]]<br /> <br /> Note that the latent structure &lt;math&gt;p(z|x, y)&lt;/math&gt; is assumed known.<br /> <br /> In general, causal inference with latent confounders can be dangerous: it uses the data twice, and thus it may bias the estimates of each arrow &lt;math&gt;X_m → Y&lt;/math&gt;. Why is this justified? This is answered below:<br /> <br /> '''Proposition 1'''. Assume the causal graph of Figure 2 (left) is correct and that the true distribution resides in some configuration of the parameters of the causal model (Figure 2 (right)). Then the posterior &lt;math&gt;p(θ | x, y)<br /> &lt;/math&gt; provides a consistent estimator of the causal mechanism &lt;math&gt;f_y&lt;/math&gt;.<br /> <br /> Proposition 1 rigorizes previous methods in the framework of probabilistic causal models. The intuition is that as more SNPs arrive (“M → ∞, N fixed”), the posterior concentrates at the true confounders &lt;math&gt;z_n&lt;/math&gt;, and thus we can estimate the causal mechanism given each data point’s confounder &lt;math&gt;z_n&lt;/math&gt;. As more data points arrive (“N → ∞, M fixed”), we can estimate the causal mechanism given any confounder &lt;math&gt;z_n&lt;/math&gt; as there is an infinity of them.<br /> <br /> ===Implicit Causal Model with a Latent Confounder===<br /> This section is the algorithm and functions to implementing an implicit causal model for GWAS.<br /> <br /> ====Generative Process of Confounders &lt;math&gt;z_n&lt;/math&gt;.====<br /> The distribution of confounders is set as standard normal. &lt;math&gt;z_n \in R^K&lt;/math&gt; , where &lt;math&gt;K&lt;/math&gt; is the dimension of &lt;math&gt;z_n&lt;/math&gt; and &lt;math&gt;K&lt;/math&gt; should make the latent space as close as possible to the true population structural. <br /> <br /> ====Generative Process of SNPs &lt;math&gt;x_{nm}&lt;/math&gt;.====<br /> Given SNP is coded for,<br /> <br /> [[File: SNP.png|300px|center]]<br /> <br /> The authors defined a &lt;math&gt;Binomial(2,\pi_{nm})&lt;/math&gt; distribution on &lt;math&gt;x_{nm}&lt;/math&gt;. And used logistic factor analysis to design the SNP matrix.<br /> <br /> [[File: gpx.png|800px|center]]<br /> <br /> A SNP matrix looks like this:<br /> [[File: SNP_matrix.png|200px|center]]<br /> <br /> <br /> Since logistic factor analysis makes strong assumptions, this paper suggests using a neural network to relax these assumptions,<br /> <br /> [[File: gpxnn.png|800px|center]]<br /> <br /> This renders the outputs to be a full &lt;math&gt;N*M&lt;/math&gt; matrix due the the variables &lt;math&gt;w_m&lt;/math&gt;, which act as principal component in PCA. Here, &lt;math&gt;\phi&lt;/math&gt; has a standard normal prior distribution. The weights &lt;math&gt;w&lt;/math&gt; and biases &lt;math&gt;\phi&lt;/math&gt; are shared over the &lt;math&gt;m&lt;/math&gt; SNPs and &lt;math&gt;n&lt;/math&gt; individuals, which makes it possible to learn nonlinear interactions between &lt;math&gt;z_n&lt;/math&gt; and &lt;math&gt;w_m&lt;/math&gt;.<br /> <br /> ====Generative Process of Traits &lt;math&gt;y_n&lt;/math&gt;.====<br /> Previously, each trait is modeled by a linear regression,<br /> <br /> [[File: gpy.png|800px|center]]<br /> <br /> This also has very strong assumptions on SNPs, interactions, and additive noise. It can also be replaced by a neural network which only outputs a scalar,<br /> <br /> [[File: gpynn.png|800px|center]]<br /> <br /> <br /> ==Likelihood-free Variational Inference==<br /> Calculating the posterior of &lt;math&gt;\theta&lt;/math&gt; is the key of applying the implicit causal model with latent confounders.<br /> <br /> [[File: eqt5.png|800px|center]]<br /> <br /> could be reduces to <br /> <br /> [[File: lfvi1.png|800px|center]]<br /> <br /> However, with implicit models, integrating over a nonlinear function could be suffered. The authors applied likelihood-free variational inference (LFVI). LFVI proposes a family of distribution over the latent variables. Here the variables &lt;math&gt;w_m&lt;/math&gt; and &lt;math&gt;z_n&lt;/math&gt; are all assumed to be Normal,<br /> <br /> [[File: lfvi2.png|700px|center]]<br /> <br /> For LFVI applied to GWAS, the algorithm which similar to the EM algorithm has been used:<br /> [[File: em.png|800px|center]]<br /> <br /> ==Empirical Study==<br /> The authors performed simulation on 100,000 SNPs, 940 to 5,000 individuals, and across 100 replications of 11 settings. <br /> Four methods were compared: <br /> <br /> * implicit causal model (ICM);<br /> * PCA with linear regression (PCA); <br /> * a linear mixed model (LMM); <br /> * logistic factor analysis with inverse regression (GCAT).<br /> <br /> The feedforward neural networks for traits and SNPs are fully connected with two hidden layers using ReLU activation function, and batch normalization. <br /> <br /> ===Simulation Study===<br /> Based on real genomic data, a true model is applied to generate the SNPs and traits for each configuration. <br /> There are four datasets used in this simulation study: <br /> <br /> # HapMap [Balding-Nichols model]<br /> # 1000 Genomes Project (TGP) [PCA]<br /> #* Human Genome Diversity project (HGDP) [PCA]<br /> #* HGDP [Pritchard-Stephens-Donelly model] <br /> # A latent spatial position of individuals for population structure [spatial]<br /> <br /> <br /> The table shows the prediction accuracy. The accuracy is calculated by the rate of the number of true positives divide the number of true positives plus false positives. True positives measure the proportion of positives that are correctly identified as such (e.g. the percentage of SNPs which are correctly identified as having the causal relation with the trait). In contrast, false positives state the SNPs has the causal relation with the trait when they don’t. The closer the rate to 1, the better the model is since false positives are considered as the wrong prediction.<br /> <br /> [[File: table_1.png|650px|center|]]<br /> <br /> The result represented above shows that the implicit causal model has the best performance among these four models in every situation. Especially, other models tend to do poorly on PSD and Spatial when &lt;math&gt;a&lt;/math&gt; is small, but the ICM achieved a significantly high rate. The only comparable method to ICM is GCAT, when applying to simpler configurations.<br /> <br /> <br /> ===Real-data Analysis===<br /> They also applied ICM to GWAS of Northern Finland Birth Cohorts, which measure 10 metabolic traits and also contain 324,160 SNPs and 5,027 individuals. The data came from the database of Genotypes and Phenotypes (dbGaP) and used the same preprocessing as Song et al. Ten implicit causal models were fitted, one for each trait to be modeled. For each of the 10 implicit causal models the dimension of the counfounders was set to be six, same as what was used in the paper by Song. The SNP network used 512 hidden units in both layers and the trait network used 32 and 256. et al. for comparable models in Table 2.<br /> <br /> [[File: table_2.png|650px|center|]]<br /> <br /> The numbers in the above table are the number of significant loci for each of the 10 traits. The number for other methods, such as GCAT, LMM, PCA, and &quot;uncorrected&quot; (association tests without accounting for hidden relatedness of study samples) are obtained from other papers. By comparison, the ICM reached the level of the best previous model for each trait.<br /> <br /> ==Conclusion==<br /> This paper introduced implicit causal models in order to account for nonlinear complex causal relationships, and applied the method to GWAS. It can not only capture important interactions between genes within an individual and among population level, but also can adjust for latent confounders by taking account of the latent variables into the model.<br /> <br /> By the simulation study, the authors proved that the implicit causal model could beat other methods by 15-45.3% on a variety of datasets with variations on parameters.<br /> <br /> The authors also believed this GWAS application is only the start of the usage of implicit causal models. The authors suggest that it might also be successfully used in the design of dynamic theories in high-energy physics or for modeling discrete choices in economics.<br /> <br /> ==Critique==<br /> This paper is an interesting and novel work. The main contribution of this paper is to connect the statistical genetics and the machine learning methodology. The method is technically sound and does indeed generalize techniques currently used in statistical genetics.<br /> <br /> The neural network used in this paper is a very simple feed-forward 2 hidden-layer neural network, but the idea of where to use the neural network is crucial and might be significant in GWAS. <br /> <br /> It has limitations as well. The empirical example in this paper is too easy, and far away from the realistic situation. Despite the simulation study showing some competing results, the Northern Finland Birth Cohort Data application did not demonstrate the advantage of using implicit causal model over the previous methods, such as GCAT or LMM.<br /> <br /> Another limitation is about linkage disequilibrium as the authors stated as well. SNPs are not completely independent of each other; usually, they have correlations when the alleles at close locus. They did not consider this complex case, rather they only considered the simplest case where they assumed all the SNPs are independent.<br /> <br /> Furthermore, one SNP maybe does not have enough power to explain the causal relationship. Recent papers indicate that causation to a trait may involve multiple SNPs.<br /> This could be a future work as well.<br /> <br /> ==References==<br /> Tran D, Blei D M. Implicit Causal Models for Genome-wide Association Studies[J]. arXiv preprint arXiv:1710.10742, 2017.<br /> <br /> Patrik O Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Prof Bernhard Schölkopf. Non- linear causal discovery with additive noise models. In Neural Information Processing Systems, 2009.<br /> <br /> Alkes L Price, Nick J Patterson, Robert M Plenge, Michael E Weinblatt, Nancy A Shadick, and David Reich. Principal components analysis corrects for stratification in genome-wide association studies. Nature Genetics, 38(8):904–909, 2006.<br /> <br /> Minsun Song, Wei Hao, and John D Storey. Testing for genetic associations in arbitrarily structured populations. Nature, 47(5):550–554, 2015.<br /> <br /> Dustin Tran, Rajesh Ranganath, and David M Blei. Hierarchical implicit models and likelihood-free variational inference. In Neural Information Processing Systems, 2017.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36288 Dynamic Routing Between Capsules STAT946 2018-04-07T20:43:09Z <p>S3wen: </p> <hr /> <div>= Presented by =<br /> <br /> Yang, Tong(Richard)<br /> <br /> = Contributions =<br /> <br /> This paper introduces the concept of &quot;capsules&quot; and an approach to implement this concept in neural networks. Capsules are groups of neurons used to represent various properties of an entity/object present in the image, such as pose, deformation, and even the existence of the entity. Instead of the obvious representation of a logistic unit for the probability of existence, the paper explores using the length of the capsule output vector to represent existence, and the orientation to represent other properties of the entity. The paper makes the following major contributions:<br /> <br /> * Proposes an alternative to max-pooling called routing-by-agreement.<br /> * Demonstrates a mathematical structure for capsule layers and a routing mechanism. Builds a prototype architecture for capsule networks. <br /> * Presented promising results that confirm the value of Capsnet as a new direction for development in deep learning.<br /> <br /> = Hinton's Critiques on CNN =<br /> <br /> In a past talk , Hinton tried to explain why max-pooling is the biggest problem with current convolutional networks. Here are some highlights from his talk. <br /> <br /> == Four arguments against pooling ==<br /> <br /> * It is a bad fit to the psychology of shape perception: It does not explain why we assign intrinsic coordinate frames to objects and why they have such huge effects. <br /> <br /> * It solves the wrong problem: We want equivariance, not invariance. Disentangling rather than discarding.<br /> <br /> * It fails to use the underlying linear structure: It does not make use of the natural linear manifold that perfectly handles the largest source of variance in images.<br /> <br /> * Pooling is a poor way to do dynamic routing: We need to route each part of the input to the neurons that know how to deal with it. Finding the best routing is equivalent to parsing the image.<br /> <br /> ===Intuition Behind Capsules ===<br /> We try to achieve viewpoint invariance in the activities of neurons by doing max-pooling. Invariance here means that by changing the input a little, the output still stays the same while the activity is just the output signal of a neuron. In other words, when in the input image we shift the object that we want to detect by a little bit, networks activities (outputs of neurons) will not change because of max pooling and the network will still detect the object. But the spacial relationships are not taken care of in this approach so instead capsules are used, because they encapsulate all important information about the state of the features they are detecting in a form of a vector. Capsules encode probability of detection of a feature as the length of their output vector. And the state of the detected feature is encoded as the direction in which that vector points to. So when detected feature moves around the image or its state somehow changes, the probability still stays the same (length of vector does not change), but its orientation changes.<br /> <br /> For example given two sets of hospital records the first of which sorts by [age, weight, height] and the second by [height, age, weight] if we apply the machine learning to this data set it would not preform very well. Capsules aims to solve this problem by routing the information (age, weight, height) to the appropriate neurons.<br /> <br /> == Equivariance ==<br /> <br /> To deal with the invariance problem of CNN, Hinton proposes the concept called equivariance, which is the foundation of capsule concept.<br /> <br /> === Two types of equivariance ===<br /> <br /> ==== Place-coded equivariance ====<br /> If a low-level part moves to a very different position it will be represented by a different capsule.<br /> <br /> ==== Rate-coded equivariance ====<br /> If a part only moves a small distance it will be represented by the same capsule but the pose outputs of the capsule will change.<br /> <br /> Higher-level capsules have bigger domains so low-level place-coded equivariance gets converted into high-level rate-coded equivariance.<br /> <br /> = Dynamic Routing =<br /> <br /> In the second section of this paper, authors give mathematical representations for two key features in routing algorithm in capsule network, which are squashing and agreement. The general setting for this algorithm is between two arbitrary capsules i and j. Capsule j is assumed to be an arbitrary capsule from the first layer of capsules, and capsule i is an arbitrary capsule from the layer below. The purpose of routing algorithm is to generate a vector output for routing decision between capsule j and capsule i. Furthermore, this vector output will be used in the decision for choice of dynamic routing. <br /> <br /> == Routing Algorithm ==<br /> <br /> The routing algorithm is as the following:<br /> <br /> [[File:DRBC_Figure_1.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> In the following sections, each part of this algorithm will be explained in details.<br /> <br /> === Log Prior Probability ===<br /> <br /> &lt;math&gt;b_{ij}&lt;/math&gt; represents the log prior probabilities that capsule i should be coupled to capsule j, and updated in each routing iteration. As line 2 suggests, the initial values of &lt;math&gt;b_{ij}&lt;/math&gt; for all possible pairs of capsules are set to 0. In the very first routing iteration, &lt;math&gt;b_{ij}&lt;/math&gt; equals to zero. For each routing iteration, &lt;math&gt;b_{ij}&lt;/math&gt; gets updated by the value of agreement, which will be explained later.<br /> <br /> === Coupling Coefficient === <br /> <br /> &lt;math&gt;c_{ij}&lt;/math&gt; represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability &lt;math&gt;b_{ij}&lt;/math&gt;. The mathematical transformation is shown below (Equation 3 in paper): <br /> <br /> \begin{align}<br /> c_{ij} = \frac{exp(b_ij)}{\sum_{k}exp(b_ik)}<br /> \end{align}<br /> <br /> &lt;math&gt;c_{ij}&lt;/math&gt; are served as weights for computing the weighted sum and probabilities. Therefore, as probabilities, they have the following properties:<br /> <br /> \begin{align}<br /> c_{ij} \geq 0, \forall i, j<br /> \end{align}<br /> <br /> and, <br /> <br /> \begin{align}<br /> \sum_{i,j}c_{ij} = 1, \forall i, j<br /> \end{align}<br /> <br /> === Predicted Output from Layer Below === <br /> <br /> &lt;math&gt;u_{i}&lt;/math&gt; are the output vector from capsule i in the lower layer, and &lt;math&gt;\hat{u}_{j|i}&lt;/math&gt; are the input vector for capsule j, which are the &quot;prediction vectors&quot; from the capsules in the layer below. &lt;math&gt;\hat{u}_{j|i}&lt;/math&gt; is produced by multiplying &lt;math&gt;u_{i}&lt;/math&gt; by a weight matrix &lt;math&gt;W_{ij}&lt;/math&gt;, such as the following:<br /> <br /> \begin{align}<br /> \hat{u}_{j|i} = W_{ij}u_i<br /> \end{align}<br /> <br /> where &lt;math&gt;W_{ij}&lt;/math&gt; encodes some spatial relationship between capsule j and capsule i.<br /> <br /> === Capsule ===<br /> <br /> By using the definitions from previous sections, the total input vector for an arbitrary capsule j can be defined as:<br /> <br /> \begin{align}<br /> s_j = \sum_{i}c_{ij}\hat{u}_{j|i}<br /> \end{align}<br /> <br /> which is a weighted sum over all prediction vectors by using coupling coefficients.<br /> <br /> === Squashing ===<br /> <br /> The length of &lt;math&gt;s_j&lt;/math&gt; is arbitrary, which is needed to be addressed with. The next step is to convert its length between 0 and 1, since we want the length of the output vector of a capsule to represent the probability that the entity represented by the capsule is present in the current input. The &quot;squashing&quot; process is shown below:<br /> <br /> \begin{align}<br /> v_j = \frac{||s_j||^2}{1+||s_j||^2}\frac{s_j}{||s_j||}<br /> \end{align}<br /> <br /> Notice that &quot;squashing&quot; is not just normalizing the vector into unit length. In addition, it does extra non-linear transformation to ensure that short vectors get shrunk to almost zero length and long vectors get shrunk to a length slightly below 1. The reason for doing this is to make decision of routing, which is called &quot;routing by agreement&quot; much easier to make between capsule layers.<br /> <br /> === Agreement ===<br /> <br /> The final step of a routing iteration is to form an routing agreement &lt;math&gt;a_{ij}&lt;/math&gt;, which is represents as a scalar product:<br /> <br /> \begin{align}<br /> a_{ij} = v_{j} \cdot \hat{u}_{j|i}<br /> \end{align}<br /> <br /> As we mentioned in &quot;squashing&quot; section, the length of &lt;math&gt;v_{j}&lt;/math&gt; is either close to 0 or close to 1, which will effect the magnitude of &lt;math&gt;a_{ij}&lt;/math&gt; in this case. Therefore, the magnitude of &lt;math&gt;a_{ij}&lt;/math&gt; indicate the how strong the routing algorithm agrees on taking the route between capsule j and capsule i. For each routing iteration, the log prior probability, &lt;math&gt;b_{ij}&lt;/math&gt; will be updated by adding the value of its agreement value, which will effect how the coupling coefficients are computed in the next routing iteration. Because of the &quot;squashing&quot; process, we will eventually end up with a capsule j with its &lt;math&gt;v_{j}&lt;/math&gt; close to 1 while all other capsules with its &lt;math&gt;v_{j}&lt;/math&gt; close to 0, which indicates that this capsule j should be activated.<br /> <br /> = CapsNet Architecture =<br /> <br /> The second part of this paper discuss the experiment results from a 3-layer CapsNet, the architecture can be divided into two parts, encoder and decoder. <br /> <br /> == Encoder == <br /> <br /> [[File:DRBC_Architecture.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> === How many routing iteration to use? === <br /> In appendix A of this paper, the authors have shown the empirical results from 500 epochs of training at different choice of routing iterations. According to their observation, more routing iterations increases the capacity of CapsNet but tends to bring additional risk of overfitting. Moreover, CapsNet with routing iterations less than three are not effective in general. As result, they suggest 3 iterations of routing for all experiments.<br /> <br /> === Marginal loss for digit existence ===<br /> <br /> The experiments performed include segmenting overlapping digits on MultiMINST data set, so the loss function has be adjusted for presents of multiple digits. The marginal lose &lt;math&gt;L_k&lt;/math&gt; for each capsule k is calculate by:<br /> <br /> \begin{align}<br /> L_k = T_k max(0, m^+ - ||v_k||)^2 + \lambda(1 - T_k) max(0, ||v_k|| - m^-)^2<br /> \end{align}<br /> <br /> where &lt;math&gt;m^+ = 0.9&lt;/math&gt;, &lt;math&gt;m^- = 0.1&lt;/math&gt;, and &lt;math&gt;\lambda = 0.5&lt;/math&gt;.<br /> <br /> &lt;math&gt;T_k&lt;/math&gt; is an indicator for presence of digit of class k, it takes value of 1 if and only if class k is presented. If class k is not presented, &lt;math&gt;\lambda&lt;/math&gt; down-weight the loss which shrinks the lengths of the activity vectors for all the digit capsules. By doing this, The loss function penalizes the initial learning for all absent digit class, since we would like the top-level capsule for digit class k to have long instantiation vector if and only if that digit class is present in the input.<br /> <br /> === Layer 1: Conv1 === <br /> <br /> The first layer of CapsNet. Similar to CNN, this is just convolutional layer that converts pixel intensities to activities of local feature detectors. <br /> <br /> * Layer Type: Convolutional Layer.<br /> * Input: &lt;math&gt;28 \times 28&lt;/math&gt; pixels.<br /> * Kernel size: &lt;math&gt;9 \times 9&lt;/math&gt;.<br /> * Number of Kernels: 256.<br /> * Activation function: ReLU.<br /> * Output: &lt;math&gt;20 \times 20 \times 256&lt;/math&gt; tensor.<br /> <br /> === Layer 2: PrimaryCapsules ===<br /> <br /> The second layer is formed by 32 primary 8D capsules. By 8D, it means that each primary capsule contains 8 convolutional units with a &lt;math&gt;9 \times 9&lt;/math&gt; kernel and a stride of 2. Each capsule will take a &lt;math&gt;20 \times 20 \times 256&lt;/math&gt; tensor from Conv1 and produce an output of a &lt;math&gt;6 \times 6 \times 8&lt;/math&gt; tensor.<br /> <br /> * Layer Type: Convolutional Layer<br /> * Input: &lt;math&gt;20 \times 20 \times 256&lt;/math&gt; tensor.<br /> * Number of capsules: 32.<br /> * Number of convolutional units in each capsule: 8.<br /> * Size of each convolutional unit: &lt;math&gt;6 \times 6&lt;/math&gt;.<br /> * Output: &lt;math&gt;6 \times 6 \times 8&lt;/math&gt; 8-dimensional vectors.<br /> <br /> === Layer 3: DigitsCaps ===<br /> <br /> The last layer has 10 16D capsules, one for each digit. Not like the PrimaryCapsules layer, this layer is fully connected. Since this is the top capsule layer, dynamic routing mechanism will be applied between DigitsCaps and PrimaryCapsules. The process begins by taking a transformation of predicted output from PrimaryCapsules layer. Each output is a 8-dimensional vector, which needed to be mapped to a 16-dimensional space. Therefore, the weight matrix, &lt;math&gt;W_{ij}&lt;/math&gt; is a &lt;math&gt;8 \times 16&lt;/math&gt; matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform &quot;squashing&quot; to get the output. <br /> <br /> * Layer Type: Fully connected layer.<br /> * Input: &lt;math&gt;6 \times 6 \times 8&lt;/math&gt; 8-dimensional vectors.<br /> * Output: &lt;math&gt;16 \times 10 &lt;/math&gt; matrix.<br /> <br /> === The loss function ===<br /> <br /> The output of the loss function would be a ten-dimensional one-hot encoded vector with 9 zeros and 1 one at the correct position.<br /> <br /> <br /> == Regularization Method: Reconstruction ==<br /> <br /> This is regularization method introduced in the implementation of CapsNet. The method is to introduce a reconstruction loss (scaled down by 0.0005) to margin loss during training. The authors argue this would encourage the digit capsules to encode the instantiation parameters the input digits. All the reconstruction during training is by using the true labels of the image input. The results from experiments also confirms that adding the reconstruction regularizer enforces the pose encoding in CapsNet and thus boots the performance of routing procedure. <br /> <br /> === Decoder ===<br /> <br /> The decoder consists of 3 fully connected layers, each layer maps pixel intensities to pixel intensities. The number of parameters in each layer and the activation functions used are indicated in the figure below:<br /> <br /> [[File:DRBC_Decoder.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> === Result ===<br /> <br /> The authors include some results for CapsNet classification test accuracy to justify the result of reconstruction. We can see that for CapsNet with 1 routing iteration and CapsNet with 3 routing iterations, implement reconstruction shows significant improvements in both MINIST and MultiMINST data set. These improvements show the importance of routing and reconstruction regularizer. <br /> <br /> [[File:DRBC_Reconstruction.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> The decision to use a 3 iteration approach came from experimental results. The image below shows the average logit difference over epochs and at the end for different numbers of routing iterations.<br /> <br /> [[File:DRBC_AvgLogitDiff.png|700px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> The above image shows that the average logit difference decreases at a logarithmic rate according to the number of iterations. As part of this, it was seen that the higher routing iterations lead to overfitting on the training dataset. The following image however, shows that when trained on CIFAR10 the training loss is much lower for the 3 iteration method over the 1 iteration method. From these two evaluations the 3 iteration approach was selected as the most ideal.<br /> <br /> [[File:DRBC_TrainLossIter.png|350px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> = Experiment Results for CapsNet = <br /> <br /> In this part, the authors demonstrate experiment results of CapsNet on different data sets, such as MINIST and different variation of MINST, such as expanded MINST, affNIST, MultiMNIST. Moreover, they also briefly discuss the performance on some other popular data set such CIFAR 10. <br /> <br /> == MINST ==<br /> <br /> === Highlights ===<br /> <br /> * CapsNet archives state-of-the-art performance on MINST with significantly fewer parameters (3-layer baseline CNN model has 35.4M parameters, compared to 8.2M for CapsNet with reconstruction network).<br /> * CapsNet with shallow structure (3 layers) achieves performance that only achieves by deeper network before.<br /> <br /> === Interpretation of Each Capsule ===<br /> <br /> The authors suggest that they found evidence that dimension of some capsule always captures some variance of the digit, while some others represents the global combinations of different variations, this would open some possibility for interpretation of capsules in the future. After computing the activity vector for the correct digit capsule, the authors fed perturbed versions of those activity vectors to the decoder to examine the effect on reconstruction. Some results from perturbations are shown below, where each row represents the reconstructions when one of the 16 dimensions in the DigitCaps representation is tweaked by intervals of 0.05 from the range [-0.25, 0.25]: <br /> <br /> [[File:DRBC_Dimension.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br /> <br /> == affNIST == <br /> <br /> affNIT data set contains different affine transformation of original MINST data set. By the concept of capsule, CapsNet should gain more robustness from its equivariance nature, and the result confirms this. Compare the baseline CNN, CapsNet achieves 13% improvement on accuracy.<br /> <br /> == MultiMNIST ==<br /> <br /> The MultiMNIST is basically the overlapped version of MINIST. An important point to notice here is that this data set is generated by overlaying a digit on top of another digit from the same set but different class. In other words, the case of stacking digits from the same class is not allowed in MultiMINST. For example, stacking a 5 on a 0 is allowed, but stacking a 5 on another 5 is not. The reason is that CapsNet suffers from the &quot;crowding&quot; effect which will be discussed in the weakness of CapsNet section.<br /> <br /> The architecture used for the training is same as the one used for MNIST dataset. However, decay step of the learning rate is 10x larger to account for the larger dataset. Even with the overlap in MultiMNIST, the network is able to segment both digits separately and it shows that the network is able to position and style of the object in the image.<br /> <br /> [[File:multimnist.PNG | 700px|thumb|center|This figure shows some sample reconstructions on the MultiMNIST dataset using CapsNet. CapsNet reconstructs both of the digits in the image in different colours (green and red). It can be seen that the right most images have incorrect classifications with the 9 being classified as a 0 and the 7 being classified as an 8. ]]<br /> <br /> == Other data sets ==<br /> <br /> CapsNet is used on other data sets such as CIFAR10, smallNORB and SVHN. The results are not comparable with state-of-the-art performance, but it is still promising since this architecture is the very first, while other networks have been development for a long time. The authors pointed out one drawback of CapsNet is that they tend to account for everything in the input images - in the CIFAR10 dataset, the image backgrounds were too varied to model in a reasonably sized network, which partly explains the poorer results.<br /> <br /> = Conclusion = <br /> <br /> This paper discuss the specific part of capsule network, which is the routing-by-agreement mechanism. <br /> <br /> The authors suggest this is a great approach to solve the current problem with max-pooling in convolutional neural network. We see that the design of the capsule builds up upon the design of artificial neuron, but expands it to the vector form to allow for more powerful representational capabilities. It also introduces matrix weights to encode important hierarchical relationships between features of different layers. The result succeeds to achieve the goal of the designer: neuronal activity equivariance with respect to changes in inputs and invariance in probabilities of feature detection. <br /> <br /> Moreover, as author mentioned, the approach mentioned in this paper is only one possible implementation of the capsule concept. Approaches like [https://openreview.net/pdf?id=HJWLfGWRb/ this] have also been proposed to test other routing techniques.<br /> <br /> The preliminary results from experiment using a simple shallow CapsNet also demonstrate unparalleled performance that indicates the capsules are a direction worth exploring.<br /> <br /> = Weakness of Capsule Network =<br /> <br /> * Routing algorithm introduces internal loops for each capsule. As number of capsules and layers increases, these internal loops may exponentially expand the training time. <br /> * Capsule network suffers a perceptual phenomenon called &quot;crowding&quot;, which is common for human vision as well. To address this weakness, capsules have to make a very strong representation assumption that at each location of the image, there is at most one instance of the type of entity that capsule represents. This is also the reason for not allowing overlaying digits from same class in generating process of MultiMINST.<br /> * Other criticisms include that the design of capsule networks requires domain knowledge or feature engineering, contrary to the abstraction-oriented goals of deep learning.<br /> <br /> = Implementations = <br /> 1) Tensorflow Implementation : https://github.com/naturomics/CapsNet-Tensorflow<br /> <br /> 2) Keras Implementation. : https://github.com/XifengGuo/CapsNet-Keras<br /> <br /> = References =<br /> # S. Sabour, N. Frosst, and G. E. Hinton, “Dynamic routing between capsules,” arXiv preprint arXiv:1710.09829v2, 2017<br /> # “XifengGuo/CapsNet-Keras.” GitHub, 14 Dec. 2017, github.com/XifengGuo/CapsNet-Keras. <br /> # “Naturomics/CapsNet-Tensorflow.” GitHub, 6 Mar. 2018, github.com/naturomics/CapsNet-Tensorflow.<br /> # Geoffrey Hinton. &quot;What is wrong with convolutional neural nets?&quot;, https://youtu.be/rTawFwUvnLE?t=612</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&diff=36287 stat946w18/Tensorized LSTMs 2018-04-07T19:32:06Z <p>S3wen: /* Introduction */</p> <hr /> <div>= Presented by =<br /> <br /> Chen, Weishi(Edward)<br /> <br /> = Introduction =<br /> <br /> Long Short-Term Memory (LSTM) is a popular approach to boosting the ability of Recurrent Neural Networks to store longer term temporal information. The capacity of an LSTM network can be increased by widening and adding layers. Increasing the width (increasing the number of units in a hidden layer) causes the number of parameters to increase quadratically which in turn increases the time required for model training and evaluation. While increasing the depth by stacking multiple LSTMs increases runtime by a proportional amount. As an alternative He et. al (2017) in ''Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning'' have proposed a model based on LSTM called the '''Tensorized LSTM''' in which the hidden states are represented by '''tensors''' and updated via a '''cross-layer convolution'''. <br /> <br /> * By increasing the tensor size, the network can be widened efficiently without additional parameters since the parameters are shared across different locations in the tensor<br /> * By delaying the output, the network can be deepened implicitly with little additional run-time since deep computations for each time step are merged into temporal computations of the sequence. <br /> <br /> <br /> Also, the paper presents experiments that were conducted on five challenging sequence learning tasks to show the potential of the proposed model.<br /> <br /> = A Quick Introduction to RNN and LSTM =<br /> <br /> We consider the time-series prediction task of producing a desired output &lt;math&gt;y_t&lt;/math&gt; at each time-step t∈ {1, ..., T} given an observed input sequence &lt;math&gt;x_{1:t} = {x_1,x_2, ···, x_t}&lt;/math&gt;, where &lt;math&gt;x_t∈R^R&lt;/math&gt; and &lt;math&gt;y_t∈R^S&lt;/math&gt; are vectors. RNNs learn how to use a hidden state vector &lt;math&gt;h_t ∈ R^M&lt;/math&gt; to encapsulate the relevant features of the entire input history x1:t (indicates all inputs from the initial time-step to the final step before predication - illustration given below) up to time-step t.<br /> <br /> \begin{align}<br /> h_{t-1}^{cat} = [x_t, h_{t-1}] \hspace{2cm} (1)<br /> \end{align}<br /> <br /> Where &lt;math&gt;h_{t-1}^{cat} ∈R^{R+M}&lt;/math&gt; is the concatenation of the current input &lt;math&gt;x_t&lt;/math&gt; and the previous hidden state &lt;math&gt;h_{t−1}&lt;/math&gt;, which expands the dimensionality of intermediate information.<br /> <br /> The update of the hidden state h_t is defined as:<br /> <br /> \begin{align}<br /> a_{t} =h_{t-1}^{cat} W^h + b^h \hspace{2cm} (2)<br /> \end{align}<br /> <br /> and<br /> <br /> \begin{align}<br /> h_t = \Phi(a_t) \hspace{2cm} (3)<br /> \end{align}<br /> <br /> &lt;math&gt;W^h∈R^{(R+M)\times M} &lt;/math&gt; guarantees each hidden state provided by the previous step is of dimension M. &lt;math&gt; a_t ∈R^M &lt;/math&gt; is the hidden activation, and φ(·) is the element-wise hyperbolic tangent. Finally, the output &lt;math&gt; y_t &lt;/math&gt; at time-step t is generated by:<br /> <br /> \begin{align}<br /> y_t = \varphi(h_{t}^{cat} W^y + b^y) \hspace{2cm} (4)<br /> \end{align}<br /> <br /> where &lt;math&gt;W^y∈R^{M×S}&lt;/math&gt; and &lt;math&gt;b^y∈R^S&lt;/math&gt;, and &lt;math&gt;\varphi(·)&lt;/math&gt; can be any differentiable function. Note that the &lt;math&gt;\phi&lt;/math&gt; is a non-linear, element-wise function which generates hidden output, while &lt;math&gt;\varphi&lt;/math&gt; generates the final network output.<br /> <br /> [[File:StdRNN.png|650px|center||Figure 1: Recurrent Neural Network]]<br /> <br /> One shortfall of RNN is the problem of vanishing/exploding gradients. This shortfall is significant, especially when modeling long-range dependencies. One alternative is to instead use LSTM (Long Short-Term Memory), which alleviates these problems by employing several gates to selectively modulate the information flow across each neuron. Since LSTMs have been successfully used in sequence models, it is natural to consider them for accommodating more complex analytical needs.<br /> <br /> [[File:LSTM_Gated.png|650px|center||Figure 2: LSTM]]<br /> <br /> = Structural Measurement of Sequential Model =<br /> <br /> We can consider the capacity of a network consists of two components: the '''width''' (the amount of information handled in parallel) and the depth (the number of computation steps). <br /> <br /> A way to '''widen''' the LSTM is to increase the number of units in a hidden layer; however, the parameter number scales quadratically with the number of units. To deepen the LSTM, the popular Stacked LSTM (sLSTM) stacks multiple LSTM layers. The drawback of sLSTM, however, is that runtime is proportional to the number of layers and information from the input is potentially lost (due to gradient vanishing/explosion) as it propagates vertically through the layers. This paper introduced a way to both widen and deepen the LSTM whilst keeping the parameter number and runtime largely unchanged. In summary, we make the following contributions:<br /> <br /> '''(a)''' Tensorize RNN hidden state vectors into higher-dimensional tensors, to enable more flexible parameter sharing and can be widened more efficiently without additional parameters.<br /> <br /> '''(b)''' Based on (a), merge RNN deep computations into its temporal computations so that the network can be deepened with little additional runtime, resulting in a Tensorized RNN (tRNN).<br /> <br /> '''(c)''' We extend the tRNN to an LSTM, namely the Tensorized LSTM (tLSTM), which integrates a novel memory cell convolution to help to prevent the vanishing/exploding gradients.<br /> <br /> = Method =<br /> <br /> == Part 1: Tensorize RNN hidden State vectors ==<br /> <br /> '''Definition:''' Tensorization is defined as the transformation or mapping of lower-order data to higher-order data. For example, the low-order data can be a vector, and the tensorized result is a matrix, a third-order tensor or a higher-order tensor. The ‘low-order’ data can also be a matrix or a third-order tensor, for example. In the latter case, tensorization can take place along one or multiple modes.<br /> <br /> [[File:VecTsor.png|320px|center||Figure 3: Vector Third-order tensorization of a vector]]<br /> <br /> '''Optimization Methodology Part 1:''' It can be seen that in an RNN, the parameter number scales quadratically with the size of the hidden state. A popular way to limit the parameter number when widening the network is to organize parameters as higher-dimensional tensors which can be factorized into lower-rank sub-tensors that contain significantly fewer elements, which is is known as tensor factorization. <br /> <br /> '''Optimization Methodology Part 2:''' Another common way to reduce the parameter number is to share a small set of parameters across different locations in the hidden state, similar to Convolutional Neural Networks (CNNs).<br /> <br /> '''Effects:''' This '''widens''' the network since the hidden state vectors are in fact broadcast to interact with the tensorized parameters. <br /> <br /> <br /> <br /> We adopt parameter sharing to cutdown the parameter number for RNNs, since compared with factorization, it has the following advantages: <br /> <br /> (i) '''Scalability,''' the number of shared parameters can be set independent of the hidden state size<br /> <br /> (ii) '''Separability,''' the information flow can be carefully managed by controlling the receptive field, allowing one to shift RNN deep computations to the temporal domain<br /> <br /> <br /> <br /> We also explicitly tensorize the RNN hidden state vectors, since compared with vectors, tensors have a better: <br /> <br /> (i) '''Flexibility,''' one can specify which dimensions to share parameters and then can just increase the size of those dimensions without introducing additional parameters<br /> <br /> (ii) '''Efficiency,''' with higher-dimensional tensors, the network can be widened faster w.r.t. its depth when fixing the parameter number (explained later). <br /> <br /> <br /> '''Illustration:''' For ease of exposition, we first consider 2D tensors (matrices): we tensorize the hidden state &lt;math&gt;h_t∈R^{M}&lt;/math&gt; to become &lt;math&gt;Ht∈R^{P×M}&lt;/math&gt;, '''where P is the tensor size,''' and '''M the channel size'''. We locally-connect the first dimension of &lt;math&gt;H_t&lt;/math&gt; (which is P - the tensor size) in order to share parameters, and fully-connect the second dimension of &lt;math&gt;H_t&lt;/math&gt; (which is M - the channel size) to allow global interactions. This is analogous to the CNN which fully-connects one dimension (e.g., the RGB channel for input images) to globally fuse different feature planes. Also, if one compares &lt;math&gt;H_t&lt;/math&gt; to the hidden state of a Stacked RNN (sRNN) (see Figure Blow). <br /> <br /> [[File:Screen_Shot_2018-03-26_at_11.28.37_AM.png|160px|center||Figure 4: Stacked RNN]]<br /> <br /> [[File:ind.png|60px|center||Figure 4: Stacked RNN]]<br /> <br /> Then P is akin to the number of stacked hidden layers (vertical length in the graph), and M the size of each hidden layer (each white node in the graph). We start to describe our model based on 2D tensors, and finally show how to strengthen the model with higher-dimensional tensors.<br /> <br /> == Part 2: Merging Deep Computations ==<br /> <br /> Since an RNN is already deep in its temporal direction, we can deepen an input-to-output computation by associating the input &lt;math&gt;x_t&lt;/math&gt; with a (delayed) future output. In doing this, we need to ensure that the output &lt;math&gt;y_t&lt;/math&gt; is separable, i.e., not influenced by any future input &lt;math&gt;x_{t^{'}}&lt;/math&gt; &lt;math&gt;(t^{'}&gt;t)&lt;/math&gt;. Thus, we concatenate the projection of &lt;math&gt;x_t&lt;/math&gt; to the top of the previous hidden state &lt;math&gt;H_{t−1}&lt;/math&gt;, then gradually shift the input information down when the temporal computation proceeds, and finally generate &lt;math&gt;y_t&lt;/math&gt; from the bottom of &lt;math&gt;H_{t+L−1}&lt;/math&gt;, where L−1 is the number of delayed time-steps for computations of depth L. <br /> <br /> An example with L= 3 is shown in Figure.<br /> <br /> [[File:tRNN.png|160px|center||Figure 5: skewed sRNN]]<br /> <br /> [[File:ind.png|60px|center||Figure 5: skewed sRNN]]<br /> <br /> <br /> This is in fact a skewed sRNN (or tRNN without feedback). However, the method does not need to change the network structure and also allows different kinds of interactions as long as the output is separable; for example, one can increase the local connections and '''use feedback''' (shown in figure below), which can be beneficial for sRNNs (or tRNN). <br /> <br /> [[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br /> <br /> [[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br /> <br /> '''In order to share parameters, we update &lt;math&gt;H_t&lt;/math&gt; using a convolution with a learnable kernel.''' In this manner we increase the complexity of the input-to-output mapping (by delaying outputs) and limit parameter growth (by sharing transition parameters using convolutions).<br /> <br /> To examine the resulting model mathematically, let &lt;math&gt;H^{cat}_{t−1}∈R^{(P+1)×M}&lt;/math&gt; be the concatenated hidden state, and &lt;math&gt;p∈Z_+&lt;/math&gt; the location at a tensor. The channel vector &lt;math&gt;h^{cat}_{t−1, p }∈R^M&lt;/math&gt; at location p of &lt;math&gt;H^{cat}_{t−1}&lt;/math&gt; (the p-th channel of H) is defined as:<br /> <br /> \begin{align}<br /> h^{cat}_{t-1, p} = x_t W^x + b^x \hspace{1cm} if p = 1 \hspace{1cm} (5)<br /> \end{align}<br /> <br /> \begin{align}<br /> h^{cat}_{t-1, p} = h_{t-1, p-1} \hspace{1cm} if p &gt; 1 \hspace{1cm} (6)<br /> \end{align}<br /> <br /> where &lt;math&gt;W^x ∈ R^{R×M}&lt;/math&gt; and &lt;math&gt;b^x ∈ R^M&lt;/math&gt; (recall the dimension of input x is R). Then, the update of tensor &lt;math&gt;H_t&lt;/math&gt; is implemented via a convolution:<br /> <br /> \begin{align}<br /> A_t = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (7)<br /> \end{align}<br /> <br /> \begin{align}<br /> H_t = \Phi{A_t} \hspace{2cm} (8)<br /> \end{align}<br /> <br /> where &lt;math&gt;W^h∈R^{K×M^i×M^o}&lt;/math&gt; is the kernel weight of size K, with &lt;math&gt;M^i =M&lt;/math&gt; input channels and &lt;math&gt;M^o =M&lt;/math&gt; output channels, &lt;math&gt;b^h ∈ R^{M^o}&lt;/math&gt; is the kernel bias, &lt;math&gt;A_t ∈ R^{P×M^o}&lt;/math&gt; is the hidden activation, and &lt;math&gt;\circledast&lt;/math&gt; is the convolution operator. Since the kernel convolves across different hidden layers, we call it the cross-layer convolution. The kernel enables interaction, both bottom-up and top-down across layers. Finally, we generate &lt;math&gt;y_t&lt;/math&gt; from the channel vector &lt;math&gt;h_{t+L−1,P}∈R^M&lt;/math&gt; which is located at the bottom of &lt;math&gt;H_{t+L−1}&lt;/math&gt;:<br /> <br /> \begin{align}<br /> y_t = \varphi(h_{t+L−1}, _PW^y + b^y) \hspace{2cm} (9)<br /> \end{align}<br /> <br /> Where &lt;math&gt;W^y ∈R^{M×S}&lt;/math&gt; and &lt;math&gt;b^y ∈R^S&lt;/math&gt;. To guarantee that the receptive field of &lt;math&gt;y_t&lt;/math&gt; only covers the current and previous inputs x1:t. (Check the Skewed sRNN again below):<br /> <br /> [[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br /> <br /> [[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br /> <br /> === Quick Summary of Set of Parameters ===<br /> <br /> '''1. &lt;math&gt; W^x&lt;/math&gt; and &lt;math&gt;b_x&lt;/math&gt;''' connect input to the first hidden node<br /> <br /> '''2. &lt;math&gt; W^h&lt;/math&gt; and &lt;math&gt;b_h&lt;/math&gt;''' convolute between layers<br /> <br /> '''3. &lt;math&gt; W^y&lt;/math&gt; and &lt;math&gt;b_y&lt;/math&gt;''' produce output of each stages<br /> <br /> <br /> == Part 3: Extending to LSTMs==<br /> <br /> Similar to standard RNN, to allow the tRNN (skewed sRNN) to capture long-range temporal dependencies, one can straightforwardly extend it<br /> to a tLSTM by replacing the tRNN tensors:<br /> <br /> \begin{align}<br /> [A^g_t, A^i_t, A^f_t, A^o_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (10)<br /> \end{align}<br /> <br /> \begin{align}<br /> [G_t, I_t, F_t, O_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t)] \hspace{2cm} (11)<br /> \end{align}<br /> <br /> Which are pretty similar to tRNN case, the main differences can be observes for memory cells of tLSTM (Ct):<br /> <br /> \begin{align}<br /> C_t= G_t \odot I_t + C_{t-1} \odot F_t \hspace{2cm} (12)<br /> \end{align}<br /> <br /> \begin{align}<br /> H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (13)<br /> \end{align}<br /> <br /> Note that since the previous memory cell &lt;math&gt;C_{t-1}&lt;/math&gt; is only gated along the temporal direction, increasing the tensor size ''P'' might result in the loss of long-range dependencies from the input to the output.<br /> <br /> Summary of the terms: <br /> <br /> 1. '''&lt;math&gt;\{W^h, b^h \}&lt;/math&gt;:''' Kernel of size K <br /> <br /> 2. '''&lt;math&gt;A^g_t, A^i_t, A^f_t, A^o_t \in \mathbb{R}^{P\times M}&lt;/math&gt;:''' Activations for the new content &lt;math&gt;G_t&lt;/math&gt;<br /> <br /> 3. '''&lt;math&gt;I_t&lt;/math&gt;:''' Input gate<br /> <br /> 4. '''&lt;math&gt;F_t&lt;/math&gt;:''' Forget gate<br /> <br /> 5. '''&lt;math&gt;O_t&lt;/math&gt;:''' Output gate<br /> <br /> 6. '''&lt;math&gt;C_t \in \mathbb{R}^{P\times M}&lt;/math&gt;:''' Memory cell<br /> <br /> Then, see graph below for illustration:<br /> <br /> [[File:tLSTM_wo_MC.png ‎|160px|center||Figure 5: tLSTM wo MC]]<br /> <br /> [[File:ind.png|60px|center||Figure 5: tLSTM wo MC]]<br /> <br /> To further evolve tLSTM, we invoke the '''Memory Cell Convolution''' to capture long-range dependencies from multiple directions, we additionally introduce a novel memory cell convolution, by which the memory cells can have a larger receptive field (figure provided below). <br /> <br /> [[File:tLSTM_w_MC.png ‎|160px|center||Figure 5: tLSTM w MC]]<br /> <br /> [[File:ind.png|60px|center||Figure 5: tLSTM w MC]]<br /> <br /> One can also dynamically generate this convolution kernel so that it is both time - and location-dependent, allowing for flexible control over long-range dependencies from different directions. Mathematically, it can be represented in with the following formulas:<br /> <br /> \begin{align}<br /> [A^g_t, A^i_t, A^f_t, A^o_t, A^q_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (14)<br /> \end{align}<br /> <br /> \begin{align}<br /> [G_t, I_t, F_t, O_t, Q_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t), ς(A^q_t)] \hspace{2cm} (15)<br /> \end{align}<br /> <br /> \begin{align}<br /> W_t^c(p) = reshape(q_{t,p}, [K, 1, 1]) \hspace{2cm} (16)<br /> \end{align}<br /> <br /> \begin{align}<br /> C_{t-1}^{conv}= C_{t-1} \circledast W_t^c(p) \hspace{2cm} (17)<br /> \end{align}<br /> <br /> \begin{align}<br /> C_t= G_t \odot I_t + C_{t-1}^{conv} \odot F_t \hspace{2cm} (18)<br /> \end{align}<br /> <br /> \begin{align}<br /> H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (19)<br /> \end{align}<br /> <br /> where the kernel &lt;math&gt;{W^h, b^h}&lt;/math&gt; has additional &lt;K&gt; output channels to generate the activation &lt;math&gt;A^q_t ∈ R^{P×&lt;K&gt;}&lt;/math&gt; for the dynamic kernel bank &lt;math&gt;Q_t∈R^{P × &lt;K&gt;}&lt;/math&gt;, &lt;math&gt;q_{t,p}∈R^{&lt;K&gt;}&lt;/math&gt; is the vectorized adaptive kernel at the location p of &lt;math&gt;Q_t&lt;/math&gt;, and &lt;math&gt;W^c_t(p) ∈ R^{K×1×1}&lt;/math&gt; is the dynamic kernel of size K with a single input/output channel, which is reshaped from &lt;math&gt;q_{t,p}&lt;/math&gt;. Each channel of the previous memory cell &lt;math&gt;C_{t-1}&lt;/math&gt; is convolved with &lt;math&gt;W_t^c(p)&lt;/math&gt; whose values vary with &lt;math&gt;p&lt;/math&gt;, to form a memory cell convolution, which produces a convolved memory cell &lt;math&gt;C_{t-1}^{conv} \in \mathbb{R}^{P\times M}&lt;/math&gt;. This convolution is defined by:<br /> <br /> \begin{align}<br /> C_{t-1,p,m}^{conv} = \sum\limits_{k=1}^K C_{t-1,p-\frac{K-1}{2}+k,m} · W_{t,k,1,1}^c(p) \hspace{2cm} (30)<br /> \end{align}<br /> <br /> where &lt;math&gt;C_{t-1}&lt;/math&gt; is padded with the boundary values to retain the stored information.<br /> <br /> Note the paper also employed a softmax function ς(·) to normalize the channel dimension of &lt;math&gt;Q_t&lt;/math&gt;. which can also stabilize the value of memory cells and help to prevent the vanishing/exploding gradients. An illustration is provided below to better illustrate the process:<br /> <br /> [[File:MCC.png ‎|240px|center||Figure 5: MCC]]<br /> <br /> Theorem 17-18 of Leifert et al.  proves the prevention of vanishing/exploding gradients for the lambda gate, which is very similar to the proposed memory cell convolution kernel. The only major differences between the two are the use of softmax for normalization and the sharing the of the kernal for all channels. Since these changes to not affect the assertions made in Theorem 17-18, it can be established that the prevention of vanishing/exploding gradients is a feature of the memory cell convolution kernel as well.<br /> <br /> To improve training, the authors introduced a new normalization technique for ''t''LSTM termed channel normalization (adapted from layer normalization), in which the channel vector are normalized at different locations with their own statistics. Note that layer normalization does not work well with ''t''LSTM, because lower level information is near the input and higher level information is near the output. Channel normalization (CN) is defined as: <br /> <br /> \begin{align}<br /> \mathrm{CN}(\mathbf{Z}; \mathbf{\Gamma}, \mathbf{B}) = \mathbf{\hat{Z}} \odot \mathbf{\Gamma} + \mathbf{B} \hspace{2cm} (20)<br /> \end{align}<br /> <br /> where &lt;math&gt;\mathbf{Z}&lt;/math&gt;, &lt;math&gt;\mathbf{\hat{Z}}&lt;/math&gt;, &lt;math&gt;\mathbf{\Gamma}&lt;/math&gt;, &lt;math&gt;\mathbf{B} \in \mathbb{R}^{P \times M^z}&lt;/math&gt; are the original tensor, normalized tensor, gain parameter and bias parameter. The &lt;math&gt;m^z&lt;/math&gt;-th channel of &lt;math&gt;\mathbf{Z}&lt;/math&gt; is normalized element-wisely: <br /> <br /> \begin{align}<br /> \hat{z_{m^z}} = (z_{m^z} - z^\mu)/z^{\sigma} \hspace{2cm} (21)<br /> \end{align}<br /> <br /> where &lt;math&gt;z^{\mu}&lt;/math&gt;, &lt;math&gt;z^{\sigma} \in \mathbb{R}^P&lt;/math&gt; are the mean and standard deviation along the channel dimension of &lt;math&gt;\mathbf{Z}&lt;/math&gt;, and &lt;math&gt;\hat{z_{m^z}} \in \mathbb{R}^P&lt;/math&gt; is the &lt;math&gt;m^z&lt;/math&gt;-th channel &lt;math&gt;\mathbf{\hat{Z}}&lt;/math&gt;. Channel normalization introduces very few additional parameters compared to the number of other parameters in the model.<br /> <br /> = Results and Evaluation =<br /> <br /> Summary of list of models tLSTM family (may be useful later):<br /> <br /> (a) sLSTM (baseline): the implementation of sLSTM with parameters shared across all layers.<br /> <br /> (b) 2D tLSTM: the standard 2D tLSTM.<br /> <br /> (c) 2D tLSTM–M: removing memory (M) cell convolutions from (b).<br /> <br /> (d) 2D tLSTM–F: removing (–) feedback (F) connections from (b).<br /> <br /> (e) 3D tLSTM: tensorizing (b) into 3D tLSTM.<br /> <br /> (f) 3D tLSTM+LN: applying (+) Layer Normalization.<br /> <br /> (g) 3D tLSTM+CN: applying (+) Channel Normalization.<br /> <br /> === Efficiency Analysis ===<br /> <br /> '''Fundaments:''' For each configuration, fix the parameter number and increase the tensor size to see if the performance of tLSTM can be boosted without increasing the parameter number. Can also investigate how the runtime is affected by the depth, where the runtime is measured by the average GPU milliseconds spent by a forward and backward pass over one timestep of a single example. <br /> <br /> '''Dataset:''' The Hutter Prize Wikipedia dataset consists of 100 million characters taken from 205 different characters including alphabets, XML markups and special symbols. We model the dataset at the character-level, and try to predict the next character of the input sequence.<br /> <br /> All configurations are evaluated with depths L = 1, 2, 3, 4. Bits-per-character(BPC) is used to measure the model performance and the results are shown in the figure below.<br /> [[File:wiki.png ‎|280px|center||Figure 5: WifiPerf]]<br /> [[File:Wiki_Performance.png ‎|480px|center||Figure 5: WifiPerf]]<br /> <br /> === Accuracy Analysis ===<br /> <br /> The MNIST dataset  consists of 50000/10000/10000 handwritten digit images of size 28×28 for training/validation/test. Two tasks are used for evaluation on this dataset:<br /> <br /> (a) '''Sequential MNIST:''' The goal is to classify the digit after sequentially reading the pixels in a scan-line order. It is therefore a 784 time-step sequence learning task where a single output is produced at the last time-step; the task requires very long range dependencies in the sequence.<br /> <br /> (b) '''Sequential Permuted MNIST:''' We permute the original image pixels in a fixed random order, resulting in a permuted MNIST (pMNIST) problem that has even longer range dependencies across pixels and is harder.<br /> <br /> In both tasks, all configurations are evaluated with M = 100 and L= 1, 3, 5. The model performance is measured by the classification accuracy and results are shown in the figure below.<br /> <br /> [[File:MNISTperf.png ‎|480px|center]]<br /> <br /> <br /> <br /> [[File:Acc_res.png ‎|480px|center||Figure 5: MNIST]]<br /> <br /> [[File:33_mnist.PNG|center|thumb|800px| This figure displays a visualization of the means of the diagonal channels of the tLSTM memory cells per task. The columns indicate the time steps and the rows indicate the diagonal locations. The values are normalized between 0 and 1.]]<br /> <br /> It can be seen in the above figure that tLSTM behaves differently with different tasks:<br /> <br /> - Wikipedia: the input can be carried to the output location with less modification if it is sufficient to determine the next character, and vice versa<br /> <br /> - addition: the first integer is gradually encoded into memories and then interacts (performs addition) with the second integer, producing the sum <br /> <br /> - memorization: the network behaves like a shift register that continues to move the input symbol to the output location at the correct timestep<br /> <br /> - sequential MNIST: the network is more sensitive to the pixel value change (representing the contour, or topology of the digit) and can gradually accumulate evidence for the final prediction <br /> <br /> - sequential pMNIST: the network is sensitive to high value pixels (representing the foreground digit), and we conjecture that this is because the permutation destroys the topology of the digit, making each high value pixel potentially important.<br /> <br /> From the figure above it can can also be observe some common phenomena in all tasks: <br /> # it is clear that wider (larger) tensors can encode more information by observing that at each timestep, the values at different tensor locations are markedly different<br /> # from the input to the output, the values become increasingly distinct and are shifted by time, revealing that deep computations are indeed performed together with temporal computations, with long-range dependencies carried by memory cells.<br /> <br /> <br /> = Conclusions =<br /> <br /> The paper introduced the Tensorized LSTM, which employs tensors to share parameters and utilizes the temporal computation to perform the deep computation for sequential tasks. Then validated the model<br /> on a variety of tasks, showing its potential over other popular approaches. The paper shows a method to widen and deepen the LSTM network at the same time and the following 3 points list their main contributions:<br /> * The RNNs are now tensorized into higher dimensional tensors which are more flexible.<br /> * RNNs' deep computation is merged into the temporal computation, referred to as the tensorizedRNN.<br /> * tRNN is extended to a LSTM architecture and a new architecture is studied: tensorizedLSTM<br /> <br /> = Critique(to be edited) =<br /> <br /> * Using tensor as hidden layer indeed increasing the capability of the network, but authors never mentioned the trade-off in terms of extra computation cost and training time.<br /> <br /> = References =<br /> #Zhen He, Shaobing Gao, Liang Xiao, Daxue Liu, Hangen He, and David Barber. &lt;Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning&gt; (2017)<br /> #Ali Ghodsi, &lt;Deep Learning: STAT 946 - Winter 2018&gt;<br /> #Gundram Leifert, Tobias Strauß, Tobias Grüning, Welf Wustlich, and Roger Labahn. Cells in multidimensional recurrent neural networks. JMLR, 17(1):3313–3349, 2016.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Do_Deep_Neural_Networks_Suffer_from_Crowding&diff=36253 Do Deep Neural Networks Suffer from Crowding 2018-04-07T01:30:12Z <p>S3wen: /* Critique */</p> <hr /> <div>= Introduction =<br /> Since the increase in popularity of Deep Neural Networks (DNNs), there has been increased research in making machines capable of recognizing objects the same way humans do. Humans can recognize objects in ways that are invariant to scale, translation, and clutter. Crowding is visual effect suffered by humans, in which an object that can be recognized in isolation can no longer be recognized when other objects, called flankers, are placed close to it. This paper focuses on studying the impact of crowding on DNNs trained for object recognition by adding clutter to the images and then analyzing which models and settings suffer less from such effects. <br /> <br /> [[File:paper25_fig_crowding_ex.png|center|600px]]<br /> The figure shows a visual example of crowding . Keep your eyes still and look at the dot in the center and try to identify the &quot;A&quot; in the two circles. You should see that it is much easier to make out the &quot;A&quot; in the right than in the left circle. The same &quot;A&quot; exists in both circles, however, the left circle contains flankers which are those line segments.<br /> <br /> Another common example to visualize the same:<br /> [[File:crowding-tigger.jpg|center|600px]]<br /> <br /> <br /> ===Drawbacks of CNNs===<br /> CNNs fall short in explaining human perceptual invariance. Firstly, CNNs typically take input at a single uniform resolution. Biological measurements suggest that resolution is not uniform across the human visual field, but rather decays with eccentricity, i.e. distance from the center of focus. The major cause of this issue is the pooling layer in CNN structure. The pooling is an efficient technique but loses important spatial information. Pooling is also not capable to capture the hierarchical structure in image, which is also crucial to view point problems. Even more importantly, CNNs rely not only on weight-sharing but also on data augmentation to achieve transformation invariance and so obviously a lot of processing is needed for CNNs. <br /> <br /> The paper investigates two types of DNNs for crowding: traditional deep convolutional neural networks (DCNN) and a multi-scale eccentricity-dependent model which is an extension of the DCNNs and inspired by the retina where the receptive field size of the convolutional filters in the model grows with increasing distance from the center of the image, called the eccentricity and is explained below. The authors focus on the dependence of crowding on image factors, such as flanker configuration, target-flanker similarity, target eccentricity and premature pooling in particular. Along with that, there is major emphasis on reducing the training time of the networks since the motive is to have a simple network capable of learning space-invariant features.<br /> <br /> = Models =<br /> The authors describe two kinds of DNN architectures: Deep Convolutional Neural Networks, and eccentricity dependent networks, with varying pooling strategies across space and scale. Of particular note is the pooling operation, as many researchers have suggested that this may be the cause of crowding in human perception.<br /> <br /> == Deep Convolutional Neural Networks ==<br /> The DCNN is a basic architecture with 3 convolutional layers, spatial 3x3 max-pooling with varying strides and a fully connected layer for classification as shown in the below figure. <br /> [[File:DCNN.png|800px|center]]<br /> <br /> The network is fed with images resized to 60x60, with mini-batches of 128 images, 32 feature channels for all convolutional layers, and convolutional filters of size 5x5 and stride 1.<br /> <br /> As highlighted earlier, the effect of pooling is into main consideration and hence three different configurations have been investigated as below: <br /> <br /> # '''No total pooling''' Feature maps sizes decrease only due to boundary effects, as the 3x3 max pooling has stride 1. The square feature maps sizes after each pool layer are 60-54-48-42.<br /> # '''Progressive pooling''' 3x3 pooling with a stride of 2 halves the square size of the feature maps, until we pool over what remains in the final layer, getting rid of any spatial information before the fully connected layer. (60-27-11-1).<br /> # '''At end pooling''' Same as no total pooling, but before the fully connected layer, max-pool over the entire feature map. (60-54-48-1).<br /> <br /> ==Eccentricity-dependent Model==<br /> In order to take care of the scale invariance in the input image, the eccentricity dependent DNN is utilized. This was proposed as a model of the human visual cortex by [https://arxiv.org/pdf/1406.1770.pdf, Poggio et al] and later further studied in . The main intuition behind this architecture is that as we increase eccentricity, the receptive fields also increase and hence the model will become invariant to changing input scales. The authors note that the width of each scale is roughly related to the amount of translation invariance for objects at that scale, simply because once the object is outside that window, the filter no longer observes it. Therefore, the authors say that the architecture emphasizes scale invariance over translation invariance, in contrast to traditional DCNNs. From a biological perspective, eye movement can compensate for the limitations of translation invariance, but compensating for scale invariance requires changing distance from the object. In this model, the input image is cropped into varying scales (11 crops increasing by a factor of &lt;math&gt;\sqrt{2}&lt;/math&gt; which are then resized to 60x60 pixels) and then fed to the network. Exponentially interpolated crops are used over linearly interpolated crops since they produce fewer boundary effects while maintaining the same behavior qualitatively. The model computes an invariant representation of the input by sampling the inverted pyramid at a discrete set of scales with the same number of filters at each scale. Since the same number of filters are used for each scale, the smaller crops will be sampled at a high resolution while the larger crops will be sampled with a low resolution. These scales are fed into the network as an input channel to the convolutional layers and share the weights across scale and space. Due to the downsampling of the input image, this is equivalent to having receptive fields of varying sizes. Intuitively, this means that the network generalizes learnings across scales and is guaranteed by during back-propagation by averaging the error derivatives over all scale channels, then using the averages to compute weight adjustments. The same set of weight adjustments to the convolutional units across different scale channels is applied.<br /> [[File:EDM.png|2000x450px|center]]<br /> <br /> <br /> The architecture of this model is the same as the previous DCNN model with the only change being the extra filters added for each of the scales, so the number of parameters remains the same as DCNN models. The authors perform spatial pooling, the aforementioned ''At end pooling'' is used here, and scale pooling which helps in reducing the number of scales by taking the maximum value of corresponding locations in the feature maps across multiple scales. It has three configurations: (1) at the beginning, in which all the different scales are pooled together after the first layer, 11-1-1-1-1 (2) progressively, 11-7-5-3-1 and (3) at the end, 11-11-11-11-1, in which all 11 scales are pooled together at the last layer.<br /> <br /> ===Contrast Normalization===<br /> Since there are multiple scales of an input image, in some experiments, normalization is performed such that the sum of the pixel intensities in each scale is in the same range [0,1] (this is to prevent smaller crops, which have more non-black pixels, from disproportionately dominating max-pooling across scales). The normalized pixel intensities are then divided by a factor proportional to the crop area [[File:sqrtf.png|60px]] where i=1 is the smallest crop.<br /> <br /> =Experiments=<br /> Targets are the set of objects to be recognized and flankers are the set of objects the model has not been trained to recognize, which act as clutter with respect to these target objects. The target objects are the even MNIST numbers having translational variance (shifted at different locations of the image along the horizontal axis), while flankers are from odd MNIST numbers, not MNIST dataset (contains alphabet letters) and Omniglot dataset (contains characters). Examples of the target and flanker configurations are shown below: <br /> [[File:eximages.png|800px|center]]<br /> <br /> The target and the object are referred to as ''a'' and ''x'' respectively with the below four configurations: <br /> # No flankers. Only the target object. (a in the plots) <br /> # One central flanker closer to the center of the image than the target. (xa) <br /> # One peripheral flanker closer to the boundary of the image that the target. (ax) <br /> # Two flankers spaced equally around the target, being both the same object, see Figure 1 above for an example (xax).<br /> <br /> Training is done using backpropagation with images of size &lt;math&gt;1920 px^2&lt;/math&gt; with embedded targets objects and flankers of size of &lt;math&gt;120 px^2&lt;/math&gt;. The training and test images are divided as per the usual MNIST configuration. To determine if there is a difference between the peripheral flankers and the central flankers, all the tests are performed in the right half image plane.<br /> <br /> ==DNNs trained with Target and Flankers==<br /> This is a constant spacing training setup where identical flankers are placed at a distance of 120 pixels either side of the target(xax) with the target having translational variance. The tests are evaluated on (i) DCNN with at the end pooling, and (ii) eccentricity-dependent model with 11-11-11-11-1 scale pooling, at the end spatial pooling and contrast normalization. The results are reported by different flanker types &lt;math&gt;(xax,ax, xa)&lt;/math&gt; at test. <br /> [[File:result1.png|x450px|center]]<br /> <br /> ===Observations===<br /> * With the flanker configuration same as the training one, models are better at recognizing objects in clutter rather than isolated objects for all image locations<br /> * If the target-flanker spacing is changed, then models perform worse<br /> * the eccentricity model is much better at recognizing objects in isolation than the DCNN because the multi-scale crops divide the image into discrete regions, letting the model learn from image parts as well as the whole image<br /> * Only the eccentricity-dependent model is robust to different flanker configurations not included in training when the target is centered.<br /> <br /> ==DNNs trained with Images with the Target in Isolation==<br /> Here the target objects are in isolation and with translational variance while the test-set is the same set of flanker configurations as used before. The constant spacing and constant eccentricity effect have been evaluated.<br /> <br /> [[File:result2.png|750x400px|center]]<br /> <br /> In addition to the evaluation of DCNNs in constant target eccentricity at 240 pixels, here they are tested with images in which the target is fixed at 720 pixels from the center of the image, as shown in Fig 3. Since the target is already at the edge of the visual field, a flanker cannot be more peripheral in the image than the target. Same results as for the 240 pixels target eccentricity can be extracted. The closer the flanker is to the target, the more accuracy decreases. Also, it can be seen that when the target is close to the image boundary, recognition is poor because of boundary effects eroding away information about the target.<br /> <br /> [[File:paper25_supplemental1.png|800px|center]]<br /> <br /> The authors also test the effect of flankers from different datasets on a DCNN model with at end pooling, with results shown in Fig. 7 below. Omniglot flankers crowd less than MNIST digits, and the authors note that this is because they are visually similar to MNIST digits, but are not actually digits, and thus activate the model's convolutional filters less than MNIST digits. The notMNIST digits however, result it more crowding. This is due to the fact that the different font style results in more high intensity pixels and edges. The intensity distributions for the 3 datasets is shown in the histograms in Fig. 12 below. The correlation between crowding and relative frequency of high intensity pixels can be seen from this figure.<br /> <br /> [[File:crowding_at_end_pooling.png|750px|center]]<br /> <br /> [[File:DCNN dataset histogram.png|750px|center]]<br /> <br /> <br /> ===DCNN Observations===<br /> * Accuracy decreases with the increase in the number of flankers.<br /> * Unsurprisingly, CNNs are capable of being invariant to translations.<br /> * In the constant target eccentricity setup, where the target is fixed at the center of the image with varying target-flanker spacing, we observe that as the distance between target and flankers increase, recognition gets better.<br /> * Spatial pooling helps the network in learning invariance.<br /> * Flankers similar to the target object helps in recognition since they activate the convolutional filter more.<br /> * notMNIST data affects leads to more crowding since they have many more edges and white image pixels which activate the convolutional layers more.<br /> <br /> ===Eccentric Model===<br /> The set-up is the same as explained earlier. The spacial pooling keeps constant. The effect of pooling across scales are investigated. The three configurations for scale pooling are (i) at the beginning, (ii)progressively and (iii) at the end. <br /> [[File:result3.png|750x400px|center]]<br /> <br /> ====Observations====<br /> * The recognition accuracy is dependent on the eccentricity of the target object.<br /> * If the target is placed at the center and no contrast normalization is done, then the recognition accuracy is high since this model concentrates the most on the central region of the image.<br /> * If contrast normalization is done, then all the scales will contribute equal amount and hence the eccentricity dependence is removed.<br /> * Early pooling is harmful since it might take away the useful information very early which might be useful to the network.<br /> <br /> Without contrast normalization, the middle portion of the image can be focused more with high resolution so the target at the center with no normalization performs well in that case. But if normalization is done, then all the segments of the image contribute to the classification and hence the overall accuracy is not that great but the system becomes robust to the changes in eccentricity.<br /> <br /> ==Complex Clutter==<br /> Here, the targets are randomly embedded into images of the Places dataset and shifted along horizontally in order to investigate model robustness when the target is not at the image center. Tests are performed on DCNN and the eccentricity model with and without contrast normalization using at end pooling. The results are shown in Figure 9 below. <br /> <br /> [[File:result4.png|750x400px|center]]<br /> <br /> ====Observations====<br /> * Only eccentricity model without contrast normalization can recognize the target and only when the target is close to the image center.<br /> * The eccentricity model does not need to be trained on different types of clutter to become robust to those types of clutter, but it needs to fixate on the relevant part of the image to recognize the target. If it can fixate on the relevant part of the image, it can still discriminate it, even at different scales. This implies that the eccentricity model is robust to clutter.<br /> <br /> =Conclusions=<br /> This paper investigates the effect of crowding on a DNN. Using a simple technique of adding clutter in the model didn't improve the performance. We often think that just training the network with data similar to the test data would achieve good results in a general scenario too but that's not the case as we trained the model with flankers and it did not give us the ideal results for the target objects. The following 4 techniques influenced crowding in DNN:<br /> *'''Flanker Configuration''': When models are trained with images of objects in isolation, adding flankers harms recognition. Adding two flankers is the same or worse than adding just one and the smaller the spacing between flanker and target, the more crowding occurs. This is because the pooling operation merges nearby responses, such as the target and flankers if they are close.<br /> *'''Similarity between target and flanker''': Flankers more similar to targets cause more crowding, because of the selectivity property of the learned DNN filters.<br /> *'''Dependence on target location and contrast normalization''': In DCNNs and eccentricity-dependent models with contrast normalization, recognition accuracy is the same across all eccentricities. In eccentricity-dependent networks without contrast normalization, recognition does not decrease despite the presence of clutter when the target is at the center of the image.<br /> *'''Effect of pooling''': adding pooling leads to better recognition accuracy of the models. Yet, in the eccentricity model, pooling across the scales too early in the hierarchy leads to lower accuracy.<br /> * The Eccentricity Dependent Models can be used for modeling the feedforward path of the primate visual cortex. <br /> * If target locations are proposed, then the system can become even more robust and hence a simple network can become robust to clutter while also reducing the amount of training data and time needed<br /> <br /> =Critique=<br /> This paper only tries to check the impact of flankers on targets as to how crowding can affect recognition but it does not propose anything novel in terms of architecture to take care of such type of crowding. The paper only shows that the eccentricity based model does better (than plain DCNN model) when the target is placed at the center of the image but maybe windowing over the frames the same way that a convolutional model passes a filter over an image, instead of taking crops starting from the middle, might help.<br /> <br /> This paper focuses on image classification. For a stronger argument, their model could be applied to the task of object detection. Perhaps crowding does not have as large of an impact when the objects of interest are localized by a region proposal network. Further the artificial crowding introduced in the paper may not be random enough for the neural network to learn to classify the object of interest as opposed to the entire cluster of objects. For example in the case of an even MNIST digit being flanked by two odd MNIST digits there are only 25 possible combinations of flankers and targets. <br /> <br /> This paper does not provide a convincing argument that the problem of crowding as experienced by humans somehow shares a similar mechanism to the problem of DNN accuracy falling when there is more clutter in the scene. The multi-scale architecture does not appear similar to the distribution of rods and cones in the retina[https://www.ncbi.nlm.nih.gov/books/NBK10848/figure/A763/?report=objectonly]. It might be that the eccentric model does well when the target is centered because it is being sampled by more scales, not because it is similar to a primate visual cortex, and primates are able to recognize an object in clutter when looking directly at it.<br /> <br /> =References=<br /> # Volokitin A, Roig G, Poggio T:&quot;Do Deep Neural Networks Suffer from Crowding?&quot; Conference on Neural Information Processing Systems (NIPS). 2017<br /> # Francis X. Chen, Gemma Roig, Leyla Isik, Xavier Boix and Tomaso Poggio: &quot;Eccentricity Dependent Deep Neural Networks for Modeling Human Vision&quot; Journal of Vision. 17. 808. 10.1167/17.10.808.<br /> # J Harrison, W &amp; W Remington, R &amp; Mattingley, Jason. (2014). Visual crowding is anisotropic along the horizontal meridian during smooth pursuit. Journal of vision. 14. 10.1167/14.1.21. http://willjharrison.com/2014/01/new-paper-visual-crowding-is-anisotropic-along-the-horizontal-meridian-during-smooth-pursuit/</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/IMPROVING_GANS_USING_OPTIMAL_TRANSPORT&diff=36252 stat946w18/IMPROVING GANS USING OPTIMAL TRANSPORT 2018-04-07T01:19:51Z <p>S3wen: /* Experiments */</p> <hr /> <div>== Introduction ==<br /> Recently, the problem of how to learn models that generate media such as images, video, audio and text has become very popular and is called Generative Modeling. One of the main benefits of such an approach is that generative models can be trained on unlabeled data that is readily available . Therefore, generative networks have a huge potential in the field of deep learning.<br /> <br /> Generative Adversarial Networks (GANs) are powerful generative models used for unsupervised learning techniques where the 2 agents compete to generate a zero-sum model. A GAN model consists of a generator and a discriminator or critic. The generator is a neural network which is trained to generate data having a distribution matched with the distribution of the real data. The critic is also a neural network, which is trained to separate the generated data from the real data. A loss function that measures the distribution distance between the generated data and the real one is important to train the generator.<br /> <br /> Optimal transport theory, which is another approach to measuring distances between distributions, evaluates the distribution distance between the generated data and the training data based on a metric, which provides another method for generator training. The main advantage of optimal transport theory over the distance measurement in GAN is its closed form solution for having a tractable training process. But the theory might also result in inconsistency in statistical estimation due to the given biased gradients if the mini-batches method is applied (Bellemare et al.,<br /> 2017).<br /> <br /> This paper presents a variant GANs named OT-GAN, which incorporates a discriminative metric called 'Mini-batch Energy Distance' into its critic in order to overcome the issue of biased gradients.<br /> <br /> == GANs and Optimal Transport ==<br /> <br /> ===Generative Adversarial Nets===<br /> Original GAN was firstly reviewed. The objective function of the GAN: <br /> <br /> [[File:equation1.png|700px]]<br /> <br /> The goal of GANs is to train the generator g and the discriminator d finding a pair of (g,d) to achieve Nash equilibrium(such that either of them cannot reduce their cost without changing the others' parameters). However, it could cause failure of converging since the generator and the discriminator are trained based on gradient descent techniques.<br /> <br /> ===Wasserstein Distance (Earth-Mover Distance)===<br /> <br /> In order to solve the problem of convergence failure, Arjovsky et. al. (2017) suggested Wasserstein distance (Earth-Mover distance) based on the optimal transport theory.<br /> <br /> [[File:equation2.png|600px]]<br /> <br /> where &lt;math&gt; \prod (p,g) &lt;/math&gt; is the set of all joint distributions &lt;math&gt; \gamma (x,y) &lt;/math&gt; with marginals &lt;math&gt; p(x) &lt;/math&gt; (real data), &lt;math&gt; g(y) &lt;/math&gt; (generated data). &lt;math&gt; c(x,y) &lt;/math&gt; is a cost function and the Euclidean distance was used by Arjovsky et. al. in the paper. <br /> <br /> The Wasserstein distance can be considered as moving the minimum amount of points between distribution &lt;math&gt; g(y) &lt;/math&gt; and &lt;math&gt; p(x) &lt;/math&gt; such that the generator distribution &lt;math&gt; g(y) &lt;/math&gt; is similar to the real data distribution &lt;math&gt; p(x) &lt;/math&gt;.<br /> <br /> Computing the Wasserstein distance is intractable. The proposed Wasserstein GAN (W-GAN) provides an estimated solution by switching the optimal transport problem into Kantorovich-Rubinstein dual formulation using a set of 1-Lipschitz functions. A neural network can then be used to obtain an estimation.<br /> <br /> [[File:equation3.png|600px]]<br /> <br /> W-GAN helps to solve the unstable training process of original GAN and it can solve the optimal transport problem approximately, but it is still intractable.<br /> <br /> ===Sinkhorn Distance===<br /> Genevay et al. (2017) proposed to use the primal formulation of optimal transport instead of the dual formulation to generative modeling. They introduced Sinkhorn distance which is a smoothed generalization of the Wasserstein distance.<br /> [[File: equation4.png|600px]]<br /> <br /> It introduced entropy restriction (&lt;math&gt; \beta &lt;/math&gt;) to the joint distribution &lt;math&gt; \prod_{\beta} (p,g) &lt;/math&gt;. This distance could be generalized to approximate the mini-batches of data &lt;math&gt; X ,Y&lt;/math&gt; with &lt;math&gt; K &lt;/math&gt; vectors of &lt;math&gt; x, y&lt;/math&gt;. The &lt;math&gt; i, j &lt;/math&gt; th entry of the cost matrix &lt;math&gt; C &lt;/math&gt; can be interpreted as the cost it needs to transport the &lt;math&gt; x_i &lt;/math&gt; in mini-batch X to the &lt;math&gt; y_i &lt;/math&gt; in mini-batch &lt;math&gt;Y &lt;/math&gt;. The resulting distance will be:<br /> <br /> [[File: equation5.png|550px]]<br /> <br /> where &lt;math&gt; M &lt;/math&gt; is a &lt;math&gt; K \times K &lt;/math&gt; matrix, each row of &lt;math&gt; M &lt;/math&gt; is a joint distribution of &lt;math&gt; \gamma (x,y) &lt;/math&gt; with positive entries. The summmation of rows or columns of &lt;math&gt; M &lt;/math&gt; is always equal to 1. <br /> <br /> This mini-batch Sinkhorn distance is not only fully tractable but also capable of solving the instability problem of GANs. However, it is not a valid metric over probability distribution when taking the expectation of &lt;math&gt; \mathcal{W}_{c} &lt;/math&gt; and the gradients are biased when the mini-batch size is fixed.<br /> <br /> ===Energy Distance (Cramer Distance)===<br /> In order to solve the above problem, Bellemare et al. proposed Energy distance:<br /> <br /> [[File: equation6.png|700px]]<br /> <br /> where &lt;math&gt; x, x' &lt;/math&gt; and &lt;math&gt; y, y'&lt;/math&gt; are independent samples from data distribution &lt;math&gt; p &lt;/math&gt; and generator distribution &lt;math&gt; g &lt;/math&gt;, respectively. Based on the Energy distance, Cramer GAN is to minimize the ED distance metric when training the generator.<br /> <br /> ==Mini-Batch Energy Distance==<br /> Salimans et al. (2016) mentioned that comparing to use distributions over individual images, mini-batch GAN is more powerful when using the distributions over mini-batches &lt;math&gt; g(X), p(X) &lt;/math&gt;. The distance measure is generated for mini-batches.<br /> <br /> ===Generalized Energy Distance===<br /> The generalized energy distance allowed to use non-Euclidean distance functions d. It is also valid for mini-batches and is considered better than working with individual data batch.<br /> <br /> [[File: equation7.png|670px]]<br /> <br /> Similarly as defined in the Energy distance, &lt;math&gt; X, X' &lt;/math&gt; and &lt;math&gt; Y, Y'&lt;/math&gt; can be the independent samples from data distribution &lt;math&gt; p &lt;/math&gt; and the generator distribution &lt;math&gt; g &lt;/math&gt;, respectively. While in Generalized engergy distance, &lt;math&gt; X, X' &lt;/math&gt; and &lt;math&gt; Y, Y'&lt;/math&gt; can also be valid for mini-batches. The &lt;math&gt; D_{GED}(p,g) &lt;/math&gt; is a metric when having &lt;math&gt; d &lt;/math&gt; as a metric. Thus, taking the triangle inequality of &lt;math&gt; d &lt;/math&gt; into account, &lt;math&gt; D(p,g) \geq 0,&lt;/math&gt; and &lt;math&gt; D(p,g)=0 &lt;/math&gt; when &lt;math&gt; p=g &lt;/math&gt;.<br /> <br /> ===Mini-Batch Energy Distance===<br /> As &lt;math&gt; d &lt;/math&gt; is free to choose, authors proposed Mini-batch Energy Distance by using entropy-regularized Wasserstein distance as &lt;math&gt; d &lt;/math&gt;. <br /> <br /> [[File: equation8.png|650px]]<br /> <br /> where &lt;math&gt; X, X' &lt;/math&gt; and &lt;math&gt; Y, Y'&lt;/math&gt; are independent sampled mini-batches from the data distribution &lt;math&gt; p &lt;/math&gt; and the generator distribution &lt;math&gt; g &lt;/math&gt;, respectively. This distance metric combines the energy distance with primal form of optimal transport over mini-batch distributions &lt;math&gt; g(Y) &lt;/math&gt; and &lt;math&gt; p(X) &lt;/math&gt;. Inside the generalized energy distance, the Sinkhorn distance is a valid metric between each mini-batches. By adding the &lt;math&gt; - \mathcal{W}_c (Y,Y')&lt;/math&gt; and &lt;math&gt; \mathcal{W}_c (X,Y)&lt;/math&gt; to equation (5) and using energy distance, the objective becomes statistically consistent (meaning the objective converges to the true parameter value for large sample sizes) and mini-batch gradients are unbiased.<br /> <br /> ==Optimal Transport GAN (OT-GAN)==<br /> <br /> The mini-batch energy distance which was proposed depends on the transport cost function &lt;math&gt;c(x,y)&lt;/math&gt;. One possibility would be to choose c to be some fixed function over vectors, like Euclidean distance, but the authors found this to perform poorly in preliminary experiments. For simple fixed cost functions like Euclidean distance, there exists many bad distributions &lt;math&gt;g&lt;/math&gt; in higher dimensions for which the mini-batch energy distance is zero such that it is difficult to tell &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; apart if the sample size is not big enough. To solve this the authors propose learning the cost function adversarially, so that it can adapt to the generator distribution &lt;math&gt;g&lt;/math&gt; and thereby become more discriminative. <br /> <br /> In practice, in order to secure the statistical efficiency (i.e. being able to tell &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; apart without requiring an enormous sample size when their distance is close to zero), authors suggested using cosine distance between vectors &lt;math&gt; v_\eta (x) &lt;/math&gt; and &lt;math&gt; v_\eta (y) &lt;/math&gt; based on the deep neural network that maps the mini-batch data to a learned latent space. Here is the transportation cost:<br /> <br /> [[File: euqation9.png|370px]]<br /> <br /> where the &lt;math&gt; v_\eta &lt;/math&gt; is chosen to maximize the resulting minibatch energy distance.<br /> <br /> Unlike the practice when using the original GANs, the generator was trained more often than the critic, which keep the cost function from degeneration. The resulting generator in OT-GAN has a well defined and statistically consistent objective through the training process.<br /> <br /> The algorithm is defined below. The backpropagation is not used in the algorithm since ignoring this gradient flow is justified by the envelope theorem (i.e. when changing the parameters of the objective function, changes in the optimizer do not contribute to a change in the objective function). Stochastic gradient descent is used as the optimization method in algorithm 1 below, although other optimizers are also possible. In fact, Adam was used in experiments. <br /> <br /> [[File: al.png|600px]]<br /> <br /> <br /> [[File: al_figure.png|600px]]<br /> <br /> ==Experiments==<br /> <br /> In order to demonstrate the supermum performance of the OT-GAN, authors compared it with the original GAN and other popular models based on four experiments: Dataset recovery; CIFAR-10 test; ImageNet test; and the conditional image synthesis test.<br /> <br /> ===Mixture of Gaussian Dataset===<br /> OT-GAN has a statistically consistent objective when it is compared with the original GAN (DC-GAN), such that the generator would not update to a wrong direction even if the signal provided by the cost function to the generator is not good. In order to prove this advantage, authors compared the OT-GAN with the original GAN loss (DAN-S) based on a simple task. The task was set to recover all of the 8 modes from 8 Gaussian mixers in which the means were arranged in a circle. MLP with RLU activation functions were used in this task. The critic was only updated for 15K iterations. The generator distribution was tracked for another 25K iteration. The results showed that the original GAN experiences the model collapse after fixing the discriminator while the OT-GAN recovered all the 8 modes from the mixed Gaussian data.<br /> <br /> [[File: 5_1.png|600px]]<br /> <br /> ===CIFAR-10===<br /> <br /> The dataset CIFAR-10 was then used for inspecting the effect of batch-size to the model training process and the image quality. OT-GAN and four other methods were compared using &quot;inception score&quot; as the criteria for comparison. Figure 3 shows the change of inceptions scores (y-axis) by the increased of the iteration number. Scores of four different batch sizes (200, 800, 3200 and 8000) were compared. The results show that a larger batch size, which would more likely cover more modes in the distribution of data, lead to a more stable model showing a larger value in inception score. However, a large batch size would also require a high-performance computational environment. The sample quality across all 5 methods, ran using a batch size of 8000, are compared in Table 1 where the OT_GAN has the best score.<br /> <br /> The OT-GAN was trained using Adam optimizer. The learning rate was set to &lt;math&gt; 3 x10^-4, β_1 = 0.5, β_2 = 0.999 &lt;/math&gt;. The introduced OT-GAN algorithm also includes two additional hyperparameters for the Sinkhorn algorithm. The first hyperparameters indicated the number of iterations to run the algorithm and the second &lt;math&gt; 1 / \lambda &lt;/math&gt; the entropy penalty of alignments. The authors found that a value of 500 worked well for both mentioned hyperparameters. The network uses the following architecture:<br /> <br /> [[File: cf10gc.png|600px]]<br /> <br /> [[File: 5_2.png|600px]]<br /> <br /> Figure 4 below shows samples generated by the OT-GAN trained with a batch size of 8000. Figure 5 below shows random samples from a model trained with the same architecture and hyperparameters, but with random matching of samples in place of optimal transport.<br /> <br /> [[File: ot_gan_cifar_10_samples.png|600px]]<br /> <br /> <br /> In order to show the advantage of learning the cost function adversarially, the CIFAR-10 experiment was re-run with the cost as follows:<br /> <br /> [[File: OTGAN_CosineDist.png|250px]]<br /> <br /> When using this fixed cost and keeping the other experiment settings constant, the max inception score dropped from 8.47 with learned to 4.93 with fixed cost functions. The results of the fixed cost are seen in Figure 8 below.<br /> <br /> [[File: OTGAN_fixedDist.png|600px]]<br /> <br /> ===ImageNet Dogs===<br /> <br /> In order to investigate the performance of OT-GAN when dealing with the high-quality images, the dog subset of ImageNet (128*128) was used to train the model. Figure 6 shows that OT-GAN produces less nonsensical images and it has a higher inception score compare to the DC-GAN. <br /> <br /> [[File: 5_3.png|600px]]<br /> <br /> <br /> To analyze mode collapse in GANs the authors trained both types of GANs for a large number of epochs. They find the DCGAN shows mode collapse as soon as 900 epochs. They trained the OT-GAN for 13000 epochs and saw no evidence of mode collapse or less diversity in the samples. Samples can be viewed in Figure 9.<br /> <br /> [[File: ModelCollapseImageNetDogs.png|600px]]<br /> <br /> ===Conditional Generation of Birds===<br /> <br /> The last experiment was to compare OT-GAN with three popular GAN models for processing the text-to-image generation demonstrating the performance on conditional image synthesis. As can be found from Table 2, OT-GAN received the highest inception score than the scores of the other three models. <br /> <br /> [[File: 5_4.png|600px]]<br /> <br /> The algorithm used to obtain the results above is conditional generation generalized from '''Algorithm 1''' to include conditional information &lt;math&gt;s&lt;/math&gt; such as some text description of an image. The modified algorithm is outlined in '''Algorithm 2'''.<br /> <br /> [[File: paper23_alg2.png|600px]]<br /> <br /> ==Conclusion==<br /> <br /> In this paper, an OT-GAN method was proposed based on the optimal transport theory. A distance metric that combines the primal form of the optimal transport and the energy distance was given was presented for realizing the OT-GAN. The results showed OT-GAN to be uniquely stable when trained with large mini batches and state of the art results were achieved on some datasets. One of the advantages of OT-GAN over other GAN models is that OT-GAN can stay on the correct track with an unbiased gradient even if the training on critic is stopped or presents a weak cost signal. The performance of the OT-GAN can be maintained when the batch size is increasing, though the computational cost has to be taken into consideration.<br /> <br /> ==Critique==<br /> <br /> The paper presents a variant of GANs by defining a new distance metric based on the primal form of optimal transport and the mini-batch energy distance. The stability was demonstrated through the four experiments that comparing OP-GAN with other popular methods. However, limitations in computational efficiency were not discussed much. Furthermore, in section 2, the paper lacks explanation on using mini-batches instead of a vector as input when applying Sinkhorn distance. It is also confusing when explaining the algorithm in section 4 about choosing M for minimizing &lt;math&gt; \mathcal{W}_c &lt;/math&gt;. Lastly, it is found that it is lack of parallel comparison with existing GAN variants in this paper. Readers may feel jumping from one algorithm to another without necessary explanations.<br /> <br /> ==Reference==<br /> Salimans, Tim, Han Zhang, Alec Radford, and Dimitris Metaxas. &quot;Improving GANs using optimal transport.&quot; (2018).</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:cf10gc.png&diff=36251 File:cf10gc.png 2018-04-07T01:18:00Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/MaskRNN:_Instance_Level_Video_Object_Segmentation&diff=36250 stat946w18/MaskRNN: Instance Level Video Object Segmentation 2018-04-07T01:06:03Z <p>S3wen: /* Ablation Study */</p> <hr /> <div>== Introduction ==<br /> Deep Learning has produced state of the art results in many computer vision tasks like image classification, object localization, object detection, object segmentation, semantic segmentation and instance level video object segmentation. Image classification classify the image based on the prominent objects. Object localization is the task of finding objects’ location in the frame. Object Segmentation task involves providing a pixel map which represents the pixel wise location of the objects in the image. Semantic segmentation task attempts at segmenting the image into meaningful parts. Instance level video object segmentation is the task of consistent object segmentation in video sequences. Deforming shapes, fast movements, and occlusion from multiple objects, are just some of the significant challenges in instance level video object segmentation.<br /> <br /> There are 2 different types of video object segmentation, Unsupervised and Semi-supervised. <br /> * In unsupervised video object segmentation, the task is to find the salient objects and track the main objects in the video. <br /> * In a semi-supervised setting, the ground truth mask of the salient objects is provided for the first frame. The task is thus simplified to only track the objects required. <br /> <br /> In this paper, the authors look at an unsupervised video object segmentation technique.<br /> <br /> == Background Papers ==<br /> Video object segmentation has been performed using spatio-temporal graphs [[https://pdfs.semanticscholar.org/7221/c3470fa89879aab3ef270570ced15cde28de.pdf 5], [http://ieeexplore.ieee.org/abstract/document/5539893/ 6], [http://openaccess.thecvf.com/content_iccv_2013/papers/Li_Video_Segmentation_by_2013_ICCV_paper.pdf 7], [https://link.springer.com/content/pdf/10.1007/s11263-011-0512-5.pdf 8]] and deep learning. The graph based methods construct 3D spatio-temporal graphs in order to model the inter and the intra-frame relationship of pixels or superpixels in a video. Hence they are computationally slower than deep learning methods and are unable to run at real-time. There are 2 main deep learning techniques for semi-supervised video object segmentation: One Shot Video Object Segmentation (OSVOS) and Learning Video Object Segmentation from Static Images (MaskTrack). Following is a brief description of the new techniques introduced by these papers for semi-supervised video object segmentation task.<br /> <br /> === OSVOS (One-Shot Video Object Segmentation) ===<br /> <br /> [[File:OSVOS.jpg | 1000px]]<br /> <br /> This paper introduces the technique of using a frame-by-frame object segmentation without any temporal information from the previous frames of the video. The paper uses a VGG-16 network with pre-trained weights from image classification task. This network is then converted into a fully-connected network (FCN) by removing the fully connected dense layers at the end and adding convolution layers to generate a segment mask of the input. This network is then trained on the DAVIS 2016 dataset.<br /> <br /> During testing, the trained VGG-16 FCN is fine-tuned using the first frame of the video using the ground truth. Because this is a semi-supervised case, the segmented mask (ground truth) for the first frame is available. The first frame data is augmented by zooming/rotating/flipping the first frame and the associated segment mask.<br /> <br /> === MaskTrack (Learning Video Object Segmentation from Static Images) ===<br /> <br /> [[File:MaskTrack.jpg | 500px]]<br /> <br /> MaskTrack takes the output of the previous frame to improve its predictions and to generate the segmentation mask for the next frame. Thus the input to the network is 4 channel wide (3 RGB channels from the frame at time &lt;math&gt;t&lt;/math&gt; plus one binary segmentation mask from frame &lt;math&gt;t-1&lt;/math&gt;). The output of the network is the binary segmentation mask for frame at time &lt;math&gt;t&lt;/math&gt;. Using the binary segmentation mask (referred to as guided object segmentation in the paper), the network is able to use some temporal information from the previous frame to improve its segmentation mask prediction for the next frame.<br /> <br /> The model of the MaskTrack network is similar to a modular VGG-16 and is referred to as MaskTrack ConvNet in the paper. The network is trained offline on saliency segmentation datasets: ECSSD, MSRA 10K, SOD and PASCAL-S. The input mask for the binary segmentation mask channel is generated via non-rigid deformation and affine transformation of the ground truth segmentation mask. Similar data-augmentation techniques are also used during online training. Just like OSVOS, MaskTrack uses the first frame as ground truth (with augmented images) to fine-tune the network to improve prediction score for the particular video sequence.<br /> <br /> A parallel ConvNet network is used to generate a predicted segment mask based on the optical flow magnitude. The optical flow between 2 frames is calculated using the EpicFlow algorithm. The output of the two networks is combined using an averaging operation to generate the final predicted segmented mask.<br /> <br /> Table 1 below gives a summary comparison of the different state of the art algorithms. The noteworthy information included in this table is that the technique presented in this paper is the only one which takes into account long-term temporal information. This is accomplished with a recurrent neural net. Furthermore, the bounding box is also estimated instead of just a segmentation mask. The authors claim that this allows the incorporation of a location prior from the tracked object.<br /> <br /> [[File:Paper19-SegmentationComp.png]]<br /> <br /> == Dataset ==<br /> The three major datasets used in this paper are DAVIS-2016, DAVIS-2017 and Segtrack v2. DAVIS-2016 dataset provides video sequences with only one segment mask for all salient objects. DAVIS-2017 improves the ground truth data by providing segmentation mask for each salient object as a separate color segment mask. Segtrack v2 also provides multiple segmentation mask for all salient objects in the video sequence. These datasets try to recreate real-life scenarios like occlusions, low resolution videos, background clutter, motion blur, fast motion etc.<br /> <br /> == MaskRNN: Introduction ==<br /> Most techniques mentioned above don’t work directly on instance level segmentation of the objects through the video sequence. The above approaches focus on image segmentation on each frame and using additional information (mask propagation and optical flow) from the preceding frame perform predictions for the current frame. To address the instance level segmentation problem, MaskRNN proposes a framework where the salient objects are tracked and segmented by capturing the temporal information in the video sequence using a recurrent neural network.<br /> <br /> == MaskRNN: Overview ==<br /> In a video sequence &lt;math&gt;I = \{I_1, I_2, …, I_T\}&lt;/math&gt;, the sequence of &lt;math&gt;T&lt;/math&gt; frames are given as input to the network, where the video sequence contains &lt;math&gt;N&lt;/math&gt; salient objects. The ground truth for the first frame &lt;math&gt;y_1^*&lt;/math&gt; is also provided for &lt;math&gt;N&lt;/math&gt; salient objects.<br /> In this paper, the problem is formulated as a time dependency problem and using a recurrent neural network, the prediction of the previous frame influences the prediction of the next frame. The approach also computes the optical flow between frames (optical flow is the apparent motion of objects between two consecutive frames in the form of a 2D vector field representing the displacement in brightness patterns for each pixel, apparent because it depends on the relative motion between the observer and the scene) and uses that as the input to the neural network. The optical flow is also used to align the output of the predicted mask. “The warped prediction, the optical flow itself, and the appearance of the current frame are then used as input for &lt;math&gt;N&lt;/math&gt; deep nets, one for each of the &lt;math&gt;N&lt;/math&gt; objects.”[1 - MaskRNN] Each deep net is a made of an object localization network and a binary segmentation network. The binary segmentation network is used to generate the segmentation mask for an object. The object localization network is used to alleviate outliers from the predictions. The final prediction of the segmentation mask is generated by merging the predictions of the 2 networks. For &lt;math&gt;N&lt;/math&gt; objects, there are N deep nets which predict the mask for each salient object. The predictions are then merged into a single prediction using an &lt;math&gt;\text{argmax}&lt;/math&gt; operation at test time.<br /> <br /> == MaskRNN: Multiple Instance Level Segmentation ==<br /> <br /> [[File:2ObjectSeg.jpg | 850px]]<br /> <br /> Image segmentation requires producing a pixel level segmentation mask and this can become a multi-class problem. Instead, using the approach from [2- Mask R-CNN] this approach is converted into a multiple binary segmentation problem. A separate segmentation mask is predicted separately for each salient object and thus we get a binary segmentation problem. The binary segments are combined using an &lt;math&gt;\text{argmax}&lt;/math&gt; operation where each pixel is assigned to the object containing the largest predicted probability.<br /> <br /> === MaskRNN: Binary Segmentation Network ===<br /> <br /> [[File:MaskRNNDeepNet.jpg | 850px]]<br /> <br /> The above picture shows a single deep net employed for predicting the segment mask for one salient object in the video frame. The network consists of 2 networks: binary segmentation network and object localization network. The binary segmentation network is split into two streams: appearance and flow stream. The input of the appearance stream is the RGB frame at time t and the wrapped prediction of the binary segmentation mask from time &lt;math&gt;t-1&lt;/math&gt;. The wrapping function uses the optical flow between frame &lt;math&gt;t-1&lt;/math&gt; and frame &lt;math&gt;t&lt;/math&gt; to generate a new binary segmentation mask for frame &lt;math&gt;t&lt;/math&gt;. The input to the flow stream is the concatenation of the optical flow magnitude between frames &lt;math&gt;t-1&lt;/math&gt; to &lt;math&gt;t&lt;/math&gt; and frames &lt;math&gt;t&lt;/math&gt; to &lt;math&gt;t+1&lt;/math&gt; and the wrapped prediction of the segmentation mask from frame &lt;math&gt;t-1&lt;/math&gt;. The magnitude of the optical flow is replicated into an RBG format before feeding it to the flow stream. The network architecture closely resembles a VGG-16 network without the pooling or fully connected layers at the end. The fully connected layers are replaced with convolutional and bilinear interpolation upsampling layers which are then linearly combined to form a feature representation that is the same size of the input image. This feature representation is then used to generate a binary segment mask. This technique is borrowed from the Fully Convolutional Network mentioned above. The output of the flow stream and the appearance stream is linearly combined and sigmoid function is applied to the result to generate binary mask for ith object. All parts of the network are fully differentiable and thus it can be fully trained in every pass.<br /> <br /> === MaskRNN: Object Localization Network: ===<br /> Using a similar technique to the Fast-RCNN method of object localization, where the region of interest (RoI) pooling of the features of the region proposals (i.e. the bounding box proposals here) is performed and passed through fully connected layers to perform regression, the Object localization network generates a bounding box of the salient object in the frame. This bounding box is enlarged by a factor of 1.25 and combined with the output of binary segmentation mask. Only the segment mask available in the bounding box is used for prediction and the pixels outside of the bounding box are marked as zero. MaskRNN uses the convolutional feature output of the appearance stream as the input to the RoI-pooling layer to generate the predicted bounding box. A pixel is classified as foreground if it is both predicted to be in the foreground by the binary segmentation net and within the enlarged estimated bounding box from the object localization net.<br /> <br /> === Training and Finetuning ===<br /> For training the network depicted in Figure 1, backpropagation through time is used in order to preserve the recurrence relationship connecting the frames of the video sequence. Predictive performance is further improved by following the algorithm for semi-supervised setting for video object segmentation with fine-tuning achieved by using the first frame segmentation mask of the ground truth. In this way, the network is further optimized using the ground truth data.<br /> <br /> == MaskRNN: Implementation Details ==<br /> === Offline Training ===<br /> The deep net is first trained offline on a set of static images. The ground truth is randomly perturbed locally to generate the imperfect mask from frame &lt;math&gt;t-1&lt;/math&gt;. Two different networks are trained offline separately for DAVIS-2016 and DAVIS-2017 datasets for a fair evaluation of both datasets. After both the object localization net and binary segmentation networks have trained, the temporal information in the network is used to further improve the segmented prediction results. Because of GPU memory constraints, the RNN is only able to backpropagate the gradients back 7 frames and learn long-term temporal information. <br /> <br /> For optical flow, a pre-trained flowNet2.0 is used to compute the optical flow between frames. (A flowNet (Dosovitskiy 2015) is a deep neural network trained to predict optical flow. The simplest form of flowNet has an architecture consisting of two parts. The first part accepts the two images between which the optical flow is to be computed as input, as applies a sequence of convolution and max-pooling operations, as in a standard convolutional neural network. In the second part, repeated up-convolution operations are applied, increasing the dimensions of the feature-maps. Besides the output of the previous upconvolution, each upconvolution is also fed as input the output of the corresponding down-convolution from the first part of the network. Thus part of the architecture resembles that of a U-net (Ronneberger, 2015). The output of the network is the predicted optical flow. ) <br /> <br /> === Online Finetuning ===<br /> The deep nets (without the RNN) are then fine-tuned during test time by online training the networks on the ground truth of the first frame and some augmentations of the first frame data. The learning rate is set to &lt;math&gt;10^{-5}&lt;/math&gt; for online training for 200 iterations and the learning rate is gradually decayed over time. Data augmentation techniques similar to those in offline training, namely random resizing, rotating, cropping and flipping is applied. Also, it should be noted that the RNN is ''not'' employed during online finetuning since only a single frame of training data is available.<br /> <br /> == MaskRNN: Experimental Results ==<br /> === Evaluation Metrics ===<br /> There are 3 different techniques for performance analysis for Video Object Segmentation techniques:<br /> <br /> 1. Region Similarity (Jaccard Index): Region similarity or Intersection-over-union is used to capture precision of the area covered by the prediction segmentation mask compared to the ground truth segmentation mask. It calculates the average across all frames of the dataset. This is particularly challenging for small sized foreground objects.<br /> <br /> \begin{equation}<br /> IoU = \frac{|M \cap G|}{|M| + |G| - |M \cap G|} <br /> \label{equation:Jaccard}<br /> \end{equation}<br /> <br /> 2. Contour Accuracy (F-score): This metric measures the accuracy in the boundary of the predicted segment mask and the ground truth segment mask, by calculating the the precision and the recall of the two sets of points on the contours of the ground truth segment and the output segment via a bipartite graph matching. It is a measure of accurate delineation of the foreground objects. <br /> <br /> [[File:Fscore.jpg | 200px|center]]<br /> <br /> 3. Temporal Stability : This estimates the degree of deformation needed to transform the segmentation masks from one frame to the next and is measured by the dissimilarity of the set of points on the contours of the segmentation between two adjacent frames.<br /> <br /> Region similarity measures the true segmented area in the prediction, while Contour Accuracy measures the accuracy of the contours/segmented mask boundary.<br /> <br /> === Ablation Study ===<br /> <br /> The ablation study summarized how the different components contributed to the algorithm evaluated on DAVIS-2016 and DAVIS-2017 datasets.<br /> <br /> [[File:MaskRNNTable2.jpg | 700px|center]]<br /> <br /> The above table presents the contribution of each component of the network to the final prediction score. Online fine-tuning improves the performance by a large margin, as the network becomes adjusted to the appearance of the specific object being tracked. Addition of RNN/Localization Net and FStream all seem to positively affect the performance of the deep net. The FStream provides information on motion boundaries which help in videos with cluttered backgrounds, the RNN provides more consistent segmentation masks over time. The localization net has a more ambiguous effect on the network; adding the bounding box regression loss decreases the performance of the segmentation net but applying the bounding box to restrict the segmentation mask improves the results over those achieved by only using the segmentation net. In other words the localization net should only be used in conjunction with the segmentation net while the segmentation net can be used by itself.<br /> <br /> === Quantitative Evaluation ===<br /> <br /> The authors use DAVIS-2016, DAVIS-2017 and Segtrack v2 to compare the performance of the proposed approach to other methods based on foreground-background video object segmentation and multiple instance-level video object segmentation.<br /> <br /> [[File:MaskRNNTable3.jpg | 700px]]<br /> <br /> The above table shows the results for contour accuracy mean and region similarity. The MaskRNN method seems to outperform all previously proposed methods. The performance gain is significant by employing a Recurrent Neural Network for learning recurrence relationship and using a object localization network to improve prediction results.<br /> <br /> The following table shows the improvements in the state of the art achieved by MaskRNN on the DAVIS-2017 and the SegTrack v2 dataset.<br /> <br /> [[File:MaskRNNTable4.jpg | 700px]]<br /> <br /> === Qualitative Evaluation ===<br /> The authors showed example qualitative results from the DAVIS and Segtrack datasets. <br /> <br /> Below are some success cases of object segmentation under complex motion, cluttered background, and/or multiple object occlusion.<br /> <br /> [[File:maskrnn_example.png | 700px]]<br /> <br /> Below are a few failure cases. The authors explain two reasons for failure: a) when similar objects of interest are contained in the frame (left two images), and b) when there are large variations in scale and viewpoint (right two images).<br /> <br /> [[File:maskrnn_example_fail.png | 700px]]<br /> <br /> == Conclusion ==<br /> In this paper a novel approach to instance level video object segmentation task is presented which performs better than current state of the art. The long-term recurrence relationship is learnt using an RNN. The object localization network is added to improve accuracy of the system. Using online fine-tuning the network is adjusted to predict better for the current video sequence.<br /> <br /> == Critique ==<br /> The paper provides a technique to track multiple objects in a video. The novelty is to add back-propagation through time to improve the tracking accuracy and using a localization network to remove any outliers in the segmented binary mask. However, the network architecture it too large and isn't able to run in real-time. There are N deep-Nets for N objects and each deep-Net contains 2 parallel VGG-16 convolutional networks.<br /> <br /> == Implementation ==<br /> <br /> The implementation of this paper was produced as part of the NIPS Paper Implementation Challenge. This implementation can be found at the following open source project: https://github.com/philferriere/tfvos.<br /> <br /> == References ==<br /> # Dosovitskiy, Alexey, et al. &quot;Flownet: Learning optical flow with convolutional networks.&quot; Proceedings of the IEEE International Conference on Computer Vision. 2015.<br /> # Hu, Y., Huang, J., &amp; Schwing, A. &quot;MaskRNN: Instance level video object segmentation&quot;. Conference on Neural Information Processing Systems (NIPS). 2017<br /> # Ferriere, P. (n.d.). Semi-Supervised Video Object Segmentation (VOS) with Tensorflow. Retrieved March 20, 2018, from https://github.com/philferriere/tfvos<br /> # Ronneberger, Olaf, Philipp Fischer, and Thomas Brox. &quot;U-net: Convolutional networks for biomedical image segmentation.&quot; International Conference on Medical image computing and computer-assisted intervention. Springer, Cham, 2015.<br /> # Lee, Yong Jae, Jaechul Kim, and Kristen Grauman. &quot;Key-segments for video object segmentation.&quot; Computer Vision (ICCV), 2011 IEEE International Conference on. IEEE, 2011.<br /> # Grundmann, Matthias, et al. &quot;Efficient hierarchical graph-based video segmentation.&quot; Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on. IEEE, 2010.<br /> # Li, Fuxin, et al. &quot;Video segmentation by tracking many figure-ground segments.&quot; Computer Vision (ICCV), 2013 IEEE International Conference on. IEEE, 2013.<br /> # Tsai, David, et al. &quot;Motion coherent tracking using multi-label MRF optimization.&quot; International journal of computer vision 100.2 (2012): 190-202.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Continuous_Adaptation_via_Meta-Learning_in_Nonstationary_and_Competitive_Environments&diff=36247 Continuous Adaptation via Meta-Learning in Nonstationary and Competitive Environments 2018-04-07T00:54:20Z <p>S3wen: /* Evaluating Adaptation Strategies */</p> <hr /> <div>= Introduction =<br /> <br /> Typically, the basic goal of machine learning is to train a model to perform a task. In meta-learning, the goal is to train a model to perform the task of training a model to perform a task. Hence, in this case, the term &quot;meta-Learning&quot; has the exact meaning one would expect; the word &quot;meta&quot; has the precise function of introducing a layer of abstraction.<br /> <br /> The meta-learning task can be made more concrete by a simple example. Consider the CIFAR-100 classification task that we used for our data competition. We can alter this task from being a 100-class classification problem to a collection of 100 binary classification problems. The goal of meta-learning here is to design and train a single binary classifier for each class that will perform well on a randomly sampled task given a limited amount of training data for that specific task. In other words, we would like to train a model to perform the following procedure:<br /> <br /> # A task is sampled. The task is &quot;Is X a dog?&quot;.<br /> # A small set of labeled training data is provided to the model. Each label is a boolean representing whether or not the corresponding image is a picture of a dog.<br /> # The model uses the training data to adjust itself to the specific task of checking whether or not an image is a picture of a dog.<br /> <br /> This example also highlights the intuition that the skill of sight is distinct and separable from the skill of knowing what a dog looks like.<br /> <br /> In this paper, a probabilistic framework for meta-learning is derived, then applied to tasks involving simulated robotic spiders. This framework generalizes the typical machine learning setup using Markov Decision Processes. This paper focuses on a multi-agent non-stationary environment which requires reinforcement learning (RL) agents to do continuous adaptation in such an environment. Non-stationarity breaks the standard assumptions and requires agents to continuously adapt, both at training and execution time, in order to earn more rewards, hence the approach is to break this into a sequence of stationary tasks and present it as a multi-task learning problem.<br /> <br /> [[File:paper19_fig1.png|600px|frame|none|alt=Alt text| '''Figure 1'''. a) Illustrates a probabilistic model for Model Agnostic Meta-Learning (MAML) in a multi-task RL setting, where the tasks &lt;math&gt;T&lt;/math&gt;, policies &lt;math&gt;\pi&lt;/math&gt;, and trajectories &lt;math&gt;\tau&lt;/math&gt; are all random variables with dependencies encoded in the edges of a given graph. b) The proposed extension to MAML suitable for continuous adaptation to a task changing dynamically due to non-stationarity of the environment. The distribution of tasks is represented by a Markov chain, whereby policies from a previous step are used to construct a new policy for the current step. c) The computation graph for the meta-update from &lt;math&gt;\phi_i&lt;/math&gt; to &lt;math&gt;\phi_{i+1}&lt;/math&gt;. Boxes represent replicas of the policy graphs with the specified parameters. The model is optimized using truncated backpropagation through time starting from &lt;math&gt;L_{T_{i+1}}&lt;/math&gt;.]]<br /> <br /> = Background =<br /> == Markov Decision Process (MDP) ==<br /> A MDP is defined by the tuple &lt;math&gt;(S,A,P,r,\gamma)&lt;/math&gt;, where S is a set of states, A is a set of actions, P is the transition probability distribution, r is the reward function, and &lt;math&gt;\gamma&lt;/math&gt; is the discount factor. More information ([https://www.cs.cmu.edu/~katef/DeepRLControlCourse/lectures/lecture2_mdps.pdf here]).<br /> <br /> <br /> = Model Agnostic Meta-Learning =<br /> <br /> An initial framework for meta-learning is given in &quot;Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks&quot; (Finn et al, 2017):<br /> <br /> &quot;In our approach, the parameters of<br /> the model are explicitly trained such that a small<br /> number of gradient steps with a small amount<br /> of training data from a new task will produce<br /> good generalization performance on that task&quot; (Finn et al, 2017).<br /> <br /> [[File:MAML.png | 500px]]<br /> <br /> In this training algorithm, the parameter vector &lt;math&gt;\theta&lt;/math&gt; belonging to the model &lt;math&gt;f_{\theta}&lt;/math&gt; is trained such that the meta-objective function &lt;math&gt;\mathcal{L} (\theta) = \sum_{\tau_i \sim P(\tau)} \mathcal{L}_{\tau_i} (f_{\theta_i' }) &lt;/math&gt; is minimized. The sum in the objective function is over a sampled batch of training tasks. &lt;math&gt;\mathcal{L}_{\tau_i} (f_{\theta_i'})&lt;/math&gt; is the training loss function corresponding to the &lt;math&gt;i^{th}&lt;/math&gt; task in the batch evaluated at the model &lt;math&gt;f_{\theta_i'}&lt;/math&gt;. The parameter vector &lt;math&gt;\theta_i'&lt;/math&gt; is obtained by updating the general parameter &lt;math&gt;\theta&lt;/math&gt; using the loss function &lt;math&gt;\mathcal{L}_{\tau_i}&lt;/math&gt; and set of K training examples specific to the &lt;math&gt;i^{th}&lt;/math&gt; task. Note that in alternate versions of this algorithm, additional testing sets are sampled from &lt;math&gt;\tau_i&lt;/math&gt; and used to update &lt;math&gt;\theta&lt;/math&gt; using testing loss functions instead of training loss functions.<br /> <br /> One of the important difference between this algorithm and more typical fine-tuning methods is that &lt;math&gt;\theta&lt;/math&gt; is explicitly trained to be easily adjusted to perform well on different tasks rather than perform well on any specific tasks then fine tuned as the environment changes. (Sutton et al., 2007). In essence, the model is trained so that gradient steps are highly productive at adapting the model parameters to a new enviroment.<br /> <br /> = Probabilistic Framework for Meta-Learning =<br /> <br /> This paper puts the meta-learning problem into a Markov Decision Process (MDP) framework common to RL, see Figure 1a. Instead of training examples &lt;math&gt;\{(x, y)\}&lt;/math&gt;, we have trajectories &lt;math&gt;\tau = (x_0, a_1, x_1, R_1, a_2,x_2,R_2, ... a_H, x_H, R_H)&lt;/math&gt;. A trajectory is sequence of states/observations &lt;math&gt;x_t&lt;/math&gt;, actions &lt;math&gt;a_t&lt;/math&gt; and rewards &lt;math&gt;R_t&lt;/math&gt; that is sampled from a task &lt;math&gt; T &lt;/math&gt; according to a policy &lt;math&gt;\pi_{\theta}&lt;/math&gt;. Included with said task is a method for assigning loss values to trajectories &lt;math&gt;L_T(\tau)&lt;/math&gt; which is typically the negative cumulative reward. A policy is a deterministic function that takes in a state and returns an action. Our goal here is to train a policy &lt;math&gt;\pi_{\theta}&lt;/math&gt; with parameter vector &lt;math&gt;\theta&lt;/math&gt;. This is analougous to training a function &lt;math&gt;f_{\theta}&lt;/math&gt; that assigns labels &lt;math&gt;y&lt;/math&gt; to feature vectors &lt;math&gt;x&lt;/math&gt;. More precisely we have the following definitions:<br /> <br /> * &lt;math&gt;\tau = (x_0, a_1, x_1, R_1, x_2, ... a_H, x_H, R_H)&lt;/math&gt; trajectories.<br /> * &lt;math&gt;T :=(L_T, P_T(x), P_T(x_t | x_{t-1}, a_{t-1}), H )&lt;/math&gt; (A Task)<br /> * &lt;math&gt;D(T)&lt;/math&gt; : A distribution over tasks.<br /> * &lt;math&gt;L_T&lt;/math&gt;: A loss function for the task T that assigns numeric loss values to trajectories.<br /> * &lt;math&gt;P_T(x), P_T(x_t | x_{t-1}, a_{t-1})&lt;/math&gt;: Probability measures specifying the markovian dynamics of the observations &lt;math&gt;x_t&lt;/math&gt;<br /> * &lt;math&gt;H&lt;/math&gt;: The horizon of the MDP. This is a fixed natural number specifying the lengths of the tasks trajectories.<br /> <br /> The paper goes further to define a Markov dynamic for sequences of tasks as shown in Figure 1b. Thus the policy that we would like to meta learn &lt;math&gt;\pi_{\theta}&lt;/math&gt;, after being exposed to a sample of K trajectories &lt;math&gt;\tau_\theta^{1:K}&lt;/math&gt; from the task &lt;math&gt;T_i&lt;/math&gt;, should produce a new policy &lt;math&gt;\pi_{\phi}&lt;/math&gt; that will perform well on the next task &lt;math&gt;T_{i+1}&lt;/math&gt;. Thus we seek to minimize the following expectation:<br /> <br /> &lt;math&gt;\mathrm{E}_{P(T_0), P(T_{i+1} | T_i)}\bigg(\sum_{i=1}^{l} \mathcal{L}_{T_i, T_{i+1}}(\theta)\bigg)&lt;/math&gt;, <br /> <br /> where &lt;math&gt;\mathcal{L}_{T_i, T_{i + 1}}(\theta) := \mathrm{E}_{\tau_{i, \theta}^{1:K} } \bigg( \mathrm{E}_{\tau_{i+1, \phi}}\Big( L_{T_{i+1}}(\tau_{i+1, \phi} | \tau_{i, \theta}^{1:K}, \theta) \Big) \bigg) &lt;/math&gt; and &lt;math&gt;l&lt;/math&gt; is the number of tasks.<br /> <br /> The meta-policy &lt;math&gt;\pi_{\theta}&lt;/math&gt; is trained and then adapted at test time using the following procedures. The computational graph is given in Figure 1c.<br /> <br /> [[File:MAML2.png | 800px]]<br /> <br /> The mathematics of calculating loss gradients is omitted.<br /> <br /> = Training Spiders to Run with Dynamic Handicaps (Robotic Locomotion in Non-Stationary Environments) =<br /> <br /> The authors used the MuJoCo physics simulator to create a simulated environment where robotic spiders with 6 legs are faced with the task of running due east as quickly as possible. The robotic spider observes the location and velocity of its body, and the angles and velocities of its legs. It interacts with the environment by exerting torque on the joints of its legs. Each leg has two joints, the joint closer to the body rotates horizontally while the joint farther from the body rotates vertically. The environment is made non-stationary by gradually paralyzing two legs of the spider across training and testing episodes. This allows the agent to adapt to new environments in each episode.<br /> Putting this example into the above probabilistic framework yields:<br /> <br /> * &lt;math&gt;T_i&lt;/math&gt;: The task of walking east with the torques of two legs scaled by &lt;math&gt; (i-1)/6 &lt;/math&gt;<br /> * &lt;math&gt;\{T_i\}_{i=1}^{7}&lt;/math&gt;: A sequence of tasks with the same two legs handicapped in each task. Note there are 15 different ways to choose such legs resulting in 15 sequences of tasks. 12 are used for training and 3 for testing.<br /> * A Markov Descision process composed of<br /> ** Observations &lt;math&gt; x_t &lt;/math&gt; containing information about the state of the spider.<br /> ** Actions &lt;math&gt; a_t &lt;/math&gt; containing information about the torques to apply to the spiders legs.<br /> ** Rewards &lt;math&gt; R_t &lt;/math&gt; corresponding to the speed at which the spider is moving east.<br /> <br /> Three differently structured policy neural networks are trained in this set up using both meta-learning and three different previously developed adaption methods.<br /> <br /> At testing time, the spiders following meta learned policies initially perform worse than the spiders using non-adaptive policies. However, by the third episode (&lt;math&gt; i=3 &lt;/math&gt;), the meta-learners perform on par. And by the sixth episode, when the selected legs are mostly immobile, the meta-learners significantly out perform. These results can be seen in the graphs below.<br /> <br /> [[File:locomotion_results.png | 800px]]<br /> <br /> = Training Spiders to Fight Each Other (Adversarial Meta-Learning) =<br /> <br /> The authors created an adversarial environment called RoboSumo where pairs of agents with 4 (named Ants), 6 (named Bugs),or 8 legs (named spiders) sumo wrestle. The agents observe the location and velocity of their bodies and the bodies of their opponent, the angles and velocities of their legs, and the forces being exerted on them by their opponent (equivalent of tactile sense). The game is organized into episodes and rounds. Episodes are single wrestling matches with 500 timesteps and win/lose/draw outcomes. Agents win by pushing their opponent out of the ring or making their opponent's body touch the ground. An episode results in a draw when neither of these things happen after 500 timesteps. Rounds are batches of episodes. Rounds have possible outcomes win, lose, and draw that are decided based on majority of episodes won. K rounds will be fought. Both agents may update their policies between rounds. The agent that wins the majority of rounds is deemed the winner of the game.<br /> <br /> == Setup ==<br /> Similar to the Robotic locomotion example, this game can be phrased in terms of the RL MDP framework.<br /> <br /> * &lt;math&gt;T_i&lt;/math&gt;: The task of fighting a round.<br /> * &lt;math&gt;\{T_i\}_{i=1}^{K}&lt;/math&gt;: A sequence of rounds against the same opponent. Note that the opponent may update their policy between rounds but the anatomy of both wrestlers will be constant across rounds.<br /> * A Markov Descision process composed of<br /> ** A horizon &lt;math&gt;H = 500*n&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of episodes per round.<br /> ** Observations &lt;math&gt; x_t &lt;/math&gt; containing information about the state of the agent and its opponent.<br /> ** Actions &lt;math&gt; a_t &lt;/math&gt; containing information about the torques to apply to the agents legs.<br /> ** Rewards &lt;math&gt; R_t &lt;/math&gt; rewards given to the agent based on its wrestling performance. &lt;math&gt;R_{500*n} = &lt;/math&gt; +2000 if win episode, -2000 if lose, and -1000 if draw. A discount factor of &lt;math&gt; \gamma = 0.995 &lt;/math&gt; is applied to the rewards.<br /> <br /> Note that the above reward set up is quite sparse. Therefore in order to encourage fast training, rewards are introduced at every time step for the following.<br /> * For staying close to the center of the ring.<br /> * For exerting force on the opponents body.<br /> * For moving towards the opponent.<br /> * For the distance of the opponent to the center of the ring.<br /> <br /> In addition to the sparse win/lose rewards, the following dense rewards are also introduced in the early training stages to encourage faster learning:<br /> * Quickly push the opponent outside - penalty proportional to the distance of there opponent from the center of the ring.<br /> * Moving towards the opponent - reward proportional to the velocity component towards the opponent.<br /> * Hit the opponent - reward proportional to square root of the total forces exerted on the opponent.<br /> * Control penalty - penalty denoted by &lt;math&gt; l_2 &lt;/math&gt; on actions which lead to jittery/unnatural movements.<br /> <br /> <br /> This makes sense intuitively as these are reasonable goals for agents to explore when they are learning to wrestle.<br /> <br /> == Training ==<br /> The same combinations of policy networks and adaptation methods that were used in the locomotion example are trained and tested here. A family of non-adaptive policies are first trained via self-play and saved at all stages. Self-play simply means the two agents in the training environment use the same policy. All policy versions are saved so that agents of various skill levels can be sampled when training meta-learners. The weights of the different insects were calibrated such that the test win rate between two insects of differing anatomy, who have been trained for the same number of epochs via self-play, is close to 50%.<br /> <br /> [[File:weight_cal.png | 800px]]<br /> <br /> We can see in the above figure that the weight of the spider had to be increased by almost four times in order for the agents to be evenly matched.<br /> <br /> [[File:robosumo_results.png | 800px]]<br /> <br /> The above figure shows testing results for various adaptation strategies. The agent and opponent both start with the self-trained policies. The opponent uses all of its testing experience to continue training. The agent uses only the last 75 episodes to adapt its policy network. This shows that metal learners need only a limited amount of experience in order to hold their own against a constantly improving opponent.<br /> <br /> == Evaluating Adaptation Strategies ==<br /> To compare the proposed strategy against existing strategies, the authors leveraged the Robosumo task to have the different adaptation strategies compete against each other. The scoring metric used is TrueSkill, a metric very similar to ELO. Each strategy starts out with a fixed amount of points. When two different strategies compete the winner gains points and the loser loses points. The number of points gained and lost depend on the different in TrueSkill. A lower ranked opponent defeating a high ranked opponent will gain much more points than a high ranked opponent defeating a lower ranked one. It is assumed that agents skill levels are initially normally distributed with mean 25, and variance 25/3. 1000 matches are generated with each match consisting of 100-round iterated adaptation games. TrueSkill scores are updated after every match. The distribution of TrueSkill after playing 1000 matches are shown below:<br /> <br /> [[File:trueskill_results.PNG | 800px]]<br /> <br /> In summary recurrent policies are dominant, and the proposed meta-update performs better than or at par with the other strategies.<br /> <br /> The authors also demonstrated the same result by starting with 1000 agents, uniformly distributed in terms of adaptation strategies, and randomly matching them against each other for several generations. The loser of each match is removed and replaced with a duplicate of the winner. After several generations, the total population consists mostly of the proposed strategy. This is shown in the figure below.<br /> <br /> [[File:natural_selection.png | 800px]]<br /> <br /> = Future Work =<br /> The authors noted that the meta-learning adaptation rule they proposed is similar to backpropagation through time with a unit time lag, so a potential area for future research would be to introduce fully-recurrent meta-updates based on the full interaction history with the environment. Secondly, the algorithm proposed involves computing second-order derivatives at training time (see Figure 1c), which resulted in much slower training processes compared to baseline models during experiments, so they suggested finding a method to utilize information from the loss function without explicit backpropagation to speed up computations. The authors also mention that their approach likely will not work well with sparse rewards. This is because the meta-updates, which use policy gradients, are very dependent on the reward signal. They mention that this is an issue they would like to address in the future. A potential solution they have outlined for this is to introduce auxiliary dense rewards which could enable meta-learning.<br /> <br /> = Sources =<br /> # Chelsea Finn, Pieter Abbeel, Sergey Levine. &quot;Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks.&quot; arXiv preprint arXiv:1703.03400v3 (2017).<br /> # Richard S Sutton, Anna Koop, and David Silver. On the role of tracking in stationary environments. In Proceedings of the 24th international conference on Machine learning, pp. 871–878. ACM, 2007.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Synthetic_and_natural_noise_both_break_neural_machine_translation&diff=36226 stat946w18/Synthetic and natural noise both break neural machine translation 2018-04-07T00:08:45Z <p>S3wen: /* Richness of Natural Noise */</p> <hr /> <div>== Introduction ==<br /> Humans have surprisingly robust language processing systems which can easily overcome typos, e.g.<br /> <br /> * &quot;Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae.&quot;<br /> <br /> A person's ability to read this text comes as no surprise to the Psychology literature<br /> # Saberi &amp; Perrott (1999) found that this robustness extends to audio as well.<br /> # Rayner et al. (2006) found that in noisier settings reading comprehension only slowed by 11%.<br /> # McCusker et al. (1981) found that the common case of swapping letters could often go unnoticed by the reader.<br /> # Mayall et al (1997) shows that we rely on word shape.<br /> # Reicher, 1969; Pelli et al., (2003) found that we can switch between whole word recognition but the first and last letter positions are required to stay constant for comprehension<br /> <br /> However, neural machine translation (NMT) systems are brittle. i.e. The Arabic word<br /> [[File:Good_morning.PNG]] means a blessing for good morning, however [[File:Hunt.PNG]] means hunt or slaughter. <br /> <br /> Facebook's MT system mistakenly confused two words that only differ by one character, a situation that is challenging for a character-based NMT system.<br /> <br /> The figure below shows the performance translating German to English as a function of the percent of German words modified. Here two types of noise are shown: (1) In blue, random permutation of the word and (2) In green, swapping a pair of adjacent letters that does not include the first or last letter of the word. The important thing to note is that even small amounts of noise lead to substantial drops in performance.<br /> <br /> [[File:BLEU_plot.PNG|center]] <br /> <br /> BLEU (bilingual evaluation understudy) is an algorithm for evaluating the quality of text which has been machine-translated from one natural language to another. Quality is considered to be the correspondence between a machine's output and that of a human: &quot;the closer a machine translation is to a professional human translation, the better it is&quot;. BLEU is between 0 and 1. BELU computes the scores for individual translated segments and then computes an average accuracy score for the whole corpus.<br /> <br /> This paper explores two simple strategies for increasing model robustness:<br /> # using structure-invariant representations (character CNN representation)<br /> # robust training on noisy data, a form of adversarial training.<br /> <br /> The goal of the paper is two-fold:<br /> # to initiate a conversation on robust training and modeling techniques in NMT<br /> # to promote the creation of better and more linguistically accurate artificial noise to be applied to new languages and tasks<br /> <br /> == Adversarial examples ==<br /> The growing literature on adversarial examples has demonstrated how dangerous it can be to have brittle machine learning systems being used so pervasively in the real world. Small changes to the input can lead to dramatic<br /> failures of deep learning models. This leads to a potential for malicious attacks using adversarial examples. An important distinction is often drawn between white-box attacks, where adversarial examples are generated with<br /> access to the model parameters, and black-box attacks, where examples are generated without such access.<br /> <br /> The paper devises simple methods for generating adversarial examples for NMT. They do not assume any access to the NMT models' gradients, instead relying on cognitively-informed and naturally occurring language errors to generate noise.<br /> <br /> == MT system ==<br /> The authors experiment with three different NMT systems with access to character information at different levels.<br /> # Use &lt;code&gt;char2char&lt;/code&gt;, the fully character-level model of (Lee et al. 2017). This model processes a sentence as a sequence of characters. The encoder works as follows: the characters are embedded as vectors, and then the sequence of vectors is fed to a convolutional layer. The sequence output by the convolutional layer is then shortened by max pooling in the time dimension. The output of the max-pooling layer is then fed to a four-layer highway network (Srivasta et al. 2015), and the output of the highway network is in turn fed to a bidirectional GRU, producing a sequence of hidden units. The sequence of hidden units is then processed by the decoder, a GRU with attention, to produce probabilities over sequences of output characters.<br /> # Use &lt;code&gt;Nematus&lt;/code&gt; (Sennrich et al., 2017), a popular NMT toolkit. It is another sequence-to-sequence model with several architecture modifications, especially operating on sub-word units using byte-pair encoding. Byte-pair encoding (Sennich et al. 2015, Gage 1994) is an algorithm according to which we begin with a list of characters as our symbols, and repeatedly fuse common combinations to create new symbols. For example, if we begin with the letters a to z as our symbol list, and we find that &quot;th&quot; is the most common two-letter combination in a corpus, then we would add &quot;th&quot; to our symbol list in the first iteration. After we have used this algorithm to create a symbol list of the desired size, we apply a standard encoder-decoder with attention.<br /> # Use an attentional sequence-to-sequence model with a word representation based on a character convolutional neural network (&lt;code&gt;charCNN&lt;/code&gt;). The &lt;code&gt;charCNN&lt;/code&gt; model is similar to &lt;code&gt;char2char&lt;/code&gt;, but uses a shallower highway network and, although it reads the input sentence as characters, it produces as output a probability distribution over words, not characters.<br /> <br /> == Data ==<br /> === MT Data ===<br /> The authors use the TED talks parallel corpus prepared for IWSLT 2016 (Cettolo et al., 2012) for testing all of the NMT systems.<br /> <br /> [[File:Table1x.PNG|center]]<br /> <br /> === Natural and Artificial Noise ===<br /> ==== Natural Noise ====<br /> The three languages, French, German, and Czech, each have their own frequent natural errors. The corpora of edits used for these languages are:<br /> <br /> # French : Wikipedia Correction and Paraphrase Corpus (WiCoPaCo)<br /> # German : RWSE Wikipedia Correction Dataset and The MERLIN corpus<br /> # Czech : CzeSL Grammatical Error Correction Dataset (CzeSL-GEC) which is a manually annotated dataset of essays written by both non-native learners of Czech and Czech pupils<br /> <br /> The authors harvested naturally occurring errors (typos, misspellings, etc.) corresponding to these three languages from available corpora of edits to build a look-up table of possible lexical replacements.<br /> <br /> They insert these errors into the source-side of the parallel data by replacing every word in the corpus with an error if one exists in our dataset. When there is more than one possible replacement to choose, words for which there is no error, are sampled uniformly and kept as is.<br /> <br /> ==== Synthetic Noise ====<br /> In addition to naturally collected sources of error, the authors also experiment with four types of synthetic noise: Swap, Middle Random, Fully Random, and Key Typo. <br /> # &lt;code&gt;Swap&lt;/code&gt;: The first and simplest source of noise is swapping two letters (do not alter the first or last letters, only apply to words of length &gt;=4).<br /> # &lt;code&gt;Middle Random&lt;/code&gt;: Randomize the order of all the letters in a word except for the first and last (only apply to words of length &gt;=4).<br /> # &lt;code&gt;Fully Random&lt;/code&gt; Completely randomized words.<br /> # &lt;code&gt;Keyboard Typo&lt;/code&gt; Randomly replace one letter in each word with an adjacent key<br /> <br /> [[File:Table3x.PNG|center]]<br /> <br /> Table 3 shows BLEU scores of models trained on clean (Vanilla) texts and tested on clean and noisy<br /> texts. All models suffer a significant drop in BLEU when evaluated on noisy texts. This is true<br /> for both natural noise and all kinds of synthetic noise. The more noise in the text, the worse the<br /> translation quality, with random scrambling producing the lowest BLEU scores.<br /> <br /> In contrast to the poor performance of these methods in the presence of noise, humans can perform very well as mentioned in the introduction. The table below shows the translations performed by a German native-speaker human, not familiar with the meme and three machine translation methods. Clearly, the machine translation methods failed. <br /> <br /> [[File:paper16_tab4.png|center]]<br /> <br /> The author also examined improvements by using a simple spell checker. The author tried correcting error through Google's spell checker by simply accepting the first suggestion on the detected mistake. There was a small improvement in French and German translations, and a small drop in accuracy for the Czech translation due to more complex grammar. The author concluded using existing spell checkers would not improve the accuracy to be comparable with vanilla text. The results are shown in the table below.<br /> <br /> <br /> [[File:paper16_tab5.png|center]]<br /> <br /> == Dealing with noise ==<br /> === Structure Invariant Representations ===<br /> The three NMT models are all sensitive to word structure. The &lt;code&gt;char2char&lt;/code&gt; and &lt;code&gt;charCNN&lt;/code&gt; models both have convolutional layers on character sequences, designed to capture character n-grams (which are sequences of characters or words, of length n). The model in &lt;code&gt;Nematus&lt;/code&gt; is based on sub-word units obtained with byte pair encoding (where common consecutive characters are replaced with a unique byte that does not occur in the data). It thus relies on character order.<br /> <br /> The simplest way to improve such a model is to take the average character embeddings as a word representation. This model, referred to as &lt;code&gt;meanChar&lt;/code&gt;, first generates a word representation by averaging character embeddings, and then proceeds with a word-level encoder similar to the &lt;code&gt;charCNN&lt;/code&gt; model.<br /> <br /> [[File:Table5x.PNG|center]]<br /> <br /> &lt;code&gt;meanChar&lt;/code&gt; is good with the other three scrambling errors (Swap, Middle Random and Fully Random), but bad with Keyboard errors and Natural errors.<br /> <br /> === Black-Box Adversarial Training ===<br /> <br /> &lt;code&gt;charCNN&lt;/code&gt; Performance<br /> [[File:Table6x.PNG|center]]<br /> <br /> Here is the result of the translation of the scrambled meme:<br /> “According to a study of Cambridge University, it doesn’t matter which technology in a word is going to get the letters in a word that is the only important thing for the first and last letter.”<br /> <br /> == Analysis ==<br /> === Learning Multiple Kinds of Noise in &lt;code&gt;charCNN&lt;/code&gt; ===<br /> <br /> As Table 6 above shows, &lt;code&gt;charCNN&lt;/code&gt; models performed quite well across different noise types on the test set when they are trained on a mix of noise types, which led the authors to speculate that filters from different convolutional layers learned to be robust to different types of noises. To test this hypothesis, they analyzed the weights learned by &lt;code&gt;charCNN&lt;/code&gt; models trained on two kinds of input: completely scrambled words (Rand) without other kinds of noise, and a mix of Rand+Key+Nat kinds of noise. For each model, they computed the variance across the filter dimension for each one of the 1000 filters and for each one of the 25 character embedding dimensions, which were then averaged across the filters to yield 25 variances. <br /> <br /> As Figure 2 below shows, the variances for the ensemble model are higher and more varied, which indicates that the filters learned different patterns and the model differentiated between different character embedding dimensions. Under the random scrambling scheme, there should be no patterns for the model to learn, so it makes sense for the filter weights to stay close uniform weights, hence the consistently lower variance measures.<br /> <br /> [[File:Table7x.PNG|center]]<br /> <br /> === Richness of Natural Noise ===<br /> <br /> The synthetic noise used in this paper appears to be very different from natural noise. This is evident because none of the modes trained only on synthetic noise demonstrated good performance on natural noise. Therefore, the authors say that the noise models used in this paper are not representative of real noise and that a more sophisticated model using explicit phonemic and linguistic knowledge is required if an error-free corpus is to be augmented with error for training. The synthetic noise analysed is lacking a two common types of typos: inserting a character that is adjacent (on the keyboard) to a letter and omitting letters.<br /> <br /> == Conclusion ==<br /> In this work, the authors have shown that character-based NMT models are extremely brittle and tend to break when presented with both natural and synthetic kinds of noise. After a comparison of the models, they found that a character-based CNN can learn to<br /> address multiple types of errors that are seen in training.<br /> For the future work, the author suggested generating more realistic synthetic noise by using phonetic and syntactic structure. Also, they suggested that a better NMT architecture could be designed which can be robust to noise without seeing it in the training data.<br /> <br /> == Criticism ==<br /> According to the [https://openreview.net/forum?id=BJ8vJebC- OpenReview thread], a major critique of this paper is that the solutions presented do not adequately solve the problem. The response to the meanChar architecture has been mostly negative and the method of noise injection has been seen as a simple start. However, the authors have acknowledged these critiques stating that they realize their solution is just a starting point. They argue that this paper has opened the discussion on dealing with noise in machine translation which has been mostly left untouched. Also these solutions/models still do not tackle the problem of natural noise as the models trained on the synthetic noise don't generalize well to natural noise.<br /> <br /> == References ==<br /> # Yonatan Belinkov and Yonatan Bisk. Synthetic and Natural Noise Both Break Neural Machine Translation. In ''International Conference on Learning Representations (ICLR)'', 2017.<br /> # Mauro Cettolo, Christian Girardi, and Marcello Federico. WIT: Web Inventory of Transcribed and Translated Talks. In ''Proceedings of the 16th Conference of the European Association for Machine Translation (EAMT)'', pp. 261–268, Trento, Italy, May 2012.<br /> # Jason Lee, Kyunghyun Cho, and Thomas Hofmann. Fully Character-Level Neural Machine Translation without Explicit Segmentation. ''Transactions of the Association for Computational Linguistics (TACL)'', 2017.<br /> # Rico Sennrich, Orhan Firat, Kyunghyun Cho, Alexandra Birch, Barry Haddow, Julian Hitschler, Marcin Junczys-Dowmunt, Samuel Laubli, Antonio Valerio Miceli Barone, Jozef Mokry, and Maria Nadejde. Nematus: a Toolkit for Neural Machine Translation. In ''Proceedings of the Software Demonstrations of the 15th Conference of the European Chapter of the Association for Computational Linguistics'', pp. 65–68, Valencia, Spain, April 2017. Association for Computational Linguistics. URL http://aclweb.org/anthology/E17-3017.<br /> # Aurlien Max and Guillaume Wisniewski. Mining Naturally-occurring Corrections and Paraphrases from Wikipedias Revision History. In Proceedings of the Seventh conference on International Language Resources and Evaluation (LREC’10), Valletta, Malta, may 2010. European Language Resources Association (ELRA). ISBN 2-9517408-6-7. URL https://wicopaco.limsi.fr.<br /> # Katrin Wisniewski, Karin Schne, Lionel Nicolas, Chiara Vettori, Adriane Boyd, Detmar Meurers, Andrea Abel, and Jirka Hana. MERLIN: An online trilingual learner corpus empirically grounding the European Reference Levels in authentic learner data, 10 2013. URL https://www.ukp.tu-darmstadt.de/data/spelling-correction/rwse-datasets.<br /> # Torsten Zesch. Measuring Contextual Fitness Using Error Contexts Extracted from the Wikipedia Revision History. In Proceedings of the 13th Conference of the European Chapter of the Association for Computational Linguistics, pp. 529–538, Avignon, France, April 2012. Association for Computational Linguistics.<br /> # Suranjana Samanta and Sameep Mehta. Towards Crafting Text Adversarial Samples. arXiv preprint arXiv:1707.02812, 2017. Karel Sebesta, Zuzanna Bedrichova, Katerina Sormov́a, Barbora Stindlov́a, Milan Hrdlicka, Tereza Hrdlickov́a, Jiŕı Hana, Vladiḿır Petkevic, Toḿas Jeĺınek, Svatava Skodov́a, Petr Janes, Katerina Lund́akov́a, Hana Skoumalov́a, Simon Sĺadek, Piotr Pierscieniak, Dagmar Toufarov́a, Milan Straka, Alexandr Rosen, Jakub Ńaplava, and Marie Poĺackova. CzeSL grammatical error correction dataset (CzeSL-GEC). Technical report, LINDAT/CLARIN digital library at the Institute of Formal and Applied Linguistics, Charles University, 2017. URL https://lindat.mff.cuni.cz/repository/xmlui/handle/11234/1-2143.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=36224 A Neural Representation of Sketch Drawings 2018-04-06T23:48:57Z <p>S3wen: /* Limitations */</p> <hr /> <div>= Introduction =<br /> <br /> There have been many recent advances in neural generative models for low resolution pixel-based images. Humans, however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). For the conditional generation mode, the authors explore the model's latent space that it uses to express the vector image. The paper also explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br /> <br /> = Related Work =<br /> <br /> Previous work related to sketch drawing generation includes methods that focused primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focused more on generation of pixel-based imagery. For example, Gatys et al.'s (2015) work focuses on separating style and content from pixel-based artwork and imagery. Some recent work has focused on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modeling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Auto-encoders to model English sentences in latent vector space.<br /> <br /> One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br /> <br /> = Methodology =<br /> <br /> === Dataset ===<br /> <br /> The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form &lt;math&gt;(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})&lt;/math&gt;. For each vector, &lt;math&gt;\Delta x&lt;/math&gt; and &lt;math&gt;\Delta y&lt;/math&gt; give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; &lt;math&gt;p_{1}&lt;/math&gt; indicates that the pen is down and a line should be drawn between the current point and the next point, &lt;math&gt;p_{2}&lt;/math&gt; indicates that the pen is up and no line should be drawn between the current point and the next point, and &lt;math&gt;p_{3}&lt;/math&gt; indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br /> <br /> === Sketch-RNN ===<br /> [[File:sketchrnn.PNG]]<br /> <br /> The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings (sequence &lt;math&gt;S&lt;/math&gt;) in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state &lt;math&gt;h&lt;/math&gt;. <br /> <br /> \begin{align}<br /> h_\rightarrow = \text{encode}_\rightarrow(S), h_\leftarrow = \text{encode}_\leftarrow(S_{\text{reverse}}), h=[h_\rightarrow; h_\leftarrow]<br /> \end{align}<br /> <br /> The concatenated hidden state &lt;math&gt;h&lt;/math&gt; is then projected into two vectors &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\hat{\sigma}&lt;/math&gt; each of size &lt;math&gt;N_{z}&lt;/math&gt; using a fully connected layer. &lt;math&gt;\hat{\sigma}&lt;/math&gt; is then converted into a non-negative standard deviation parameter &lt;math&gt;\sigma&lt;/math&gt; using an exponential operator. These two parameters &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt; are then used along with an IID Gaussian vector distributed as &lt;math&gt;\mathcal{N}(0, I)&lt;/math&gt; of size &lt;math&gt;N_{z}&lt;/math&gt; to construct a random vector &lt;math&gt;z \in ℝ^{N_{z}}&lt;/math&gt;, similar to the method used for VAE:<br /> \begin{align}<br /> \mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br /> \end{align}<br /> <br /> The decoder model is an autoregressive RNN that samples output sketches from the latent vector &lt;math&gt;z&lt;/math&gt;. The initial hidden states of each recurrent neuron are determined using &lt;math&gt;[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})&lt;/math&gt;. Each step of the decoder RNN accepts the previous point &lt;math&gt;S_{i-1}&lt;/math&gt; and the latent vector &lt;math&gt;z&lt;/math&gt; as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point &lt;math&gt;S_{i}&lt;/math&gt;. Outputs &lt;math&gt;\Delta x&lt;/math&gt; and &lt;math&gt;\Delta y&lt;/math&gt; are modeled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states &lt;math&gt;(q_{1}, q_{2}, q_{3})&lt;/math&gt; modelled as a categorical distribution with one-hot encoding.<br /> \begin{align}<br /> P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br /> \end{align}<br /> <br /> For each of the M distributions in the GMM, parameters &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt; are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter &lt;math&gt;\rho_{xy}&lt;/math&gt; signifying correlation of each bivariate normal distribution. An additional vector &lt;math&gt;\Pi&lt;/math&gt; is an output giving the mixture weights for the GMM. The output &lt;math&gt;S_{i}&lt;/math&gt; is determined from each of the mixture models using softmax sampling from these distributions.<br /> <br /> One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter &lt;math&gt;N_{max}&lt;/math&gt; which basically is the length of the longest sketch in the dataset and limits the number of points per drawing to being no more than &lt;math&gt;N_{max}&lt;/math&gt;, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br /> <br /> To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time &lt;math&gt;N_{max}&lt;/math&gt;. The authors also introduce a “temperature” parameter &lt;math&gt;\tau&lt;/math&gt; that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br /> <br /> \begin{align}<br /> \hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br /> \end{align}<br /> <br /> This parameter &lt;math&gt;\tau&lt;/math&gt; lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br /> <br /> === Unconditional Generation ===<br /> <br /> [[File:paper15_Unconditional_Generation.png|800px|]]<br /> <br /> The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable &lt;math&gt;z&lt;/math&gt;. Figure 3 above shows different sketches that are sampled from the network by only varying the temperature parameter &lt;math&gt;\tau&lt;/math&gt; between 0.2 and 0.9.<br /> <br /> === Training ===<br /> The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss &lt;math&gt;L_{R}&lt;/math&gt; and KL Divergence Loss &lt;math&gt;L_{KL}&lt;/math&gt;. The reconstruction loss term is composed of two terms; &lt;math&gt;L_{s}&lt;/math&gt;, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;L_{p}&lt;/math&gt; which is the log loss of the pen state terms.<br /> \begin{align}<br /> L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br /> \end{align}<br /> \begin{align}<br /> L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br /> \end{align}<br /> \begin{align}<br /> L_{R} = L_{s} + L{p}<br /> \end{align}<br /> <br /> The KL divergence loss &lt;math&gt;L_{KL}&lt;/math&gt; measures the difference between the latent vector &lt;math&gt;z&lt;/math&gt; and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions &lt;math&gt;N_{z}&lt;/math&gt; is calculated as:<br /> \begin{align}<br /> L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br /> \end{align}<br /> <br /> The loss for the entire model is thus the weighted sum:<br /> \begin{align}<br /> Loss = L_{R} + w_{KL}L_{KL}<br /> \end{align}<br /> <br /> The value of the weight parameter &lt;math&gt;w_{KL}&lt;/math&gt; has the effect that as &lt;math&gt;w_{KL} \rightarrow 0&lt;/math&gt;, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder. As with VAEs, there is a trade-off between optimizing for the two loss terms (i.e. between how precisely the model can regenerate training data &lt;math&gt;S&lt;/math&gt; and how closely the latent vector &lt;math&gt;z&lt;/math&gt; follows a standard normal distribution) - smaller values of &lt;math&gt;w_{KL}&lt;/math&gt; lead to better &lt;math&gt;L_R&lt;/math&gt; and worse &lt;math&gt;L_{KL}&lt;/math&gt; compared to bigger values of &lt;math&gt;w_{KL}&lt;/math&gt;. Also for unconditional generation, the model is a standalone decoder, so there will be no &lt;math&gt;L_{KL}&lt;/math&gt; term as only &lt;math&gt;L_{R}&lt;/math&gt; is optimized for. This trade-off is illustrated in Figure 4 showing different settings of &lt;math&gt;w_{KL}&lt;/math&gt; and the resulting &lt;math&gt;L_{KL}&lt;/math&gt; and &lt;math&gt;L_{R}&lt;/math&gt;, as well as just &lt;math&gt;L_{R}&lt;/math&gt; in the case of unconditional generation with only a standalone decoder.<br /> <br /> [[File:paper15_fig4.png|600px]]<br /> <br /> === Model Configuration ===<br /> In the given model, the encoder and decoder RNNs consist of 512 and 2048 nodes respectively. Also, M = 20 mixture components are used for the decoder RNN. Layer Normalization is applied to the model, and during training recurrent dropout is applied with a keep probability of 90%. The model is trained with batch sizes of 100 samples, using Adam with a learning rate of 0.0001 and gradient clipping of 1.0. During training, simple data augmentation is performed by multiplying the offset columns by two IID random factors. <br /> <br /> = Experiments =<br /> The authors trained multiple conditional and unconditional models using varying values of &lt;math&gt;w_{KL}&lt;/math&gt; and recorded the different &lt;math&gt;L_{R}&lt;/math&gt; and &lt;math&gt;L_{KL}&lt;/math&gt; values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks. (A HyperLSTM consists of two coupled LSTMS: an auxiliary LSTM and a main LSTM. At every time step, the auxiliary LSTM reads the previous hidden state and the current input vector, and computes an intermediate vector &lt;math display=&quot;inline&quot;&gt; z &lt;/math&gt;. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector &lt;math display=&quot;inline&quot;&gt; z &lt;/math&gt;. That is, the weights of the main LSTM are allowed to vary between time steps as a function of the output of the auxiliary LSTM. See Ha et al. (2016) for details)<br /> <br /> === Conditional Reconstruction ===<br /> [[File:conditional_generation.PNG]]<br /> <br /> The authors qualitatively assessed the reconstructed images &lt;math&gt;S’&lt;/math&gt; given input sketch &lt;math&gt;S&lt;/math&gt; using different values for the temperature hyperparameter &lt;math&gt;\tau&lt;/math&gt;. The figure above shows the results for different values of &lt;math&gt;\tau&lt;/math&gt; starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes are reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush are reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br /> <br /> === Latent Space Interpolation ===<br /> [[File:latent_space_interp.PNG]]<br /> <br /> The latent space vectors &lt;math&gt;z&lt;/math&gt; have few “gaps” between encoded latent space vectors due to the enforcement of a Gaussian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors. A model trained with higher &lt;math&gt;w_{KL}&lt;/math&gt; is expected to produce images closer to the data manifold, and the figure above shows reconstructed images from latent vector interpolation between the original images. Results from the model trained with higher &lt;math&gt;w_{KL}&lt;/math&gt; seem to produce more coherent images.<br /> <br /> === Sketch Drawing Analogies ===<br /> Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br /> <br /> === Predicting Different Endings of Incomplete Sketches ===<br /> [[File:predicting_endings.PNG]]<br /> <br /> Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch with &lt;math&gt;\tau&lt;/math&gt; set to 0.8.<br /> <br /> = Applications and Future Work =<br /> Sketch-RNN may enable the production of several creative applications. These might include suggesting ways an artist could finish a sketch, enabling artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allowing the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output from a different class might be useful for producing sketches that morph the two different object classes into one sketch. For example, the image below was trained on drawing cats, but a chair was used as the input. This results in a chair looking cat.<br /> <br /> [[File:cat-chair.png]]<br /> <br /> Sketch-RNN may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to proceed to finish a sketch or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br /> <br /> The authors noted that Sketch-RNN is not as effective at generating coherent sketches when trained on a large number of classes simultaneously (experiments mostly used datasets consisting of one or two object classes), and plan to use class information outside the latent space to try to model a greater number of classes.<br /> <br /> Finally, the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the Sketch-RNN model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br /> <br /> = Limitations =<br /> The authors note a major limitation to the model is the training time relative to the number of data points. When sketches surpass 300 data points the model is difficult to train. To counteract this effect the Ramer-Douglas-Peucker algorithm was used to reduce the number of data points per sketch. This algorithm attempts to significantly reduce the number of data points while keeping the sketch as close to the original as possible.<br /> <br /> Another limitation is the effectiveness of generating sketches as the complexity of the class increases. Below are sketches of a few classes which show how the less complex classes such as cats and crabs are more accurately generated. Frogs (more complex) tend to have overly smooth lines drawn which do not seem to be part of realistic frog samples.<br /> <br /> [[File:paper15_classcomplexity.png]]<br /> <br /> A further limitation is the need to train individual neural networks for each class of drawings. While the former is useful in sketch completion with labeled incomplete sketches it may produce low quality results when the starting sketch is very different than any part of the learned representation. Further work can be done to extend the model to account for both prediction of the label and sketch completion.<br /> <br /> = Conclusion =<br /> The authors presented Sketch-RNN, a RNN model for modelling and generating vector-based sketch drawings with a goal to abstract concepts in the images similar to the way humans think. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br /> <br /> = Criticisms =<br /> The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br /> <br /> One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input &lt;math&gt;z&lt;/math&gt; by manipulating the &lt;math&gt;\tau&lt;/math&gt; hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt;, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br /> <br /> The authors contribute to a novel dataset but fail to evaluate the quality of the dataset, including generalized metrics for evaluation. They also provide no comparisons of their method on this dataset with other baseline sequence generation approaches.<br /> <br /> Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br /> <br /> == Major Contributions ==<br /> <br /> The paper provides with intuition of their approach and detailed below are the major contributions of this paper:<br /> * For images composed by sequence of lines, such as hand drawing, this paper proposed a framework to generate such image in vector format, conditionally and unconditionally. <br /> * Provided a unique training procedure that targets vector images, which makes training procedures more robust.<br /> * Composed large dataset of hand drawn vector images which benefits future development.<br /> * Discussed several potential applications of this methodology, such as drawing assist for artists and educational tool for students.<br /> <br /> = Implementation =<br /> Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn<br /> <br /> = Source =<br /> <br /> # Ha, D., &amp; Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).<br /> # Tensorflow/magenta. (n.d.). Retrieved March 25, 2018, from https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn<br /> # Graves et al, 2013, https://arxiv.org/pdf/1308.0850.pdf</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=36223 A Neural Representation of Sketch Drawings 2018-04-06T23:40:51Z <p>S3wen: /* Latent Space Interpolation */</p> <hr /> <div>= Introduction =<br /> <br /> There have been many recent advances in neural generative models for low resolution pixel-based images. Humans, however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). For the conditional generation mode, the authors explore the model's latent space that it uses to express the vector image. The paper also explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br /> <br /> = Related Work =<br /> <br /> Previous work related to sketch drawing generation includes methods that focused primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focused more on generation of pixel-based imagery. For example, Gatys et al.'s (2015) work focuses on separating style and content from pixel-based artwork and imagery. Some recent work has focused on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modeling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Auto-encoders to model English sentences in latent vector space.<br /> <br /> One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br /> <br /> = Methodology =<br /> <br /> === Dataset ===<br /> <br /> The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form &lt;math&gt;(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})&lt;/math&gt;. For each vector, &lt;math&gt;\Delta x&lt;/math&gt; and &lt;math&gt;\Delta y&lt;/math&gt; give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; &lt;math&gt;p_{1}&lt;/math&gt; indicates that the pen is down and a line should be drawn between the current point and the next point, &lt;math&gt;p_{2}&lt;/math&gt; indicates that the pen is up and no line should be drawn between the current point and the next point, and &lt;math&gt;p_{3}&lt;/math&gt; indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br /> <br /> === Sketch-RNN ===<br /> [[File:sketchrnn.PNG]]<br /> <br /> The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings (sequence &lt;math&gt;S&lt;/math&gt;) in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state &lt;math&gt;h&lt;/math&gt;. <br /> <br /> \begin{align}<br /> h_\rightarrow = \text{encode}_\rightarrow(S), h_\leftarrow = \text{encode}_\leftarrow(S_{\text{reverse}}), h=[h_\rightarrow; h_\leftarrow]<br /> \end{align}<br /> <br /> The concatenated hidden state &lt;math&gt;h&lt;/math&gt; is then projected into two vectors &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\hat{\sigma}&lt;/math&gt; each of size &lt;math&gt;N_{z}&lt;/math&gt; using a fully connected layer. &lt;math&gt;\hat{\sigma}&lt;/math&gt; is then converted into a non-negative standard deviation parameter &lt;math&gt;\sigma&lt;/math&gt; using an exponential operator. These two parameters &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt; are then used along with an IID Gaussian vector distributed as &lt;math&gt;\mathcal{N}(0, I)&lt;/math&gt; of size &lt;math&gt;N_{z}&lt;/math&gt; to construct a random vector &lt;math&gt;z \in ℝ^{N_{z}}&lt;/math&gt;, similar to the method used for VAE:<br /> \begin{align}<br /> \mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br /> \end{align}<br /> <br /> The decoder model is an autoregressive RNN that samples output sketches from the latent vector &lt;math&gt;z&lt;/math&gt;. The initial hidden states of each recurrent neuron are determined using &lt;math&gt;[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})&lt;/math&gt;. Each step of the decoder RNN accepts the previous point &lt;math&gt;S_{i-1}&lt;/math&gt; and the latent vector &lt;math&gt;z&lt;/math&gt; as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point &lt;math&gt;S_{i}&lt;/math&gt;. Outputs &lt;math&gt;\Delta x&lt;/math&gt; and &lt;math&gt;\Delta y&lt;/math&gt; are modeled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states &lt;math&gt;(q_{1}, q_{2}, q_{3})&lt;/math&gt; modelled as a categorical distribution with one-hot encoding.<br /> \begin{align}<br /> P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br /> \end{align}<br /> <br /> For each of the M distributions in the GMM, parameters &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt; are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter &lt;math&gt;\rho_{xy}&lt;/math&gt; signifying correlation of each bivariate normal distribution. An additional vector &lt;math&gt;\Pi&lt;/math&gt; is an output giving the mixture weights for the GMM. The output &lt;math&gt;S_{i}&lt;/math&gt; is determined from each of the mixture models using softmax sampling from these distributions.<br /> <br /> One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter &lt;math&gt;N_{max}&lt;/math&gt; which basically is the length of the longest sketch in the dataset and limits the number of points per drawing to being no more than &lt;math&gt;N_{max}&lt;/math&gt;, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br /> <br /> To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time &lt;math&gt;N_{max}&lt;/math&gt;. The authors also introduce a “temperature” parameter &lt;math&gt;\tau&lt;/math&gt; that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br /> <br /> \begin{align}<br /> \hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br /> \end{align}<br /> <br /> This parameter &lt;math&gt;\tau&lt;/math&gt; lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br /> <br /> === Unconditional Generation ===<br /> <br /> [[File:paper15_Unconditional_Generation.png|800px|]]<br /> <br /> The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable &lt;math&gt;z&lt;/math&gt;. Figure 3 above shows different sketches that are sampled from the network by only varying the temperature parameter &lt;math&gt;\tau&lt;/math&gt; between 0.2 and 0.9.<br /> <br /> === Training ===<br /> The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss &lt;math&gt;L_{R}&lt;/math&gt; and KL Divergence Loss &lt;math&gt;L_{KL}&lt;/math&gt;. The reconstruction loss term is composed of two terms; &lt;math&gt;L_{s}&lt;/math&gt;, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;L_{p}&lt;/math&gt; which is the log loss of the pen state terms.<br /> \begin{align}<br /> L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br /> \end{align}<br /> \begin{align}<br /> L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br /> \end{align}<br /> \begin{align}<br /> L_{R} = L_{s} + L{p}<br /> \end{align}<br /> <br /> The KL divergence loss &lt;math&gt;L_{KL}&lt;/math&gt; measures the difference between the latent vector &lt;math&gt;z&lt;/math&gt; and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions &lt;math&gt;N_{z}&lt;/math&gt; is calculated as:<br /> \begin{align}<br /> L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br /> \end{align}<br /> <br /> The loss for the entire model is thus the weighted sum:<br /> \begin{align}<br /> Loss = L_{R} + w_{KL}L_{KL}<br /> \end{align}<br /> <br /> The value of the weight parameter &lt;math&gt;w_{KL}&lt;/math&gt; has the effect that as &lt;math&gt;w_{KL} \rightarrow 0&lt;/math&gt;, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder. As with VAEs, there is a trade-off between optimizing for the two loss terms (i.e. between how precisely the model can regenerate training data &lt;math&gt;S&lt;/math&gt; and how closely the latent vector &lt;math&gt;z&lt;/math&gt; follows a standard normal distribution) - smaller values of &lt;math&gt;w_{KL}&lt;/math&gt; lead to better &lt;math&gt;L_R&lt;/math&gt; and worse &lt;math&gt;L_{KL}&lt;/math&gt; compared to bigger values of &lt;math&gt;w_{KL}&lt;/math&gt;. Also for unconditional generation, the model is a standalone decoder, so there will be no &lt;math&gt;L_{KL}&lt;/math&gt; term as only &lt;math&gt;L_{R}&lt;/math&gt; is optimized for. This trade-off is illustrated in Figure 4 showing different settings of &lt;math&gt;w_{KL}&lt;/math&gt; and the resulting &lt;math&gt;L_{KL}&lt;/math&gt; and &lt;math&gt;L_{R}&lt;/math&gt;, as well as just &lt;math&gt;L_{R}&lt;/math&gt; in the case of unconditional generation with only a standalone decoder.<br /> <br /> [[File:paper15_fig4.png|600px]]<br /> <br /> === Model Configuration ===<br /> In the given model, the encoder and decoder RNNs consist of 512 and 2048 nodes respectively. Also, M = 20 mixture components are used for the decoder RNN. Layer Normalization is applied to the model, and during training recurrent dropout is applied with a keep probability of 90%. The model is trained with batch sizes of 100 samples, using Adam with a learning rate of 0.0001 and gradient clipping of 1.0. During training, simple data augmentation is performed by multiplying the offset columns by two IID random factors. <br /> <br /> = Experiments =<br /> The authors trained multiple conditional and unconditional models using varying values of &lt;math&gt;w_{KL}&lt;/math&gt; and recorded the different &lt;math&gt;L_{R}&lt;/math&gt; and &lt;math&gt;L_{KL}&lt;/math&gt; values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks. (A HyperLSTM consists of two coupled LSTMS: an auxiliary LSTM and a main LSTM. At every time step, the auxiliary LSTM reads the previous hidden state and the current input vector, and computes an intermediate vector &lt;math display=&quot;inline&quot;&gt; z &lt;/math&gt;. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector &lt;math display=&quot;inline&quot;&gt; z &lt;/math&gt;. That is, the weights of the main LSTM are allowed to vary between time steps as a function of the output of the auxiliary LSTM. See Ha et al. (2016) for details)<br /> <br /> === Conditional Reconstruction ===<br /> [[File:conditional_generation.PNG]]<br /> <br /> The authors qualitatively assessed the reconstructed images &lt;math&gt;S’&lt;/math&gt; given input sketch &lt;math&gt;S&lt;/math&gt; using different values for the temperature hyperparameter &lt;math&gt;\tau&lt;/math&gt;. The figure above shows the results for different values of &lt;math&gt;\tau&lt;/math&gt; starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes are reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush are reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br /> <br /> === Latent Space Interpolation ===<br /> [[File:latent_space_interp.PNG]]<br /> <br /> The latent space vectors &lt;math&gt;z&lt;/math&gt; have few “gaps” between encoded latent space vectors due to the enforcement of a Gaussian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors. A model trained with higher &lt;math&gt;w_{KL}&lt;/math&gt; is expected to produce images closer to the data manifold, and the figure above shows reconstructed images from latent vector interpolation between the original images. Results from the model trained with higher &lt;math&gt;w_{KL}&lt;/math&gt; seem to produce more coherent images.<br /> <br /> === Sketch Drawing Analogies ===<br /> Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br /> <br /> === Predicting Different Endings of Incomplete Sketches ===<br /> [[File:predicting_endings.PNG]]<br /> <br /> Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch with &lt;math&gt;\tau&lt;/math&gt; set to 0.8.<br /> <br /> = Applications and Future Work =<br /> Sketch-RNN may enable the production of several creative applications. These might include suggesting ways an artist could finish a sketch, enabling artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allowing the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output from a different class might be useful for producing sketches that morph the two different object classes into one sketch. For example, the image below was trained on drawing cats, but a chair was used as the input. This results in a chair looking cat.<br /> <br /> [[File:cat-chair.png]]<br /> <br /> Sketch-RNN may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to proceed to finish a sketch or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br /> <br /> The authors noted that Sketch-RNN is not as effective at generating coherent sketches when trained on a large number of classes simultaneously (experiments mostly used datasets consisting of one or two object classes), and plan to use class information outside the latent space to try to model a greater number of classes.<br /> <br /> Finally, the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the Sketch-RNN model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br /> <br /> = Limitations =<br /> The authors note a major limitation to the model is the training time relative to the number of data points. When sketches surpass 300 data points the model is difficult to train. To counteract this effect the Ramer-Douglas-Peucker algorithm was used to reduce the number of data points per sketch. This algorithm attempts to significantly reduce the number of data points while keeping the sketch as close to the original as possible.<br /> <br /> Another limitation is the effectiveness of generating sketches as the complexity of the class increases. Below are sketches of a few classes which show how the less complex classes such as cats and crabs are more accurately generated. Frogs (more complex) tend to have overly smooth lines drawn which do not seem to be part of realistic frog samples.<br /> <br /> [[File:paper15_classcomplexity.png]]<br /> <br /> = Conclusion =<br /> The authors presented Sketch-RNN, a RNN model for modelling and generating vector-based sketch drawings with a goal to abstract concepts in the images similar to the way humans think. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br /> <br /> = Criticisms =<br /> The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br /> <br /> One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input &lt;math&gt;z&lt;/math&gt; by manipulating the &lt;math&gt;\tau&lt;/math&gt; hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like &lt;math&gt;\mu&lt;/math&gt; and &lt;math&gt;\sigma&lt;/math&gt;, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br /> <br /> The authors contribute to a novel dataset but fail to evaluate the quality of the dataset, including generalized metrics for evaluation. They also provide no comparisons of their method on this dataset with other baseline sequence generation approaches.<br /> <br /> Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br /> <br /> == Major Contributions ==<br /> <br /> The paper provides with intuition of their approach and detailed below are the major contributions of this paper:<br /> * For images composed by sequence of lines, such as hand drawing, this paper proposed a framework to generate such image in vector format, conditionally and unconditionally. <br /> * Provided a unique training procedure that targets vector images, which makes training procedures more robust.<br /> * Composed large dataset of hand drawn vector images which benefits future development.<br /> * Discussed several potential applications of this methodology, such as drawing assist for artists and educational tool for students.<br /> <br /> = Implementation =<br /> Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn<br /> <br /> = Source =<br /> <br /> # Ha, D., &amp; Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).<br /> # Tensorflow/magenta. (n.d.). Retrieved March 25, 2018, from https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn<br /> # Graves et al, 2013, https://arxiv.org/pdf/1308.0850.pdf</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Self_Normalizing_Neural_Networks&diff=36221 stat946w18/Self Normalizing Neural Networks 2018-04-06T23:35:23Z <p>S3wen: /* Experimental Results */</p> <hr /> <div>==Introduction and Motivation==<br /> <br /> While neural networks have been making a lot of headway in improving benchmark results and narrowing the gap with human-level performance, success has been fairly limited to visual and sequential processing tasks through advancements in convolutional network and recurrent network structures. Most data science competitions outside of these domains are still outperformed by algorithms such as gradient boosting and random forests. The traditional (densely connected) feed-forward neural networks (FNNs) are rarely used competitively, and when they do win on rare occasions, they have very shallow network architectures with just up to four layers .<br /> <br /> The authors, Klambauer et al., believe that what prevents FNNs from becoming more competitively useful is the inability to train a deeper FNN structure, which would allow the network to learn more levels of abstract representations. Primarily, the difficulty arises due to the instability of gradients in very deep FNNs leading to problems like gradient vanishing/explosion. To have a deeper network, oscillations in the distribution of activations need to be kept under control so that stable gradients can be obtained during training. Several techniques are available to normalize activations, including batch normalization , layer normalization  and weight normalization . These methods work well with CNNs and RNNs, but not so much with FNNs because backpropagating through normalization parameters introduces additional variance to the gradients, and regularization techniques like dropout further perturb the normalization effect. CNNs and RNNs are less sensitive to such perturbations, presumably due to their weight sharing architecture, but FNNs do not have such a property and thus suffer from high variance in training errors, which hinders learning. Furthermore, the aforementioned normalization techniques involve adding external layers to the model and can slow down computation, which may already be slow when working with very deep FNNs. <br /> <br /> Therefore, the authors were motivated to develop a new FNN implementation that can achieve the intended effect of normalization techniques that works well with stochastic gradient descent and dropout. Self-normalizing neural networks (SNNs) are based on the idea of scaled exponential linear units (SELU), a new activation function introduced in this paper, whose output distribution is proved to converge to a fixed point, thus making it possible to train deeper networks.<br /> <br /> ==Notations==<br /> <br /> As the paper (primarily in the supplementary materials) comes with lengthy proofs, important notations are listed first.<br /> <br /> Consider two fully-connected layers, let &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; denote the inputs to the second layer, then &lt;math display=&quot;inline&quot;&gt;z = Wx&lt;/math&gt; represents the network inputs of the second layer, and &lt;math display=&quot;inline&quot;&gt;y = f(z)&lt;/math&gt; represents the activations in the second layer.<br /> <br /> Assume that all &lt;math display=&quot;inline&quot;&gt;x_i&lt;/math&gt;'s, &lt;math display=&quot;inline&quot;&gt;1 \leqslant i \leqslant n&lt;/math&gt;, have mean &lt;math display=&quot;inline&quot;&gt;\mu := \mathrm{E}(x_i)&lt;/math&gt; and variance &lt;math display=&quot;inline&quot;&gt;\nu := \mathrm{Var}(x_i)&lt;/math&gt; and that each &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; has mean &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu} := \mathrm{E}(y)&lt;/math&gt; and variance &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} := \mathrm{Var}(y)&lt;/math&gt;, then let &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; be the set of functions that maps &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt; to &lt;math display=&quot;inline&quot;&gt;(\widetilde{\mu}, \widetilde{\nu})&lt;/math&gt;. <br /> <br /> For the weight vector &lt;math display=&quot;inline&quot;&gt;w&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;n&lt;/math&gt; times the mean of the weight vector is &lt;math display=&quot;inline&quot;&gt;\omega := \sum_{i = 1}^n \omega_i&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;n&lt;/math&gt; times the second moment is &lt;math display=&quot;inline&quot;&gt;\tau := \sum_{i = 1}^{n} w_i^2&lt;/math&gt;.<br /> <br /> ==Key Concepts==<br /> <br /> ===Self-Normalizing Neural-Net (SNN)===<br /> <br /> ''A neural network is self-normalizing if it possesses a mapping &lt;math display=&quot;inline&quot;&gt;g: \Omega \rightarrow \Omega&lt;/math&gt; for each activation &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; that maps mean and variance from one layer to the next and has a stable and attracting fixed point depending on &lt;math display=&quot;inline&quot;&gt;(\omega, \tau)&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;. Furthermore, the mean and variance remain in the domain &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, that is &lt;math display=&quot;inline&quot;&gt;g(\Omega) \subseteq \Omega&lt;/math&gt;, where &lt;math display=&quot;inline&quot;&gt;\Omega = \{ (\mu, \nu) | \mu \in [\mu_{min}, \mu_{max}], \nu \in [\nu_{min}, \nu_{max}] \}&lt;/math&gt;. When iteratively applying the mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt;, each point within &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt; converges to this fixed point.''<br /> <br /> In other words, in SNNs, if the inputs from an earlier layer (&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;) already have their mean and variance within a predefined interval &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, then the activations to the next layer (&lt;math display=&quot;inline&quot;&gt;y = f(z = Wx)&lt;/math&gt;) should remain within those intervals. This is true across all pairs of connecting layers as the normalizing effect gets propagated through the network, hence why the term self-normalizing. When the mapping is applied iteratively, it should draw the mean and variance values closer to a fixed point within &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, the value of which depends on &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt; (recall that they are from the weight vector).<br /> <br /> We will design a FNN then construct a g that takes the mean and variance of each layer to those of the next and is a contraction mapping i.e. &lt;math&gt;g(\mu_i, \nu_i) = (\mu_{i+1}, \nu_{i+1}) \forall i &lt;/math&gt;. It should be noted that although the g required in the SNN definition depends on &lt;math display=&quot;inline&quot;&gt;(\omega, \tau)&lt;/math&gt; of an individual layer, the FNN that we construct will have the same values of &lt;math display=&quot;inline&quot;&gt;(\omega, \tau)&lt;/math&gt; for each layer. Intuitively this definition can be interpreted as saying that the mean and variance of the final layer of a sufficiently deep SNN will not change when the mean and variance of the input data change. This is because the mean and variance are passing through a contraction mapping at each layer, converging to the mapping's fixed point.<br /> <br /> The activation function that makes an SNN possible should meet the following four conditions:<br /> <br /> # It can take on both negative and positive values, so it can normalize the mean;<br /> # It has a saturation region, so it can dampen variances that are too large;<br /> # It has a slope larger than one, so it can increase variances that are too small; and<br /> # It is a continuous curve, which is necessary for the fixed point to exist (see the definition of Banach fixed point theorem to follow).<br /> <br /> Commonly used activation functions such as rectified linear units (ReLU), sigmoid, tanh, leaky ReLUs and exponential linear units (ELUs) do not meet all four criteria, therefore, a new activation function is needed.<br /> <br /> ===Scaled Exponential Linear Units (SELUs)===<br /> <br /> One of the main ideas introduced in this paper is the SELU function. As the name suggests, it is closely related to ELU ,<br /> <br /> $\mathrm{elu}(x) = \begin{cases} x &amp; x &gt; 0 \\<br /> \alpha e^x - \alpha &amp; x \leqslant 0<br /> \end{cases}$<br /> <br /> but further builds upon it by introducing a new scale parameter $\lambda$ and proving the exact values that $\alpha$ and $\lambda$ should take on to achieve self-normalization. SELU is defined as:<br /> <br /> $\mathrm{selu}(x) = \lambda \begin{cases} x &amp; x &gt; 0 \\<br /> \alpha e^x - \alpha &amp; x \leqslant 0<br /> \end{cases}$<br /> <br /> SELUs meet all four criteria listed above - it takes on positive values when &lt;math display=&quot;inline&quot;&gt;x &gt; 0&lt;/math&gt; and negative values when &lt;math display=&quot;inline&quot;&gt;x &lt; 0&lt;/math&gt;, it has a saturation region when &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; is a larger negative value, the value of &lt;math display=&quot;inline&quot;&gt;\lambda&lt;/math&gt; can be set to greater than one to ensure a slope greater than one, and it is continuous at &lt;math display=&quot;inline&quot;&gt;x = 0&lt;/math&gt;. <br /> <br /> Figure 1 below gives an intuition for how SELUs normalize activations across layers. As shown, a variance dampening effect occurs when inputs are negative and far away from zero, and a variance increasing effect occurs when inputs are close to zero.<br /> <br /> [[File:snnf1.png|500px]]<br /> <br /> Figure 2 below plots the progression of training error on the MNIST and CIFAR10 datasets when training with SNNs versus FNNs with batch normalization at varying model depths. As shown, FNNs that adopted the SELU activation function exhibited lower and less variable training loss compared to using batch normalization, even as the depth increased to 16 and 32 layers.<br /> <br /> [[File:snnf2.png|600px]]<br /> <br /> === Banach Fixed Point Theorem and Contraction Mappings ===<br /> <br /> The underlying theory behind SNNs is the Banach fixed point theorem, which states the following: ''Let &lt;math display=&quot;inline&quot;&gt;(X, d)&lt;/math&gt; be a non-empty complete metric space with a contraction mapping &lt;math display=&quot;inline&quot;&gt;f: X \rightarrow X&lt;/math&gt;. Then &lt;math display=&quot;inline&quot;&gt;f&lt;/math&gt; has a unique fixed point &lt;math display=&quot;inline&quot;&gt;x_f \subseteq X&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;f(x_f) = x_f&lt;/math&gt;. Every sequence &lt;math display=&quot;inline&quot;&gt;x_n = f(x_{n-1})&lt;/math&gt; with starting element &lt;math display=&quot;inline&quot;&gt;x_0 \subseteq X&lt;/math&gt; converges to the fixed point: &lt;math display=&quot;inline&quot;&gt;x_n \underset{n \rightarrow \infty}\rightarrow x_f&lt;/math&gt;.''<br /> <br /> A contraction mapping is a function &lt;math display=&quot;inline&quot;&gt;f: X \rightarrow X&lt;/math&gt; on a metric space &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; with distance &lt;math display=&quot;inline&quot;&gt;d&lt;/math&gt;, such that for all points &lt;math display=&quot;inline&quot;&gt;\mathbf{u}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathbf{v}&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt;: &lt;math display=&quot;inline&quot;&gt;d(f(\mathbf{u}), f(\mathbf{v})) \leqslant \delta d(\mathbf{u}, \mathbf{v})&lt;/math&gt;, for a &lt;math display=&quot;inline&quot;&gt;0 \leqslant \delta \leqslant 1&lt;/math&gt;.<br /> <br /> The easiest way to prove a contraction mapping is usually to show that the spectral norm  of its Jacobian is less than 1 , as was done for this paper.<br /> <br /> ==Proving the Self-Normalizing Property==<br /> <br /> ===Mean and Variance Mapping Function===<br /> <br /> &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is derived under the assumption that &lt;math display=&quot;inline&quot;&gt;x_i&lt;/math&gt;'s are independent but not necessarily having the same mean and variance [[#Footnotes |(2)]]. Under this assumption (and recalling earlier notation of &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt;),<br /> <br /> \begin{align}<br /> \mathrm{E}(z = \mathbf{w}^T \mathbf{x}) = \sum_{i = 1}^n w_i \mathrm{E}(x_i) = \mu \omega<br /> \end{align}<br /> <br /> \begin{align}<br /> \mathrm{Var}(z) = \mathrm{Var}(\sum_{i = 1}^n w_i x_i) = \sum_{i = 1}^n w_i^2 \mathrm{Var}(x_i) = \nu \sum_{i = 1}^n w_i^2 = \nu\tau \textrm{ .}<br /> \end{align}<br /> <br /> When the weight terms are normalized, &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; can be viewed as a weighted sum of &lt;math display=&quot;inline&quot;&gt;x_i&lt;/math&gt;'s. Wide neural net layers with a large number of nodes is common, so &lt;math display=&quot;inline&quot;&gt;n&lt;/math&gt; is usually large, and by the Central Limit Theorem, &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; approaches a normal distribution &lt;math display=&quot;inline&quot;&gt;\mathcal{N}(\mu\omega, \sqrt{\nu\tau})&lt;/math&gt;. <br /> <br /> Using the above property, the exact form for &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; can be obtained using the definitions for mean and variance of continuous random variables: <br /> <br /> [[File:gmapping.png|600px|center]]<br /> <br /> Analytical solutions for the integrals can be obtained as follows: <br /> <br /> [[File:gintegral.png|600px|center]]<br /> <br /> The authors are interested in the fixed point &lt;math display=&quot;inline&quot;&gt;(\mu, \nu) = (0, 1)&lt;/math&gt; as these are the parameters associated with the common standard normal distribution. The authors also proposed using normalized weights such that &lt;math display=&quot;inline&quot;&gt;\omega = \sum_{i = 1}^n = 0&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau = \sum_{i = 1}^n w_i^2= 1&lt;/math&gt; as it gives a simpler, cleaner expression for &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; in the calculations in the next steps. This weight scheme can be achieved in several ways, for example, by drawing from a normal distribution &lt;math display=&quot;inline&quot;&gt;\mathcal{N}(0, \frac{1}{n})&lt;/math&gt; or from a uniform distribution &lt;math display=&quot;inline&quot;&gt;U(-\sqrt{3}, \sqrt{3})&lt;/math&gt;.<br /> <br /> At &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu} = \mu = 0&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} = \nu = 1&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\omega = 0&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau = 1&lt;/math&gt;, the constants &lt;math display=&quot;inline&quot;&gt;\lambda&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\alpha&lt;/math&gt; from the SELU function can be solved for - &lt;math display=&quot;inline&quot;&gt;\lambda_{01} \approx 1.0507&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\alpha_{01} \approx 1.6733&lt;/math&gt;. These values are used throughout the rest of the paper whenever an expression calls for &lt;math display=&quot;inline&quot;&gt;\lambda&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\alpha&lt;/math&gt;.<br /> <br /> ===Details of Moment-Mapping Integrals ===<br /> Consider the moment-mapping integrals:<br /> \begin{align}<br /> \widetilde{\mu} &amp; = \int_{-\infty}^\infty \mathrm{selu} (z) p_N(z; \mu \omega, \sqrt{\nu \tau})dz\\<br /> \widetilde{\nu} &amp; = \int_{-\infty}^\infty \mathrm{selu} (z)^2 p_N(z; \mu \omega, \sqrt{\nu \tau}) dz-\widetilde{\mu}^2.<br /> \end{align}<br /> <br /> The equation for &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; can be expanded as <br /> \begin{align}<br /> \widetilde{\mu} &amp; = \frac{\lambda}{2}\left( 2\alpha\int_{-\infty}^0 (\exp(z)-1) p_N(z; \mu \omega, \sqrt{\nu \tau})dz +2\int_{0}^\infty z p_N(z; \mu \omega, \sqrt{\nu \tau})dz \right)\\<br /> &amp;= \frac{\lambda}{2}\left( 2 \alpha \frac{1}{\sqrt{2\pi\tau\nu}} \int_{-\infty}^0 (\exp(z)-1) \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz +2\frac{1}{\sqrt{2\pi\tau\nu}}\int_{0}^\infty z \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 dz \right)\\<br /> &amp;= \frac{\lambda}{2}\left( 2 \alpha\frac{1}{\sqrt{2\pi\tau\nu}}\int_{-\infty}^0 \exp(z) \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz - 2 \alpha\frac{1}{\sqrt{2\pi\tau\nu}}\int_{-\infty}^0 \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz +2\frac{1}{\sqrt{2\pi\tau\nu}}\int_{0}^\infty z \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 dz \right)\\<br /> \end{align}<br /> <br /> The first integral can be simplified via the substituiton<br /> \begin{align}<br /> q:= \frac{1}{\sqrt{2\tau \nu}}(z-\mu \omega -\tau \nu).<br /> \end{align}<br /> While the second and third can be simplified via the substitution<br /> \begin{align}<br /> q:= \frac{1}{\sqrt{2\tau \nu}}(z-\mu \omega ).<br /> \end{align}<br /> Using the definitions of &lt;math display=&quot;inline&quot;&gt;\mathrm{erf}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathrm{erfc}&lt;/math&gt; then yields the result of the previous section.<br /> <br /> ===Self-Normalizing Property Under Normalized Weights===<br /> <br /> Assuming the the weights normalized with &lt;math display=&quot;inline&quot;&gt;\omega=0&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau=1&lt;/math&gt;, it is possible to calculate the exact value for the spectral norm  of &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt;'s Jacobian around the fixed point &lt;math display=&quot;inline&quot;&gt;(\mu, \nu) = (0, 1)&lt;/math&gt;, which turns out to be &lt;math display=&quot;inline&quot;&gt;0.7877&lt;/math&gt;. Thus, at initialization, SNNs have a stable and attracting fixed point at &lt;math display=&quot;inline&quot;&gt;(0, 1)&lt;/math&gt;, which means that when &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is applied iteratively to a pair &lt;math display=&quot;inline&quot;&gt;(\mu_{new}, \nu_{new})&lt;/math&gt;, it should draw the points closer to &lt;math display=&quot;inline&quot;&gt;(0, 1)&lt;/math&gt;. The rate of convergence is determined by the spectral norm , whose value depends on &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt;.<br /> <br /> [[File:paper10_fig2.png|600px|frame|none|alt=Alt text|The figure illustrates, in the scenario described above, the mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; of mean and variance &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt; to &lt;math display=&quot;inline&quot;&gt;(\mu_{new}, \nu_{new})&lt;/math&gt;. The arrows show the direction &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt; is mapped by &lt;math display=&quot;inline&quot;&gt;g: (\mu, \nu)\mapsto(\mu_{new}, \nu_{new})&lt;/math&gt;. One can clearly see the fixed point mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is at &lt;math display=&quot;inline&quot;&gt;(0, 1)&lt;/math&gt;.]]<br /> <br /> ===Self-Normalizing Property Under Unnormalized Weights===<br /> <br /> As weights are updated during training, there is no guarantee that they would remain normalized. The authors addressed this issue through the first key theorem presented in the paper, which states that a fixed point close to (0, 1) can still be obtained if &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt; are restricted to a specified range. <br /> <br /> Additionally, there is no guarantee that the mean and variance of the inputs would stay within the range given by the first theorem, which led to the development of theorems #2 and #3. These two theorems established an upper and lower bound on the variance of inputs if the variance of activations from the previous layer are above or below the range specified, respectively. This ensures that the variance would not explode or vanish after being propagated through the network.<br /> <br /> The theorems come with lengthy proofs in the supplementary materials for the paper. High-level proof sketches are presented here.<br /> <br /> ====Theorem 1: Stable and Attracting Fixed Points Close to (0, 1)====<br /> <br /> '''Definition:''' We assume &lt;math display=&quot;inline&quot;&gt;\alpha = \alpha_{01}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\lambda = \lambda_{01}&lt;/math&gt;. We restrict the range of the variables to the domain &lt;math display=&quot;inline&quot;&gt;\mu \in [-0.1, 0.1]&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\omega \in [-0.1, 0.1]&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\nu \in [0.8, 1.5]&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\tau \in [0.9, 1.1]&lt;/math&gt;. For &lt;math display=&quot;inline&quot;&gt;\omega = 0&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau = 1&lt;/math&gt;, the mapping has the stable fixed point &lt;math display=&quot;inline&quot;&gt;(\mu, \nu) = (0, 1&lt;/math&gt;. For other &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt;, g has a stable and attracting fixed point depending on &lt;math display=&quot;inline&quot;&gt;(\omega, \tau)&lt;/math&gt; in the &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt;-domain: &lt;math display=&quot;inline&quot;&gt;\mu \in [-0.03106, 0.06773]&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu \in [0.80009, 1.48617]&lt;/math&gt;. All points within the &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt;-domain converge when iteratively applying the mapping to this fixed point.<br /> <br /> '''Proof:''' In order to show the the mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; has a stable and attracting fixed point close to &lt;math display=&quot;inline&quot;&gt;(0, 1)&lt;/math&gt;, the authors again applied Banach's fixed point theorem, which states that a contraction mapping on a nonempty complete metric space that does not map outside its domain has a unique fixed point, and that all points in the &lt;math display=&quot;inline&quot;&gt;(\mu, \nu)&lt;/math&gt;-domain converge to the fixed point when &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is iteratively applied. <br /> <br /> The two requirements are proven as follows:<br /> <br /> '''1. g is a contraction mapping.'''<br /> <br /> For &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; to be a contraction mapping in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt; with distance &lt;math display=&quot;inline&quot;&gt;||\cdot||_2&lt;/math&gt;, there must exist a Lipschitz constant &lt;math display=&quot;inline&quot;&gt;M &lt; 1&lt;/math&gt; such that: <br /> <br /> \begin{align} <br /> \forall \mu, \nu \in \Omega: ||g(\mu) - g(\nu)||_2 \leqslant M||\mu - \nu||_2 <br /> \end{align}<br /> <br /> As stated earlier, &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is a contraction mapping if the spectral norm  of the Jacobian &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; [[#Footnotes | (3)]] is below one, or equivalently, if the the largest singular value of &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; is less than 1.<br /> <br /> To find the singular values of &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt;, the authors used an explicit formula derived by Blinn  for &lt;math display=&quot;inline&quot;&gt;2\times2&lt;/math&gt; matrices, which states that the largest singular value of the matrix is &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}(\sqrt{(a_{11} + a_{22}) ^ 2 + (a_{21} - a{12})^2} + \sqrt{(a_{11} - a_{22}) ^ 2 + (a_{21} + a{12})^2})&lt;/math&gt;.<br /> <br /> For &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt;, an expression for the largest singular value of &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt;, made up of the first-order partial derivatives of the mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; with respect to &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt;, can be derived given the analytical solutions for &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; (and denoted &lt;math display=&quot;inline&quot;&gt;S(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt;).<br /> <br /> From the mean value theorem, we know that for a &lt;math display=&quot;inline&quot;&gt;t \in [0, 1]&lt;/math&gt;, <br /> <br /> [[File:seq.png|600px|center]]<br /> <br /> Therefore, the distance of the singular value at &lt;math display=&quot;inline&quot;&gt;S(\mu, \omega, \nu, \tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})&lt;/math&gt; and at &lt;math display=&quot;inline&quot;&gt;S(\mu + \Delta\mu, \omega + \Delta\omega, \nu + \Delta\nu, \tau \Delta\tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})&lt;/math&gt; can be bounded above by <br /> <br /> [[File:seq2.png|600px|center]]<br /> <br /> An upper bound was obtained for each partial derivative term above, mainly through algebraic reformulations and by making use of the fact that many of the functions are monotonically increasing or decreasing on the variables they depend on in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt; (see pages 17 - 25 in the supplementary materials).<br /> <br /> The &lt;math display=&quot;inline&quot;&gt;\Delta&lt;/math&gt; terms were then set (rather arbitrarily) to be: &lt;math display=&quot;inline&quot;&gt;\Delta \mu=0.0068097371&lt;/math&gt;,<br /> &lt;math display=&quot;inline&quot;&gt;\Delta \omega=0.0008292885&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\Delta \nu=0.0009580840&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\Delta \tau=0.0007323095&lt;/math&gt;. Plugging in the upper bounds on the absolute values of the derivative terms for &lt;math display=&quot;inline&quot;&gt;S&lt;/math&gt; and the &lt;math display=&quot;inline&quot;&gt;\Delta&lt;/math&gt; terms yields<br /> <br /> $S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) - S(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) &lt; 0.008747$<br /> <br /> Next, the largest singular value is found from a computer-assisted fine grid-search [[#Footnotes | (1)]] over the domain &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, with grid lengths &lt;math display=&quot;inline&quot;&gt;\Delta \mu=0.0068097371&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\Delta \omega=0.0008292885&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\Delta \nu=0.0009580840&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\Delta \tau=0.0007323095&lt;/math&gt;, which turned out to be &lt;math display=&quot;inline&quot;&gt;0.9912524171058772&lt;/math&gt;. Therefore, <br /> <br /> $S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) \leq 0.9912524171058772 + 0.008747 &lt; 1$<br /> <br /> Since the largest singular value is smaller than 1, &lt;math display=&quot;inline&gt;g&lt;/math&gt; is a contraction mapping.<br /> <br /> '''2. g does not map outside its domain.'''<br /> <br /> To prove that &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; does not map outside of the domain &lt;math display=&quot;inline&quot;&gt;\mu \in [-0.1, 0.1]&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu \in [0.8, 1.5]&lt;/math&gt;, lower and upper bounds on &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; were obtained to show that they stay within &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;. <br /> <br /> First, it was shown that the derivatives of &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\xi}&lt;/math&gt; with respect to &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt; are either positive or have the sign of &lt;math display=&quot;inline&quot;&gt;\omega&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, so the minimum and maximum points are found at the borders. In &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, it then follows that<br /> <br /> \begin{align}<br /> -0.03106 &lt;\widetilde{\mu}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq &amp; \widetilde{\mu} \leq \widetilde{\mu}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) &lt; 0.06773<br /> \end{align}<br /> <br /> and <br /> <br /> \begin{align}<br /> 0.80467 &lt;\widetilde{\xi}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq &amp; \widetilde{\xi} \leq \widetilde{\xi}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) &lt; 1.48617.<br /> \end{align}<br /> <br /> Since &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} = \widetilde{\xi} - \widetilde{\mu}^2&lt;/math&gt;, <br /> <br /> \begin{align}<br /> 0.80009 &amp; \leqslant \widetilde{\nu} \leqslant 1.48617<br /> \end{align}<br /> <br /> The bounds on &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; are narrower than those for &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt; set out in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, therefore &lt;math display=&quot;inline&quot;&gt;g(\Omega) \subseteq \Omega&lt;/math&gt;.<br /> <br /> ==== Theorem 2: Decreasing Variance from Above ====<br /> <br /> '''Definition:''' For &lt;math display=&quot;inline&quot;&gt;\lambda = \lambda_{01}&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\alpha = \alpha_{01}&lt;/math&gt;, and the domain &lt;math display=&quot;inline&quot;&gt;\Omega^+: -1 \leqslant \mu \leqslant 1, -0.1 \leqslant \omega \leqslant 0.1, 3 \leqslant \nu \leqslant 16&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;0.8 \leqslant \tau \leqslant 1.25&lt;/math&gt;, we have for the mapping of the variance &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt; under &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt;: &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) &lt; \nu&lt;/math&gt;.<br /> <br /> Theorem 2 states that when &lt;math display=&quot;inline&quot;&gt;\nu \in [3, 16]&lt;/math&gt;, the mapping &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; draws it to below 3 when applied across layers, thereby establishing an upper bound of &lt;math display=&quot;inline&quot;&gt;\nu &lt; 3&lt;/math&gt; on variance.<br /> <br /> '''Proof:''' The authors proved the inequality by showing that &lt;math display=&quot;inline&quot;&gt;g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) = \widetilde{\xi}(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) - \nu &lt; 0&lt;/math&gt;, since the second moment should be greater than or equal to variance &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt;. The behavior of &lt;math display=&quot;inline&quot;&gt;\frac{\partial }{\partial \mu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\frac{\partial }{\partial \omega } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\frac{\partial }{\partial \nu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\frac{\partial }{\partial \tau } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt; are used to find the bounds on &lt;math display=&quot;inline&quot;&gt;g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01})&lt;/math&gt; (see pages 9 - 13 in the supplementary materials). Again, the partial derivative terms were monotonic, which made it possible to find the upper bound at the board values. It was shown that the maximum value of &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; does not exceed &lt;math display=&quot;inline&quot;&gt;-0.0180173&lt;/math&gt;.<br /> <br /> ==== Theorem 3: Increasing Variance from Below ====<br /> <br /> '''Definition''': We consider &lt;math display=&quot;inline&quot;&gt;\lambda = \lambda_{01}&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\alpha = \alpha_{01}&lt;/math&gt;, and the domain &lt;math display=&quot;inline&quot;&gt;\Omega^-: -0.1 \leqslant \mu \leqslant 0.1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;-0.1 \leqslant \omega \leqslant 0.1&lt;/math&gt;. For the domain &lt;math display=&quot;inline&quot;&gt;0.02 \leqslant \nu \leqslant 0.16&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;0.8 \leqslant \tau \leqslant 1.25&lt;/math&gt; as well as for the domain &lt;math display=&quot;inline&quot;&gt;0.02 \leqslant \nu \leqslant 0.24&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;0.9 \leqslant \tau \leqslant 1.25&lt;/math&gt;, the mapping of the variance &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt; increases: &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) &gt; \nu&lt;/math&gt;.<br /> <br /> Theorem 3 states that the variance &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; increases when variance is smaller than in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;. The lower bound on variance is &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} &gt; 0.16&lt;/math&gt; when &lt;math display=&quot;inline&quot;&gt;0.8 \leqslant \tau&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} &gt; 0.24&lt;/math&gt; when &lt;math display=&quot;inline&quot;&gt;0.9 \leqslant \tau&lt;/math&gt; under the proposed mapping.<br /> <br /> '''Proof:''' According to the mean value theorem, for a &lt;math display=&quot;inline&quot;&gt;t \in [0, 1]&lt;/math&gt;,<br /> <br /> [[File:th3.png|700px|center]]<br /> <br /> Similar to the proof for Theorem 2 (except we are interested in the smallest &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; instead of the biggest), the lower bound for &lt;math display=&quot;inline&quot;&gt;\frac{\partial }{\partial \nu} \widetilde{\xi}(\mu,\omega,\nu+t(\nu_{\mathrm{min}}-\nu),\tau,\lambda_{\rm 01},\alpha_{\rm 01})&lt;/math&gt; can be derived, and substituted into the relationship &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu} = \widetilde{\xi}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) - (\widetilde{\mu}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}))^2&lt;/math&gt;. The lower bound depends on &lt;math display=&quot;inline&quot;&gt;\tau&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt;, and in the &lt;math display=&quot;inline&quot;&gt;\Omega^{-1}&lt;/math&gt; listed, it is slightly above &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt;.<br /> <br /> == Implementation Details ==<br /> <br /> === Initialization ===<br /> <br /> As previously explained, SNNs work best when inputs to the network are standardized, and the weights are initialized with mean of 0 and variance of &lt;math display=&quot;inline&quot;&gt;\frac{1}{n}&lt;/math&gt; to help converge to the fixed point &lt;math display=&quot;inline&quot;&gt;(\mu, \nu) = (0, 1)&lt;/math&gt;.<br /> <br /> === Dropout Technique ===<br /> <br /> The authors reason that regular dropout, randomly setting activations to 0 with probability &lt;math display=&quot;inline&quot;&gt;1 - q&lt;/math&gt;, is not compatible with SELUs. This is because the low variance region in SELUs is at &lt;math display=&quot;inline&quot;&gt;\lim_{x \rightarrow -\infty} = -\lambda \alpha&lt;/math&gt;, not 0. Contrast this with ReLUs, which work well with dropout since they have &lt;math display=&quot;inline&quot;&gt;\lim_{x \rightarrow -\infty} = 0&lt;/math&gt; as the saturation region. Therefore, a new dropout technique for SELUs was needed, termed ''alpha dropout''.<br /> <br /> With alpha dropout, activations are randomly set to &lt;math display=&quot;inline&quot;&gt;-\lambda\alpha = \alpha'&lt;/math&gt;, which for this paper corresponds to the constant &lt;math display=&quot;inline&quot;&gt;1.7581&lt;/math&gt;, with probability &lt;math display=&quot;inline&quot;&gt;1 - q&lt;/math&gt;.<br /> <br /> The updated mean and variance of the activations are now:<br /> $\mathrm{E}(xd + \alpha'(1 - d)) = \mu q + \alpha'(1 - q)$ <br /> <br /> and<br /> <br /> $\mathrm{Var}(xd + \alpha'(1 - d)) = q((1-q)(\alpha' - \mu)^2 + \nu)$<br /> <br /> Activations need to be transformed (e.g. scaled) after dropout to maintain the same mean and variance. In regular dropout, conserving the mean and variance correlates to scaling activations by a factor of 1/q while training. To ensure that mean and variance are unchanged after alpha dropout, the authors used an affine transformation &lt;math display=&quot;inline&quot;&gt;a(xd + \alpha'(1 - d)) + b&lt;/math&gt;, and solved for the values of &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; to give &lt;math display=&quot;inline&quot;&gt;a = (\frac{\nu}{q((1-q)(\alpha' - \mu)^2 + \nu)})^{\frac{1}{2}}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b = \mu - a(q\mu + (1-q)\alpha'))&lt;/math&gt;. As the values for &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\nu&lt;/math&gt; are set to &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; throughout the paper, these expressions can be simplified into &lt;math display=&quot;inline&quot;&gt;a = (q + \alpha'^2 q(1-q))^{-\frac{1}{2}}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b = -(q + \alpha^2 q (1-q))^{-\frac{1}{2}}((1 - q)\alpha')&lt;/math&gt;, where &lt;math display=&quot;inline&quot;&gt;\alpha' \approx 1.7581&lt;/math&gt;.<br /> <br /> Empirically, the authors found that dropout rates (1-q) of &lt;math display=&quot;inline&quot;&gt;0.05&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;0.10&lt;/math&gt; worked well with SNNs.<br /> <br /> === Optimizers ===<br /> <br /> Through experiments, the authors found that stochastic gradient descent, momentum, Adadelta and Adamax work well on SNNs. For Adam, configuration parameters &lt;math display=&quot;inline&quot;&gt;\beta_2 = 0.99&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\epsilon = 0.01&lt;/math&gt; were found to be more effective.<br /> <br /> ==Experimental Results==<br /> <br /> Three sets of experiments were conducted to compare the performance of SNNs to six other FNN structures and to other machine learning algorithms, such as support vector machines and random forests. The experiments were carried out on (1) 121 UCI Machine Learning Repository datasets, (2) the Tox21 chemical compounds toxicity effects dataset (with 12,000 compounds and 270,000 features), and (3) the HTRU2 dataset of statistics on radio wave signals from pulsar candidates (with 18,000 observations and eight features). In each set of experiment, hyperparameter search was conducted on a validation set to select parameters such as the number of hidden units, number of hidden layers, learning rate, regularization parameter, and dropout rate (see pages 95 - 107 of the supplementary material for exact hyperparameters considered). Whenever models of different setups gave identical results on the validation data, preference was given to the structure with more layers, lower learning rate and higher dropout rate.<br /> <br /> The six FNN structures considered were: (1) FNNs with ReLU activations, no normalization and “Microsoft weight initialization” (MSRA)  to control the variance of input signals ; (2) FNNs with batch normalization , in which normalization is applied to activations of the same mini-batch; (3) FNNs with layer normalization , in which normalization is applied on a per layer basis for each training example; (4) FNNs with weight normalization , whereby each layer’s weights are normalized by learning the weight’s magnitude and direction instead of the weight vector itself; (5) highway networks, in which layers are not restricted to being sequentially connected ; and (6) an FNN-version of residual networks , with residual blocks made up of two or three densely connected layers.<br /> <br /> On the Tox21 dataset, the authors demonstrated the self-normalizing effect by comparing the distribution of neural inputs &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; at initialization and after 40 epochs of training to that of the standard normal. As Figure 3 show, the distribution of &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; remained similar to a normal distribution.<br /> <br /> [[File:snnf3.png|600px]]<br /> <br /> On all three sets of classification tasks, the authors demonstrated that SNN outperformed the other FNN counterparts on accuracy and AUC measures. <br /> <br /> SNN achieves close to the state-of-the-art results on the Tox21 dataset with an 8-layer network. The challenge requires prediction of toxic effects of 12000 chemicals based on their chemical structures. SNN with 8 layers had the best performance.<br /> <br /> [[File:tox21.png|600px]]<br /> <br /> <br /> On UCI datasets with fewer than 1,000 observations, SNNs did not outperform SVMs or random forests in terms of average rank in accuracy, but on datasets with at least 1,000 observations, SNNs showed the best overall performance (average rank of 5.8, compared to 6.1 for support vector machines and 6.6 for random forests). Through hyperparameter tuning, it was also discovered that the average depth of FNNs is 10.8 layers, more than the other FNN architectures tried.<br /> <br /> <br /> <br /> In pulsar predictions with the HTRU dataset SNNs produced a new state-of-the-art AUC by a small margin (achieving an AUC 0.98, averaged over 10 cross-validation folds, versus the previous record of 0.976).<br /> [[File:htru2.png|600px]]<br /> <br /> ==Future Work==<br /> <br /> Although not the focus of this paper, the authors also briefly noted that their initial experiments with applying SELUs on relatively simple CNN structures showed promising results, which is not surprising given that ELUs, which do not have the self-normalizing property, has already been shown to work well with CNNs, demonstrating faster convergence than ReLU networks and even pushing the state-of-the-art error rates on CIFAR-100 at the time of publishing in 2015 .<br /> <br /> Since the paper was published, SELUs have been adopted by several researchers, not just with FNNs [https://github.com/bioinf-jku/SNNs see link], but also with CNNs, GANs, autoencoders, reinforcement learning and RNNs. In a few cases, researchers for those papers concluded that networks trained with SELUs converged faster than those trained with ReLUs, and that SELUs have the same convergence quality as batch normalization. There is potential for SELUs to be incorporated into more architectures in the future.<br /> <br /> ==Critique==<br /> <br /> Overall, the authors presented a convincing case for using SELUs (along with proper initialization and alpha dropout) on FNNs. FNNs trained with SELU have more layers than those with other normalization techniques, so the work here provides a promising direction for making traditional FNNs more powerful. There are not as many well-established benchmark datasets to evaluate FNNs, but the experiments carried out, particularly on the larger Tox21 dataset, showed that SNNs can be very effective at classification tasks.<br /> <br /> The proofs provide a satisfactory explanation for why SELUs have a self-normalising property within the specified domain, but during their introduction the authors give 4 criteria that an activation function must satisfy in order to be self-normalising. Those criteria make intuitive sense, but there is a lack of firm justification which creates some confusion. For example, they state that SNNs cannot be derived from tanh units, even though &lt;math&gt; tanh(\lambda x) &lt;/math&gt; satisfies all 4 criteria if &lt;math&gt; \lambda &lt;/math&gt; is larger than 1. Assuming the authors did not overlook such a simple modification, there must be some additional criteria for an activation function to have a self-normalising property.<br /> <br /> The only question I have with the proofs is the lack of explanation for how the domains, &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\Omega^-&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\Omega^+&lt;/math&gt; are determined, which is an important consideration because they are used for deriving the upper and lower bounds on expressions needed for proving the three theorems. The ranges appear somewhat set through trial-and-error and heuristics to ensure the numbers work out (e.g. make the spectral norm  of &lt;math display=&quot;inline&quot;&gt;\mathcal{J}&lt;/math&gt; as large as can be below 1 so as to ensure &lt;math display=&quot;inline&quot;&gt;g&lt;/math&gt; is a contraction mapping), so it is not clear whether they are unique conditions, or that the parameters will remain within those prespecified ranges throughout training; and if the parameters can stray away from the ranges provided, then the issue of what will happen to the self-normalizing property was not addressed. Perhaps that is why the authors gave preference to models with a deeper structure and smaller learning rate during experiments to help the parameters stay within their domains. Further, in addition to the hyperparameters considered, it would be helpful to know the final values that went into the best-performing models, for a better understanding of what range of values work better for SNNs empirically.<br /> <br /> ==Conclusion==<br /> <br /> The SNN structure proposed in this paper is built on the traditional FNN structure with a few modifications, including the use of SELUs as the activation function (with &lt;math display=&quot;inline&quot;&gt;\lambda \approx 1.0507&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\alpha \approx 1.6733&lt;/math&gt;), alpha dropout, network weights initialized with mean of zero and variance &lt;math display=&quot;inline&quot;&gt;\frac{1}{n}&lt;/math&gt;, and inputs normalized to mean of zero and variance of one. It is simple to implement while being backed up by detailed theory. <br /> <br /> When properly initialized, SELUs will draw neural inputs towards a fixed point of zero mean and unit variance as the activations are propagated through the layers. The self-normalizing property is maintained even when weights deviate from their initial values during training (under mild conditions). When the variance of inputs goes beyond the prespecified range imposed, they are still bounded above and below so SNNs do not suffer from exploding and vanishing gradients. This self-normalizing property allows SNNs to be more robust to perturbations in stochastic gradient descent, so deeper structures with better prediction performance can be built. <br /> <br /> In the experiments conducted, the authors demonstrated that SNNs outperformed FNNs trained with other normalization techniques, such as batch, layer and weight normalization, and specialized architectures, such as highway or residual networks, on several classification tasks, including on the UCI Machine Learning Repository datasets. SELUs help in reducing the computation time for normalizing the network relative to RELU+BN and hence is promising. The adoption of SELUs by other researchers also lends credence to the potential for SELUs to be implemented in more neural network architectures.<br /> <br /> ==References==<br /> <br /> # Ba, Kiros and Hinton. &quot;Layer Normalization&quot;. arXiv:1607.06450. (2016).<br /> # Blinn. &quot;Consider the Lowly 2X2 Matrix.&quot; IEEE Computer Graphics and Applications. (1996).<br /> # Clevert, Unterthiner, Hochreiter. &quot;Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs).&quot; arXiv: 1511.07289. (2015).<br /> # He, Zhang, Ren and Sun. &quot;Deep Residual Learning for Image Recognition.&quot; arXiv:1512.03385. (2015).<br /> # He, Zhang, Ren and Sun. &quot;Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification.&quot; arXiv:1502.01852. (2015). <br /> # Ioffe and Szegedy. &quot;Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariance Shift.&quot; arXiv:1502.03167. (2015).<br /> # Klambauer, Unterthiner, Mayr and Hochreiter. &quot;Self-Normalizing Neural Networks.&quot; arXiv: 1706.02515. (2017).<br /> # Salimans and Kingma. &quot;Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks.&quot; arXiv:1602.07868. (2016).<br /> # Srivastava, Greff and Schmidhuber. &quot;Highway Networks.&quot; arXiv:1505.00387 (2015).<br /> # Unterthiner, Mayr, Klambauer and Hochreiter. &quot;Toxicity Prediction Using Deep Learning.&quot; arXiv:1503.01445. (2015). <br /> # https://en.wikipedia.org/wiki/Central_limit_theorem <br /> # http://mathworld.wolfram.com/SpectralNorm.html <br /> # https://www.math.umd.edu/~petersd/466/fixedpoint.pdf<br /> <br /> ==Online Resources==<br /> https://github.com/bioinf-jku/SNNs (GitHub repository maintained by some of the paper's authors)<br /> <br /> ==Footnotes==<br /> <br /> 1. Error propagation analysis: The authors performed an error analysis to quantify the potential numerical imprecisions propagated through the numerous operations performed. The potential imprecision &lt;math display=&quot;inline&quot;&gt;\epsilon&lt;/math&gt; was quantified by applying the mean value theorem<br /> <br /> $|f(x + \Delta x - f(x)| \leqslant ||\triangledown f(x + t\Delta x|| ||\Delta x|| \textrm{ for } t \in [0, 1]\textrm{.}$ <br /> <br /> The error propagation rules, or &lt;math display=&quot;inline&quot;&gt;|f(x + \Delta x - f(x)|&lt;/math&gt;, was first obtained for simple operations such as addition, subtraction, multiplication, division, square root, exponential function, error function and complementary error function. Them, the error bounds on the compound terms making up &lt;math display=&quot;inline&quot;&gt;\Delta (S(\mu, \omega, \nu, \tau, \lambda, \alpha)&lt;/math&gt; were found by decomposing them into the simpler expressions. If each of the variables have a precision of &lt;math display=&quot;inline&quot;&gt;\epsilon&lt;/math&gt;, then it turns out &lt;math display=&quot;inline&quot;&gt;S&lt;/math&gt; has a precision better than &lt;math display=&quot;inline&quot;&gt;292\epsilon&lt;/math&gt;. For a machine with a precision of &lt;math display=&quot;inline&quot;&gt;2^{-56}&lt;/math&gt;, the rounding error is &lt;math display=&quot;inline&quot;&gt;\epsilon \approx 10^{-16}&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;292\epsilon &lt; 10^{-13}&lt;/math&gt;. In addition, all computations are correct up to 3 ulps (“unit in last place”) for the hardware architectures and GNU C library used, with 1 ulp being the highest precision that can be achieved.<br /> <br /> 2. Independence Assumption: The classic definition of central limit theorem requires &lt;math display=&quot;inline&quot;&gt;x_i&lt;/math&gt;’s to be independent and identically distributed, which is not guaranteed to hold true in a neural network layer. However, according to the Lyapunov CLT, the &lt;math display=&quot;inline&quot;&gt;x_i&lt;/math&gt;’s do not need to be identically distributed as long as the &lt;math display=&quot;inline&quot;&gt;(2 + \delta)&lt;/math&gt;th moment exists for the variables and meet the Lyapunov condition for the rate of growth of the sum of the moments . In addition, CLT has also shown to be valid under weak dependence under mixing conditions . Therefore, the authors argue that the central limit theorem can be applied with network inputs.<br /> <br /> 3. &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; versus &lt;math display=&quot;inline&quot;&gt;\mathcal{J}&lt;/math&gt; Jacobians: In solving for the largest singular value of the Jacobian &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; for the mapping &lt;math display=&quot;inline&quot;&gt;g: (\mu, \nu)&lt;/math&gt;, the authors first worked with the terms in the Jacobian &lt;math display=&quot;inline&quot;&gt;\mathcal{J}&lt;/math&gt; for the mapping &lt;math display=&quot;inline&quot;&gt;h: (\mu, \nu) \rightarrow (\widetilde{\mu}, \widetilde{\xi})&lt;/math&gt; instead, because the influence of &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; on &lt;math display=&quot;inline&quot;&gt;\widetilde{\nu}&lt;/math&gt; is small when &lt;math display=&quot;inline&quot;&gt;\widetilde{\mu}&lt;/math&gt; is small in &lt;math display=&quot;inline&quot;&gt;\Omega&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; can be easily expressed as terms in &lt;math display=&quot;inline&quot;&gt;\mathcal{J}&lt;/math&gt;. &lt;math display=&quot;inline&quot;&gt;\mathcal{J}&lt;/math&gt; was referenced in the paper, but I used &lt;math display=&quot;inline&quot;&gt;\mathcal{H}&lt;/math&gt; in the summary here to avoid confusion.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:htru2.png&diff=36220 File:htru2.png 2018-04-06T23:32:30Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=36214 Word translation without parallel data 2018-04-06T22:41:31Z <p>S3wen: /* Validation criterion for unsupervised model selection */</p> <hr /> <div>[[File:Toy_example.png]]<br /> <br /> = Presented by =<br /> <br /> Xia Fan<br /> <br /> = Introduction =<br /> <br /> Many successful methods for learning relationships between languages stem from the hypothesis that there is a relationship between the context of words and their meanings. This means that if an adequate representation of a language is found in a high dimensional space (this is called an embedding), then words similar to a given word are close to one another in this space (ex. some norm can be minimized to find a word with similar context). Historically, another significant hypothesis is that these embedding spaces show similar structures over different languages. That is to say that given an embedding space for English and one for Spanish, a mapping could be found that aligns the two spaces and such a mapping could be used as a tool for translation. Many papers exploit these hypotheses, but use large parallel datasets for training. Recently, to remove the need for supervised training, methods have been implemented that utilize identical character strings (ex. letters or digits) in order to try to align the embeddings. The downside of this approach is that the two languages need to be similar to begin with as they need to have some shared basic building block. The method proposed in this paper uses an adversarial method to find this mapping between the embedding spaces of two languages without the use of large parallel datasets.<br /> <br /> The contributions of this paper can be listed as follows: <br /> <br /> 1. This paper introduces a model that either is on par, or outperforms supervised state-of-the-art methods, without employing any cross-lingual annotated data such as bilingual dictionaries or parallel corpora (large and structured sets of texts). This method uses an idea similar to GANs: it leverages adversarial training to learn a linear mapping from a source to distinguish between the mapped source embeddings and the target embeddings, while the mapping is jointly trained to fool the discriminator. <br /> <br /> 2. Second, this paper extracts a synthetic dictionary from the resulting shared embedding space and fine-tunes the mapping with the closed-form Procrustes solution from Schonemann (1966). <br /> <br /> 3. Third, this paper also introduces an unsupervised selection metric that is highly correlated with the mapping quality and that the authors use both as a stopping criterion and to select the best hyper-parameters. <br /> <br /> 4. Fourth, they introduce a cross-domain similarity adaptation to mitigate the so-called hubness problem (points tending to be nearest neighbors of many points in high-dimensional spaces).<br /> <br /> 5. They demonstrate the effectiveness of our method using an example of a low-resource language pair where parallel corpora are not available (English-Esperanto) for which their method is particularly suited.<br /> <br /> This paper is published in ICLR 2018.<br /> <br /> = Model =<br /> <br /> <br /> === Estimation of Word Representations in Vector Space ===<br /> <br /> This model focuses on learning a mapping between the two sets such that translations are close in the shared space. Before talking about the model it used, a model which can exploit the similarities of monolingual embedding spaces should be introduced. Mikolov et al.(2013) use a known dictionary of n=5000 pairs of words &lt;math&gt; \{x_i,y_i\}_{i\in{1,n}} &lt;/math&gt;. and learn a linear mapping W between the source and the target space such that <br /> <br /> \begin{align}<br /> W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F \hspace{1cm} (1)<br /> \end{align}<br /> <br /> where d is the dimension of the embeddings, &lt;math&gt; M_d(R) &lt;/math&gt; is the space of d*d matrices of real numbers, and X and Y are two aligned matrices of size d*n containing the embeddings of the words in the parallel vocabulary. <br /> <br /> Xing et al. (2015) showed that these results are improved by enforcing orthogonality constraint on W. In that case, equation (1) boils down to the Procrustes problem, a matrix approximation problem for which the goal is to find an orthogonal matrix that best maps two given matrices on the measure of the Frobenius norm. It advantageously offers a closed form solution obtained from the singular value decomposition (SVD) of &lt;math&gt; YX^T &lt;/math&gt; :<br /> <br /> \begin{align}<br /> W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F=UV^T\textrm{, with }U\Sigma V^T=SVD(YX^T).<br /> \end{align}<br /> <br /> <br /> This can be proven as follows. First note that <br /> \begin{align}<br /> &amp;||WX-Y||_F\\<br /> &amp;= \langle WX-Y, WX-Y\rangle_F\\ <br /> &amp;= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br /> &amp;= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br /> \end{align}<br /> <br /> where &lt;math display=&quot;inline&quot;&gt; \langle \cdot, \cdot \rangle_F &lt;/math&gt; denotes the Frobenius inner-product and we have used the orthogonality of &lt;math display=&quot;inline&quot;&gt; W &lt;/math&gt;. It follows that we need only maximize the inner-product above. Let &lt;math display=&quot;inline&quot;&gt; u_1, \ldots, u_d &lt;/math&gt; denote the columns of &lt;math display=&quot;inline&quot;&gt; U &lt;/math&gt;. Let &lt;math display=&quot;inline&quot;&gt; v_1, \ldots , v_d &lt;/math&gt; denote the columns of &lt;math display=&quot;inline&quot;&gt; V &lt;/math&gt;. Let &lt;math display=&quot;inline&quot;&gt; \sigma_1, \ldots, \sigma_d &lt;/math&gt; denote the diagonal entries of &lt;math display=&quot;inline&quot;&gt; \Sigma &lt;/math&gt;. We have<br /> \begin{align}<br /> &amp;\langle W X, Y \rangle_F \\<br /> &amp;= \text{Tr} (W^T Y X^T)\\<br /> &amp; =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br /> &amp;=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br /> &amp;=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br /> &amp;\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br /> &amp;= \sum_i \sigma_i<br /> \end{align}<br /> where we have used the invariance of trace under cyclic permutations, Cauchy-Schwarz, and the orthogonality of the columns of U and V. Note that choosing <br /> \begin{align}<br /> W=UV^T<br /> \end{align}<br /> achieves the bound. This completes the proof.<br /> <br /> === Domain-adversarial setting ===<br /> <br /> This paper shows how to learn this mapping W without cross-lingual supervision. An illustration of the approach is given in Fig. 1. First, this model learns an initial proxy of W by using an adversarial criterion. Then, it uses the words that match the best as anchor points for Procrustes. Finally, it improves performance over less frequent words by changing the metric of the space, which leads to spread more of those points in dense region. <br /> <br /> [[File:Toy_example.png |frame|none|alt=Alt text|Figure 1: Toy illustration of the method. (A) There are two distributions of word embeddings, English words in red denoted by X and Italian words in blue denoted by Y , which we want to align/translate. Each dot represents a word in that space. The size of the dot is proportional to the frequency of the words in the training corpus of that language. (B) Using adversarial learning, we learn a rotation matrix W which roughly aligns the two distributions. The green stars are randomly selected words that are fed to the discriminator to determine whether the two word embeddings come from the same distribution. (C) The mapping W is further refined via Procrustes. This method uses frequent words aligned by the previous step as anchor points, and minimizes an energy function that corresponds to a spring system between anchor points. The refined mapping is then used to map all words in the dictionary. (D) Finally, we translate by using the mapping W and a distance metric, dubbed CSLS, that expands the space where there is high density of points (like the area around the word “cat”), so that “hubs” (like the word “cat”) become less close to other word vectors than they would otherwise (compare to the same region in panel (A)).]]<br /> <br /> Let &lt;math&gt; X={x_1,...,x_n} &lt;/math&gt; and &lt;math&gt; Y={y_1,...,y_m} &lt;/math&gt; be two sets of n and m word embeddings coming from a source and a target language respectively. A model is trained is trained to discriminate between elements randomly sampled from &lt;math&gt; WX={Wx_1,...,Wx_n} &lt;/math&gt; and Y, We call this model the discriminator. W is trained to prevent the discriminator from making accurate predictions. As a result, this is a two-player adversarial game, where the discriminator aims at maximizing its ability to identify the origin of an embedding, and W aims at preventing the discriminator from doing so by making WX and Y as similar as possible. This approach is in line with the work of Ganin et al.(2016), who proposed to learn latent representations invariant to the input domain, where in this case, a domain is represented by a language(source or target).<br /> <br /> 1. Discriminator objective<br /> <br /> Refer to the discriminator parameters as &lt;math&gt; \theta_D &lt;/math&gt;. Consider the probability &lt;math&gt; P_{\theta_D}(source = 1|z) &lt;/math&gt; that a vector z is the mapping of a source embedding (as opposed to a target embedding) according to the discriminator. The discriminator loss can be written as:<br /> <br /> \begin{align}<br /> L_D(\theta_D|W)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=1|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=0|y_i)<br /> \end{align}<br /> <br /> 2. Mapping objective <br /> <br /> In the unsupervised setting, W is now trained so that the discriminator is unable to accurately predict the embedding origins: <br /> <br /> \begin{align}<br /> L_W(W|\theta_D)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=0|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=1|y_i)<br /> \end{align}<br /> <br /> 3. Learning algorithm <br /> To train the model, the authors follow the standard training procedure of deep adversarial networks of Goodfellow et al. (2014). For every input sample, the discriminator and the mapping matrix W are trained successively with stochastic gradient updates to respectively minimize &lt;math&gt; L_D &lt;/math&gt; and &lt;math&gt; L_W &lt;/math&gt;<br /> <br /> === Refinement procedure ===<br /> <br /> The matrix W obtained with adversarial training gives good performance (see Table 1), but the results are still not on par with the supervised approach. In fact, the adversarial approach tries to align all words irrespective of their frequencies. However, rare words have embeddings that are less updated and are more likely to appear in different contexts in each corpus, which makes them harder to align. Under the assumption that the mapping is linear, it is then better to infer the global mapping using only the most frequent words as anchors. Besides, the accuracy on the most frequent word pairs is high after adversarial training.<br /> To refine the mapping, this paper build a synthetic parallel vocabulary using the W just learned with adversarial training. Specifically, this paper consider the most frequent words and retain only mutual nearest neighbors to ensure a high-quality dictionary. Subsequently, this paper apply the Procrustes solution in (2) on this generated dictionary. Considering the improved solution generated with the Procrustes algorithm, it is possible to generate a more accurate dictionary and apply this method iteratively, similarly to Artetxe et al. (2017). However, given that the synthetic dictionary obtained using adversarial training is already strong, this paper only observe small improvements when doing more than one iteration, i.e., the improvements on the word translation task are usually below 1%.<br /> <br /> === Cross-Domain Similarity Local Scaling (CSLS) ===<br /> <br /> This paper considers a bi-partite neighborhood graph, in which each word of a given dictionary is connected to its K nearest neighbors in the other language. &lt;math&gt; N_T(Wx_s) &lt;/math&gt; is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding &lt;math&gt; Wx_s &lt;/math&gt;. All K elements of &lt;math&gt; N_T(Wx_s) &lt;/math&gt; are words from the target language. Similarly we denote by &lt;math&gt; N_S(y_t) &lt;/math&gt; the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding &lt;math&gt; x_s &lt;/math&gt; to its target neighborhood as<br /> <br /> \begin{align}<br /> r_T(Wx_s)=\frac{1}{K}\sum_{y\in N_T(Wx_s)}cos(Wx_s,y_t)<br /> \end{align}<br /> <br /> where cos(.,.) is the cosine similarity. Likewise, the mean similarity of a target word &lt;math&gt; y_t &lt;/math&gt; to its neighborhood is denoted as &lt;math&gt; r_S(y_t) &lt;/math&gt;. This is used to define similarity measure CSLS(.,.) between mapped source words and target words as <br /> <br /> \begin{align}<br /> CSLS(Wx_s,y_t)=2cos(Wx_s,y_t)-r_T(Wx_s)-r_S(y_t)<br /> \end{align}<br /> <br /> This process increases the similarity associated with isolated word vectors, but decreases the similarity of vectors lying in dense areas. <br /> <br /> CSLS represents an improved measure for producing reliable matching words between two languages (i.e. neighbors of a word in one language should ideally correspond to the same words in the second language). The nearest neighbors algorithm is asymmetric, and in high-dimensional spaces, it suffers from the problem of hubness, in which some points are nearest neighbors to exceptionally many points, while others are not nearest neighbors to any points. Existing approaches for combating the effect of hubness on word translation retrieval involve performing similarity updates one language at a time without consideration for the other language in the pair (Dinu et al., 2015, Smith et al., 2017). Consequently, they yielded less accurate results when compared to CSLS in experiments conducted in this paper (Table 1).<br /> <br /> = Training and architectural choices =<br /> === Architecture ===<br /> <br /> This paper use unsupervised word vectors that were trained using fastText2. These correspond to monolingual embeddings of dimension 300 trained on Wikipedia corpora; therefore, the mapping W has size 300 × 300. Words are lower-cased, and those that appear less than 5 times are discarded for training. As a post-processing step, only the first 200k most frequent words were selected in the experiments.<br /> For the discriminator, it use a multilayer perceptron with two hidden layers of size 2048, and Leaky-ReLU activation functions. The input to the discriminator is corrupted with dropout noise with a rate of 0.1. As suggested by Goodfellow (2016), a smoothing coefficient s = 0.2 is included in the discriminator predictions. This paper use stochastic gradient descent with a batch size of 32, a learning rate of 0.1 and a decay of 0.95 both for the discriminator and W . <br /> <br /> === Discriminator inputs ===<br /> The embedding quality of rare words is generally not as good as the one of frequent words (Luong et al., 2013), and it is observed that feeding the discriminator with rare words had a small, but not negligible negative impact. As a result, this paper only feed the discriminator with the 50,000 most frequent words. At each training step, the word embeddings given to the discriminator are sampled uniformly. Sampling them according to the word frequency did not have any noticeable impact on the results.<br /> <br /> === Orthogonality===<br /> In this work, the authors propose to use a simple update step to ensure that the matrix W stays close to an orthogonal matrix during training (Cisse et al. (2017)). Specifically, the following update rule on the matrix W is used:<br /> <br /> \begin{align}<br /> W \leftarrow (1+\beta)W-\beta(WW^T)W<br /> \end{align}<br /> <br /> where β = 0.01 is usually found to perform well. This method ensures that the matrix stays close to the manifold of orthogonal matrices after each update.<br /> <br /> This update rule can be justified as follows. Consider the function <br /> \begin{align}<br /> g: \mathbb{R}^{d\times d} \to \mathbb{R}^{d \times d}<br /> \end{align}<br /> defined by<br /> \begin{align}<br /> g(W)= W^T W -I.<br /> \end{align}<br /> <br /> The derivative of g at W is is the linear map<br /> \begin{align}<br /> Dg[W]: \mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}<br /> \end{align}<br /> defined by<br /> \begin{align}<br /> Dg[W](H)= H^T W + W^T H.<br /> \end{align}<br /> <br /> The adjoint of this linear map is<br /> <br /> \begin{align}<br /> D^\ast g[W](H)= WH^T +WH.<br /> \end{align}<br /> <br /> Now consider the function f<br /> \begin{align}<br /> f: \mathbb{R}^{d \times d} \to \mathbb{R}<br /> \end{align}<br /> <br /> defined by<br /> <br /> \begin{align}<br /> f(W)=||g(W) ||_F^2=||W^TW -I ||_F^2.<br /> \end{align}<br /> <br /> f has gradient:<br /> \begin{align}<br /> \nabla f (W) = 2D^\ast g[W] (g(W ) ) =2W(W^TW-I) +2W(W^TW-I)=4W W^TW-4W.<br /> \end{align}<br /> or<br /> \begin{align}<br /> \nabla f (W) = \nabla||W^TW-I||_F = \nabla\text{Tr}(W^TW-I)(W^TW-I)=4(\nabla(W^TW-I))(W^TW-I)=4W(W^TW-I)\text{ (check derivative of trace function)}<br /> \end{align}<br /> <br /> Thus the update<br /> \begin{align}<br /> W \leftarrow (1+\beta)W-\beta(WW^T)W<br /> \end{align}<br /> amounts to a step in the direction opposite the gradient of f. That is, a step toward the set of orthogonal matrices.<br /> <br /> === Dictionary generation ===<br /> The refinement step requires the generation of a new dictionary at each iteration. In order for the Procrustes solution to work well, it is best to apply it on correct word pairs. As a result, the CSLS method is used to select more accurate translation pairs in the dictionary. To further increase the quality of the dictionary, and ensure that W is learned from correct translation pairs, only mutual nearest neighbors were considered, i.e. pairs of words that are mutually nearest neighbors of each other according to CSLS. This significantly decreases the size of the generated dictionary, but improves its accuracy, as well as the overall performance.<br /> <br /> === Validation criterion for unsupervised model selection ===<br /> <br /> This paper consider the 10k most frequent source words, and use CSLS to generate a translation for each of them, then compute the average cosine similarity between these deemed translations, and use this average as a validation metric. The choice of using the 10 thousand most frequent source words is requires more justification since we would expect those to be the best trained words and may not accurately represent the entire data set. Perhaps a k-fold cross validation approach should be used instead. Figure 2 below shows the correlation between the evaluation score and this unsupervised criterion (without stabilization by learning rate shrinkage)<br /> <br /> <br /> <br /> [[File:fig2_fan.png |frame|none|alt=Alt text|Figure 2: Unsupervised model selection.<br /> Correlation between the unsupervised validation criterion (black line) and actual word translation accuracy (blue line). In this particular experiment, the selected model is at epoch 10. Observe how the criterion is well correlated with translation accuracy.]]<br /> <br /> = Results =<br /> <br /> The results on word translation retrieval using the bilingual dictionaries are presented in Table 1, and a comparison to previous work in shown in Table 2 where the unsupervised model significantly outperforms previous approaches. The results on the sentence translation retrieval task are presented in Table 3, and the cross-lingual word similarity task in Table 4. Finally, the results on word-by-word translation for English-Esperanto are presented in Table 5. The bilingual dictionary used here does not account for words with multiple meanings.<br /> <br /> [[File:table1_fan.png |frame|none|alt=Alt text|Table 1: Word translation retrieval P@1 for the released vocabularies in various language pairs. The authors consider 1,500 source test queries, and 200k target words for each language pair. The authors use fastText embeddings trained on Wikipedia. NN: nearest neighbors. ISF: inverted softmax. (’en’ is English, ’fr’ is French, ’de’ is German, ’ru’ is Russian, ’zh’ is classical Chinese and ’eo’ is Esperanto)]]<br /> <br /> <br /> [[File:table2_fan.png |frame|none|alt=Alt text|English-Italian word translation average precisions (@1, @5, @10) from 1.5k source word queries using 200k target words. Results marked with the symbol † are from Smith et al. (2017). Wiki means the embeddings were trained on Wikipedia using fastText. Note that the method used by Artetxe et al. (2017) does not use the same supervision as other supervised methods, as they only use numbers in their ini- tial parallel dictionary.]]<br /> <br /> [[File:table3_fan.png |frame|none|alt=Alt text|Table 3: English-Italian sentence translation retrieval. The authors report the average P@k from 2,000 source queries using 200,000 target sentences. The authors use the same embeddings as in Smith et al. (2017). Their results are marked with the symbol †.]]<br /> <br /> [[File:table4_fan.png |frame|none|alt=Alt text|Table 4: Cross-lingual wordsim task. NASARI<br /> (Camacho-Collados et al. (2016)) refers to the official SemEval2017 baseline. The authors report Pearson correlation.]]<br /> <br /> [[File:table5_fan.png |frame|none|alt=Alt text|Table 5: BLEU score on English-Esperanto.<br /> Although being a naive approach, word-by- word translation is enough to get a rough idea of the input sentence. The quality of the gener- ated dictionary has a significant impact on the BLEU score.]]<br /> <br /> [[File:paper9_fig3.png |frame|none|alt=Alt text|Figure 3: The paper also investigated the impact of monolingual embeddings. It was found that model from this paper can align embeddings obtained through different methods, but not embeddings obtained from different corpora, which explains the large performance increase in Table 2 due to the corpus change from WaCky to Wiki using CBOW embedding. This is conveyed in this figure which displays English to English world alignment accuracies with regard to word frequency. Perfect alignment is achieved using the same model and corpora (a). Also good alignment using different model and corpora, although CSLS consistently has better results (b). Worse results due to use of different corpora (c). Even worse results when both embedding model and corpora are different.]]<br /> <br /> = Conclusion =<br /> It is clear that one major downfall of this method when it actually comes to translation is the restriction that the two languages must have similar intrinsic structures to allow for the embeddings to align. However, given this assumption, this paper shows for the first time that one can align word embedding spaces without any cross-lingual supervision, i.e., solely based on unaligned datasets of each language, while reaching or outperforming the quality of previous supervised approaches in several cases. Using adversarial training, the model is able to initialize a linear mapping between a source and a target space, which is also used to produce a synthetic parallel dictionary. It is then possible to apply the same techniques proposed for supervised techniques, namely a Procrustean optimization.<br /> <br /> = Open source code =<br /> The source code for the paper is provided at the following Github link: https://github.com/facebookresearch/MUSE. The repository provides the source code as written in PyTorch by the authors of this paper.<br /> <br /> = Source =<br /> Dinu, Georgiana; Lazaridou, Angeliki; Baroni, Marco<br /> | Improving zero-shot learning by mitigating the hubness problem<br /> | arXiv:1412.6568<br /> <br /> Lample, Guillaume; Denoyer, Ludovic; Ranzato, Marc'Aurelio <br /> | Unsupervised Machine Translation Using Monolingual Corpora Only<br /> | arXiv: 1701.04087<br /> <br /> Smith, Samuel L; Turban, David HP; Hamblin, Steven; Hammerla, Nils Y<br /> | Offline bilingual word vectors, orthogonal transformations and the inverted softmax<br /> | arXiv:1702.03859<br /> <br /> Lample, G. (n.d.). Facebookresearch/MUSE. Retrieved March 25, 2018, from https://github.com/facebookresearch/MUSE</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Rethinking_the_Smaller-Norm-Less-Informative_Assumption_in_Channel_Pruning_of_Convolutional_Layers&diff=36213 stat946w18/Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolutional Layers 2018-04-06T22:18:53Z <p>S3wen: /* Results */</p> <hr /> <div>== Introduction ==<br /> <br /> With the recent and ongoing surge in low-power, intelligent agents (such as wearables, smartphones, and IoT devices), there exists a growing need for machine learning models to work well in resource-constrained environments. Deep learning models have achieved state-of-the-art on a broad range of tasks; however, they are difficult to deploy in their original forms. For example, AlexNet (Krizhevsky et al., 2012), a model for image classification, contains 61 million parameters and requires 1.5 billion floating point operations (FLOPs) in one inference pass. A more accurate model, ResNet-50 (He et al., 2016), has 25 million parameters but requires 4.08 FLOPs. A high-end desktop GPU such as a Titan Xp is capable of [https://www.nvidia.com/en-us/titan/titan-xp/ (12 TFLOPS (tera-FLOPs per second))], while the Adreno 540 GPU used in a Samsung Galaxy S8 is only capable of [https://gflops.surge.sh (567 GFLOPS)] which is less than 5% of the Titan Xp. Clearly, it would be difficult to deploy and run these models on low-power devices.<br /> <br /> In general, model compression can be accomplished using four main, not mutually exclusive methods (Cheng et al., 2017): weight pruning, quantization, matrix transformations, and weight tying. By not mutually exclusive, we mean that these methods can be used not only separately but also in combination for compressing a single model; the use of one method does not exclude any of the other methods from being viable. <br /> <br /> Ye et al. (2018) explores pruning entire channels in a convolutional neural network (CNN). Past work has mostly focused on norm[based or error-based heuristics to prune channels; instead, Ye et al. (2018) show that their approach is easily reproducible and has favorable qualities from an optimization standpoint. In other words, they argue that the norm-based assumption is not as informative or theoretically justified as their approach, and provide strong empirical evidence of these findings.<br /> <br /> == Motivation ==<br /> <br /> Some previous works on pruning channel filters (Li et al., 2016; Molchanov et al., 2016) have focused on using the L1 norm to determine the importance of a channel. Ye et al. (2018) show that, in the deep linear convolution case, penalizing the per-layer norm is coarse-grained; they argue that one cannot assign different coefficients to L1 penalties associated with different layers without risking the loss function being susceptible to trivial re-parameterizations. As an example, consider the following deep linear convolutional neural network with modified LASSO loss:<br /> <br /> $$\min \mathbb{E}_D \lVert W_{2n} * \dots * W_1 x - y\rVert^2 + \lambda \sum_{i=1}^n \lVert W_{2i} \rVert_1$$<br /> <br /> where W are the weights and * is convolution. Here we have chosen the coefficient 0 for the L1 penalty associated with odd-numbered layers and the coefficient 1 for the L1 penalty associated with even-numbered layers. This loss is susceptible to trivial re-parameterizations: without affecting the least-squares loss, we can always reduce the LASSO loss by halving the weights of all even-numbered layers and doubling the weights of all odd-numbered layers.<br /> <br /> Furthermore, batch normalization (Ioffe, 2015) is incompatible with this method of weight regularization. Consider batch normalization at the &lt;math&gt;l&lt;/math&gt;-th layer.<br /> <br /> &lt;center&gt;&lt;math&gt;x^{l+1} = max\{\gamma \cdot BN_{\mu,\sigma,\epsilon}(W^l * x^l) + \beta, 0\}&lt;/math&gt;&lt;/center&gt;<br /> <br /> Due to the batch normalization, any uniform scaling of &lt;math&gt;W^l&lt;/math&gt; which would change &lt;math&gt;l_1&lt;/math&gt; and &lt;math&gt;l_2&lt;/math&gt; norms, but has no have no effect on &lt;math&gt;x^{l+1}&lt;/math&gt;. Thus, when trying to minimize weight norms of multiple layers, it is unclear how to properly choose penalties for each layer. Therefore, penalizing the norm of a filter in a deep convolutional network is hard to justify from a theoretical perspective.<br /> <br /> In contrast with these existing approaches, the authors focus on enforcing sparsity of a tiny set of parameters in CNN — scale parameter &lt;math&gt;\gamma&lt;/math&gt; in all batch normalization. Not only placing sparse constraints on &lt;math&gt;\gamma&lt;/math&gt; is simpler and easier to monitor, but more importantly, they put forward two reasons:<br /> <br /> 1. Every &lt;math&gt;\gamma&lt;/math&gt; always multiplies a normalized random variable, thus the channel importance becomes comparable across different layers by measuring the magnitude values of &lt;math&gt;\gamma&lt;/math&gt;;<br /> <br /> 2. The reparameterization effect across different layers is avoided if its subsequent convolution layer is also batch-normalized. In other words, the impacts from the scale changes of &lt;math&gt;\gamma&lt;/math&gt; parameter are independent across different layers.<br /> <br /> Thus, although not providing a complete theoretical guarantee on loss, Ye et al. (2018) develop a pruning technique that claims to be more justified than norm-based pruning is.<br /> <br /> == Method ==<br /> <br /> At a high level, Ye et al. (2018) propose that, instead of discovering sparsity via penalizing the per-filter or per-channel norm, penalize the batch normalization scale parameters ''gamma'' instead. The reasoning is that by having fewer parameters to constrain and working with normalized values, sparsity is easier to enforce, monitor, and learn. Having sparse batch normalization terms has the effect of pruning '''entire''' channels: if ''gamma'' is zero, then the output at that layer becomes constant (the bias term), and thus the preceding channels can be pruned.<br /> <br /> === Summary ===<br /> <br /> The basic algorithm can be summarized as follows:<br /> <br /> 1. Penalize the L1-norm of the batch normalization scaling parameters in the loss<br /> <br /> 2. Train until loss plateaus<br /> <br /> 3. Remove channels that correspond to a downstream zero in batch normalization<br /> <br /> 4. Fine-tune the pruned model using regular learning<br /> <br /> === Details ===<br /> <br /> There still exist a few problems that this summary has not addressed so far. Sub-gradient descent is known to have inverse square root convergence rate on subdifferentials (Gordon et al., 2012), so the sparsity gradient descent update may be suboptimal. Furthermore, the sparse penalty needs to be normalized with respect to previous channel sizes, since the penalty should be roughly equally distributed across all convolution layers.<br /> <br /> ==== Slow Convergence ====<br /> To address the issue of slow convergence, Ye et al. (2018) use an iterative shrinking-thresholding algorithm (ISTA) (Beck &amp; Teboulle, 2009) to update the batch normalization scale parameter. The intuition for ISTA is that the structure of the optimization objective can be taken advantage of. Consider: $$L(x) = f(x) + g(x).$$<br /> <br /> Let ''f'' be the model loss and ''g'' be the non-differentiable penalty (LASSO). ISTA is able to use the structure of the loss and converge in O(1/n), instead of O(1/sqrt(n)) when using subgradient descent, which assumes no structure about the loss. Even though ISTA is used in convex settings, Ye et. al (2018) argue that it still performs better than gradient descent.<br /> <br /> ==== Penalty Normalization ====<br /> <br /> In the paper, Ye et al. (2018) normalize the per-layer sparse penalty with respect to the global input size, the current layer kernel areas, the previous layer kernel areas, and the local input feature map area.<br /> <br /> [[File:Screenshot_from_2018-02-28_17-06-41.png]] (Ye et al., 2018)<br /> <br /> To control the global penalty, a hyperparamter ''rho'' is multiplied with all the per-layer ''lambda'' in the final loss.<br /> <br /> === Steps ===<br /> <br /> The final algorithm can be summarized as follows:<br /> <br /> 1. Compute the per-layer normalized sparse penalty constant &lt;math&gt;\lambda&lt;/math&gt;<br /> <br /> 2. Compute the global LASSO loss with global scaling constant &lt;math&gt;\rho&lt;/math&gt;<br /> <br /> 3. Until convergence, train scaling parameters using ISTA and non-scaling parameters using regular gradient descent.<br /> <br /> 4. Remove channels that correspond to a downstream zero in batch normalization<br /> <br /> 5. Fine-tune the pruned model using regular learning<br /> <br /> == Results ==<br /> <br /> The authors show state-of-the-art performance, compared with other channel-pruning approaches. It is important to note that it would be unfair to compare against general pruning approaches; channel pruning specifically removes channels without introducing '''intra-kernel sparsity''', whereas other pruning approaches introduce irregular kernel sparsity and hence computational inefficiencies.<br /> <br /> === CIFAR-10 Experiment ===<br /> <br /> Model A is trained with a sparse penalty of &lt;math&gt;\rho = 0.0002&lt;/math&gt; for 30 thousand steps, and then increased to &lt;math&gt;\rho = 0.001&lt;/math&gt;. Model B is trained by taking Model A and increasing the sparse penalty up to 0.002. Similarly Model C is a continuation of Model B with a penalty of 0.008. <br /> <br /> [[File:Screenshot_from_2018-02-28_17-24-25.png]]<br /> <br /> For the convNet, reducing the number of parameters in the base model increased the accuracy in model A. This suggests that the base model is over-parameterized. Otherwise, there would be a trade-off of accuracy and model efficiency.<br /> <br /> === ILSVRC2012 Experiment ===<br /> <br /> The authors note that while ResNet-101 takes hundreds of epochs to train, pruning only takes 5-10, with fine-tuning adding another 2, giving an empirical example how long pruning might take in practice. Both models were trained with an aggressive sparsity penalty of 0.1.<br /> <br /> [[File:Screenshot_from_2018-02-28_17-24-36.png]]<br /> <br /> === Image Foreground-Background Segmentation Experiment ===<br /> <br /> The authors note that it is common practice to take a network with pre-trained on a large task and fine-tune it to apply it to a different, smaller task. One might expect there might be some extra channels that while useful for the large task, can be omitted for the simpler task. This experiment replicated that use-case by taking a NN originally trained on multiple datasets and applying the proposed pruning method. The authors note that the pruned network actually improves over the original network in all but the most challenging test dataset, which is in line with the initial expectation. The model was trained with a sparsity penalty of 0.5 and the results are shown in table below<br /> <br /> [[File:paper8_Segmentation.png|700px]]<br /> <br /> == Conclusion ==<br /> <br /> Pruning large neural architectures to fit on low-power devices is an important task. For a real quantitative measure of efficiency, it would be interesting to conduct actual power measurements on the pruned models versus baselines; reduction in FLOPs doesn't necessarily correspond with vastly reduced power since memory accesses dominate energy consumption (Han et al., 2015). However, the reduction in the number of FLOPs and parameters is encouraging, so moderate power savings should be expected.<br /> <br /> It would also be interesting to combine multiple approaches, or &quot;throw the whole kitchen sink&quot; at this task. Han et al. (2015) sparked much recent interest by successfully combining weight pruning, quantization, and Huffman coding without loss in accuracy. However, their approach introduced irregular sparsity in the convolutional layers, so a direct comparison cannot be made.<br /> <br /> In conclusion, this novel, theoretically-motivated interpretation of channel pruning was successfully applied to several important tasks.<br /> <br /> == Implementation == <br /> A PyTorch implementation is available here: https://github.com/jack-willturner/batchnorm-pruning<br /> <br /> <br /> == References ==<br /> <br /> * Krizhevsky, A., Sutskever, I., &amp; Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems (pp. 1097-1105).<br /> * He, K., Zhang, X., Ren, S., &amp; Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 770-778).<br /> * Cheng, Y., Wang, D., Zhou, P., &amp; Zhang, T. (2017). A Survey of Model Compression and Acceleration for Deep Neural Networks. arXiv preprint arXiv:1710.09282.<br /> * Ye, J., Lu, X., Lin, Z., &amp; Wang, J. Z. (2018). Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolution Layers. arXiv preprint arXiv:1802.00124.<br /> * Li, H., Kadav, A., Durdanovic, I., Samet, H., &amp; Graf, H. P. (2016). Pruning filters for efficient convnets. arXiv preprint arXiv:1608.08710.<br /> * Molchanov, P., Tyree, S., Karras, T., Aila, T., &amp; Kautz, J. (2016). Pruning convolutional neural networks for resource efficient inference.<br /> * Ioffe, S., &amp; Szegedy, C. (2015, June). Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International conference on machine learning (pp. 448-456).<br /> * Gordon, G., &amp; Tibshirani, R. (2012). Subgradient method. https://www.cs.cmu.edu/~ggordon/10725-F12/slides/06-sg-method.pdf<br /> * Beck, A., &amp; Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2(1), 183-202.<br /> * Han, S., Mao, H., &amp; Dally, W. J. (2015). Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning&diff=36209 One-Shot Imitation Learning 2018-04-06T21:43:31Z <p>S3wen: /* Manipulation network */</p> <hr /> <div>= Introduction =<br /> Robotic systems can be used for many applications, but to truly be useful for complex applications, they need to overcome 2 challenges: having the intent of the task at hand communicated to them, and being able to perform the manipulations necessary to complete this task. It is preferable to use demonstration to teach the robotic systems rather than natural language, as natural language may often fail to convey the details and intricacies required for the task. However, current work on learning from demonstrations is only successful with large amounts of feature engineering or a large number of demonstrations. The proposed model aims to achieve 'one-shot' imitation learning, ie. learning to complete a new task from just a single demonstration of it without any other supervision. As input, the proposed model takes the observation of the current instance of a task, and a demonstration of successfully solving a different instance of the same task. Strong generalization was achieved by using a soft attention mechanism on both the sequence of actions and states that the demonstration consists of, as well as on the vector of element locations within the environment. The success of this proposed model at completing a series of block stacking tasks can be viewed at http://bit.ly/nips2017-oneshot.<br /> <br /> = Related Work =<br /> While one-shot imitation learning is a novel combination of ideas, each of the components has previously been studied.<br /> * Imitation Learning: <br /> ** Behavioural learning uses supervised learning to map from observations to actions (e.g. [https://papers.nips.cc/paper/95-alvinn-an-autonomous-land-vehicle-in-a-neural-network.pdf (Pomerleau 1988)], [https://arxiv.org/pdf/1011.0686.pdf (Ross et. al 2011)])<br /> ** Inverse reinforcement learning estimates a reward function that considers demonstrations as optimal behavior (e.g. [http://ai.stanford.edu/~ang/papers/icml00-irl.pdf (Ng et. al 2000)])<br /> * One-Shot Learning:<br /> ** Typically a form of meta-learning<br /> ** Previously used for variety of tasks but all domain-specific<br /> ** [https://arxiv.org/abs/1703.03400 (Finn et al. 2017)] proposed a generic solution but excluded imitation learning<br /> * Reinforcement Learning:<br /> ** Demonstrated to work on variety of tasks and environments, in particular on games and robotic control<br /> ** Requires large amount of trials and a user-specified reward function<br /> * Multi-task/Transfer Learning:<br /> ** Shown to be particularly effective at computer vision tasks<br /> ** Not meant for one-shot learning<br /> * Attention Modelling:<br /> ** The proposed model makes use of the attention model from [https://arxiv.org/abs/1409.0473 (Bahdanau et al. 2016)]<br /> ** The attention modelling over demonstration is similar in nature to the seq2seq models from the well known [https://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf (Sutskever et al. 2014)]<br /> <br /> = One-Shot Imitation Learning =<br /> <br /> [[File:oneshot1.jpg|1000px]]<br /> <br /> The figure above shows the differences between traditional and one-shot imitation learning. In a), the traditional method may require training different policies for performing similar tasks that are similar in nature. For example, stacking blocks to a height of 2 and to a height of 3. In b), the one-shot imitation learning allows the same policy to be used for these tasks given a single demonstration, achieving good performance without any additional system interactions. In c), the policy is trained by using a set of different training tasks, with enough examples so that the learned results can be generalized to other similar tasks. Each task has a set of successful demonstrations. Each iteration of training uses two demonstrations from a task, one is used as the input passing into the algorithm and the other is used at the output, the results from the two are then conditioned to produce the correct action.<br /> <br /> == Problem Formalization ==<br /> The problem is briefly formalized with the authors describing a distribution of tasks, an individual task, a distribution of demonstrations for this task, and a single demonstration respectively as $T, \: t\sim T, \: D(t), \: d\sim D(t)$<br /> In addition, an action, an observation, parameters, and a policy are respectively defined as $a, o, \theta, \pi_\theta(a|o,d)$<br /> In particular, a demonstration is a sequence of observation and action pairs $d = [(o_1, a_1),(o_2, a_2), . . . ,(o_H , a_H )]$<br /> Assuming that &lt;math&gt;H &lt;/math&gt;, the length or horizon of a demonstration, and some evaluation function $$R_t(d): R^H \rightarrow R$$ are given, and that succesful demonstrations are available for each task, then the objective is to maximize expectation of the policy performance over $t\sim T, d\sim D(t)$.<br /> <br /> == Block Stacking Tasks ==<br /> The tasks that the authors focus on is block stacking. A user specifies in what final configuration cubic blocks should be stacked, and the goal is to use a 7-DOF Fetch robotic arm to arrange the blocks in this configuration. The number of blocks, and their desired configuration (ie. number of towers, the height of each tower, and order of blocks within each tower) can be varied and encoded as a string. For example, 'abc def' would signify 2 towers of height 3, with block A on block B on block C in one tower, and block D on block E on block F in a second tower. To add complexity, the initial configuration of the blocks can vary and is encoded as a set of 3-dimensional vectors describing the position of each block relative to the robotic arm.<br /> <br /> == Algorithm ==<br /> To avoid needing to specify a reward function, the authors use behavioral cloning and DAGGER, 2 imitation learning methods that require only demonstrations, for training. In each training step, a list of tasks is sampled, and for each, a demonstration with injected noise along with some observation-action pairs are sampled. Given the current observation and demonstration as input, the policy is trained against the sampled actions by minimizing L2 norm for continuous actions, and cross-entropy for discrete ones. Adamax is used as the optimizer with a learning rate of 0.001.<br /> <br /> = Architecture =<br /> The authors propose a novel architecture for imitation learning, consisting of 3 networks.<br /> <br /> While, in principle, a generic neural network could learn the mapping from demonstration and current observation to appropriate action, the authors propose the following architecture which they claim as one of the main contributions of this paper, and believe it would be useful for complex tasks in the future.<br /> The proposed architecture consists of three modules: the demonstration network, the context network, and the manipulation network.<br /> <br /> [[File:oneshot2.jpg|1000px|center]]<br /> <br /> == Demonstration Network ==<br /> This network takes a demonstration as input and produces an embedding with size linearly proportional to the number of blocks and the size of the demonstration.<br /> === Temporal Dropout ===<br /> Since a demonstration for block stacking can be very long, the authors randomly discard 95% of the time steps, a process they call 'temporal dropout'. The reduced size of the demonstrations allows multiple trajectories to be explored during testing to calculate an ensemble estimate. Dilated temporal convolutions and neighborhood attention are then repeatedly applied to the downsampled demonstrations.<br /> <br /> === Neighborhood Attention ===<br /> Since demonstration sizes can vary, a mechanism is needed that is not restricted to fixed-length inputs. While soft attention is one such mechanism, the problem with it is that there may be increasingly large amounts of information lost if soft attention is used to map longer demonstrations to the same fixed length as shorter demonstrations. As a solution, the authors propose having the same number of outputs as inputs, but with attention performed on other inputs relative to the current input.<br /> <br /> A query &lt;math&gt;q&lt;/math&gt;, a list of context vectors &lt;math&gt;\{c_j\}&lt;/math&gt;, and a list of memory vectors &lt;math&gt;\{m_j\}&lt;/math&gt; are given as input to soft attention. Each attention weight is given by the product of a learned weight vector and a nonlinearity applied to the sum of the query and corresponding context vector. Softmaxed weights applied to the corresponding memory vector form the output of the soft attention.<br /> <br /> $Inputs: q, \{c_j\}, \{m_j\}$<br /> $Weights: w_i \leftarrow v^Ttanh(q+c_i)$<br /> $Output: \sum_i{m_i\frac{\exp(w_i)}{\sum_j{\exp(w_j)}}}$<br /> <br /> A list of same-length embeddings, coming from a previous neighbourhood attention layer or a projection from the list of block coordinates, is given as input to neighborhood attention. For each block, two separate linear layers produce a query vector and a context vector, while a memory vector is a list of tuples that describe the position of each block joined with the input embedding for that block. Soft attention is then performed on this query, context vector, and memory vector. The authors claim that the intuition behind this process is to allow each block to provide information about itself relative to the other blocks in the environment. Finally, for each block, a linear transformation is performed on the vector composed by concatenating the input embedding, the result of the soft attention for that block, and the robot's state.<br /> <br /> For an environment with B blocks:<br /> $State: s$<br /> $Block_i: b_i \leftarrow (x_i, y_i, z_i)$<br /> $Embeddings: h_1^{in}, ..., h_B^{in}$ <br /> $Query_i: q_i \leftarrow Linear(h_i^{in})$<br /> $Context_i: c_i \leftarrow Linear(h_i^{in})$<br /> $Memory_i: m_i \leftarrow (b_i, h_i^{in})$<br /> $Result_i: result_i \leftarrow SoftAttn(q_i, \{c_j\}_{j=1}^B, \{m_k\}_{k=1}^B)$<br /> $Output_i: output_i \leftarrow Linear(concat(h_i^{in}, result_i, b_i, s))$<br /> <br /> == Context network ==<br /> This network takes the current state and the embedding produced by the demonstration network as inputs and outputs a fixed-length &quot;context embedding&quot; which captures only the information relevant for the manipulation network at this particular step.<br /> === Attention over demonstration ===<br /> The current state is used to compute a query vector which is then used for attending over all the steps of the embedding. Since at each time step there are multiple blocks, the weights for each are summed together to produce a scalar for each time step. Neighbourhood attention is then applied several times, using an LSTM with untied weights, since the information at each time steps needs to be propagated to each block's embedding. <br /> <br /> Performing attention over the demonstration yields a vector whose size is independent of the demonstration size; however, it is still dependent on the number of blocks in the environment, so it is natural to now attend over the state in order to get a fixed-length vector.<br /> === Attention over current state ===<br /> The authors propose that in general, within each subtask, only a limited number of blocks are relevant for performing the subtask. If the subtask is to stack A on B, then intuitively, one would suppose that only block A and B are relevant, and perhaps any blocks that may be blocking access to either A or B. This is not enforced during training, but once soft attention is applied to the current state to produce a fixed-length context embedding, the authors believe that the model does indeed learn in this way.<br /> <br /> == Manipulation network ==<br /> Given the context embedding as input, this simple feedforward network decides on the particular action needed, to complete the subtask of stacking one particular 'source' block on top of another 'target' block. The manipulation network uses an MLP network. Since the network in the paper can only takes into account the source and target block it may take subobtimal paths. For example changing [ABC, D] to [C, ABD] can be done in one motion if it was possible to manipulate two blocks at once. The manipulation network is the simplest part of the network and leaves room to expand upon in future work.<br /> <br /> = Experiments = <br /> The proposed model was tested on the block stacking tasks. the experiments were designed at answering the following questions:<br /> * How does training with behavioral cloning compare with DAGGER?<br /> * How does conditioning on the entire demonstration compare to conditioning on the final state?<br /> * How does conditioning on the entire demonstration compare to conditioning on a “snapshot” of the trajectory?<br /> * Can the authors' framework generalize to tasks that it has never seen during training?<br /> For the experiments, 140 training tasks and 43 testing tasks were collected, each with between 2 to 10 blocks and a different, desired final layout. Over 1000 demonstrations for each task were collected using a hard-coded policy rather than a human user. The authors compare 4 different architectures in these experiments:<br /> * Behavioural cloning used to train the proposed model<br /> * DAGGER used to train the proposed model<br /> * The proposed model, trained with DAGGER, but conditioned on the desired final state rather than an entire demonstration<br /> * The proposed model, trained with DAGGER, but conditioned on a 'snapshot' of the environment at the end of each subtask (ie. every time a block is stacked on another block)<br /> <br /> == Performance Evaluation ==<br /> [[File:oneshot3.jpg|1000px]]<br /> <br /> The most confident action at each timestep is chosen in 100 different task configurations, and results are averaged over tasks that had the same number of blocks. The results suggest that the performance of each of the architectures is comparable to that of the hard-coded policy which they aim to imitate. Performance degrades similarly across all architectures and the hard-coded policy as the number of blocks increases. On the harder tasks, conditioning on the entire demonstration led to better performance than conditioning on snapshots or on the final state. The authors believe that this may be due to the lack of information when conditioning only on the final state as well as due to regularization caused by temporal dropout which leads to data augmentation when conditioning on the full demonstration but is omitted when conditioning only on the snapshots or final state. Both DAGGER and behavioral cloning performed comparably well. As mentioned above, noise injection was used in training to improve performance; in practice, additional noise can still be injected but some may already come from other sources.<br /> <br /> == Visualization ==<br /> The authors visualize the attention mechanisms underlying the main policy architecture to have a better understanding about how it operates. There are two kinds of attention that the authors are mainly interested in, one where the policy attends to different time steps in the demonstration, and the other where the policy attends to different blocks in the current state. The figures below show some of the policy attention heatmaps over time.<br /> <br /> [[File:paper6_Visualization.png|800px]]<br /> <br /> = Conclusions =<br /> The proposed model successfully learns to complete new instances of a new task from just a single demonstration. The model was demonstrated to work on a series of block stacking tasks. The authors propose several extensions including enabling few-shot learning when one demonstration is insufficient, using image data as the demonstrations, and attempting many other tasks aside from block stacking.<br /> <br /> = Criticisms =<br /> While the paper shows an incredibly impressive result: the ability to learn a new task from just a single demonstration, there are a few points that need clearing up.<br /> Firstly, the authors use a hard-coded policy in their experiments rather than a human. It is clear that the performance of this policy begins to degrade quickly as the complexity of the task increases. It would be useful to know what this hard-coded policy actually was, and if the proposed model could still have comparable performance if a more successful demonstration, perhaps one by a human user, were performed. Give the current popularity of adversarial examples, it would also be interesting to see the performance when conditioned on an &quot;adversarial&quot; demonstration, that achieves the correct final state, but intentionally performs complex or obfuscated steps to get there.<br /> Second, it would be useful to see the model's performance on a more complex family of tasks than block stacking, since although each block stacking task is slightly different, the differences may turn out be insignificant compared to other tasks that this model should work on if it is to be a general imitation learning architecture; intuitively, the space of all possible moves and configurations is not large for the task. Also it is a bit misleading as there seems to be a need for more demonstrations to first get a reasonable policy that can generalize, leading to generic policy and then use just one demonstration on a new task expecting the policy to generalize. So it seems there is some sort of pre training involved here. Regardless, this work is a big step forward for imitation learning, permitting a wider range of tasks for which there is little training data and no reward function available, to still be successfully solved.<br /> <br /> = Illustrative Example: Particle Reaching =<br /> <br /> [[File:f1.png]]<br /> <br /> Figure 1: [Left] Agent, [Middle] Orange square is target, [Right] Green triangle is target .<br /> <br /> Another simple yet insightful example of the One-Shot Imitation Learning is the particle reaching problem which provides a relatively simple suite of tasks from which the network needs to solve an arbitrary one. The problem is formulated such that for each task: there is an agent which can move based on a 2D force vector, and n landmarks at varying 2D locations (n varies from task to task) with the goal of moving the agent to the specific landmark reached in the demonstration. This is illustrated in Figure 1. <br /> <br /> [[File:f2.png|450px]]<br /> <br /> Figure 2: Experimental results .<br /> <br /> Some insight comes from the use of different network architectures to solve this problem. The three architectures to compare (described below) are plain LSTM, LSTM with attention, and final state with attention. The key insight is that the architectures go from generic to specific, with the best generalization performance achieved with the most specific architecture, final state with attention, as seen in Figure 2. It is important to note that this conclusion does not carry forward to more complicated tasks such as the block stacking task.<br /> *Plain LSTM: 512 hidden units, with the input being the demonstration trajectory (the position of the agent changes over time and approaches one of the targets). Output of the LSTM with the current state (from the task needed to be solved) is the input for a multi-layer perceptron (MLP) for finding the solution.<br /> *LSTM with attention: Output of LSTM is now a set of weights for the different targets during training. These weights and the test state are used in the test task. The, now, 2D output is the input for an MLP as before.<br /> *Final state with attention: Looks only at the final state of the demonstration since it can sufficiently provide the needed detail of which target to reach (trajectory is not required). Similar to previous architecture, produces weights used by MLP.<br /> <br /> = Source =<br /> # Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. &quot;Neural machine translation by jointly learning to align and translate.&quot; arXiv preprint arXiv:1409.0473 (2014).<br /> # Duan, Yan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. &quot;One-shot imitation learning.&quot; In Advances in neural information processing systems, pp. 1087-1098. 2017.<br /> # Y. Duan, M. Andrychowicz, B. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba. One-shot imitation learning. arXiv preprint arXiv:1703.07326, 2017. (Newer revision)<br /> # Finn, Chelsea, Pieter Abbeel, and Sergey Levine. &quot;Model-agnostic meta-learning for fast adaptation of deep networks.&quot; arXiv preprint arXiv:1703.03400 (2017).<br /> # Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. &quot;Sequence to sequence learning with neural networks.&quot; Advances in neural information processing systems. 2014.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wasserstein_Auto-Encoders&diff=36192 Wasserstein Auto-Encoders 2018-04-05T21:18:36Z <p>S3wen: /* Experimental Results */</p> <hr /> <div><br /> = Introduction =<br /> Recent years have seen a convergence of two previously distinct approaches: representation learning from high dimensional data, and unsupervised generative modeling. In the field that formed at their intersection, Variational Auto-Encoders (VAEs) and Generative Adversarial Networks (GANs) have emerged to become well-established. VAEs are theoretically elegant but with the drawback that they tend to generate blurry samples when applied to natural images. GANs on the other hand produce better visual quality of sampled images, but come without an encoder, are harder to train and suffer from the mode-collapse problem when the trained model is unable to capture all the variability in the true data distribution. There has been recent research in generating encoder-decoder GANs where an encoder is trained in parallel with the generator, based on the intuition that this will allow the GAN to learn meaningful mapping from the compressed representation to the original image; however, these models also suffer from mode-collapse and perform comparable to vanilla GANs. Thus there has been a push to come up with the best way to combine them together, but a principled unifying framework is yet to be discovered.<br /> <br /> This work proposes a new family of regularized auto-encoders called the Wasserstein Auto-Encoder (WAE). The proposed method provides a novel theoretical insight into setting up an objective function for auto-encoders from the point of view of of optimal transport (OT). This theoretical formulation leads the authors to examine adversarial and maximum mean discrepancy based regularizers for matching a prior and the distribution of encoded data points in the latent space. An empirical evaluation is performed on MNIST and CelebA datasets, where WAE is found to generate samples of better quality than VAE while preserving training stability, encoder-decoder structure and nice latent manifold structure.<br /> <br /> The main contribution of the proposed algorithm is to provide theoretical foundations for using optimal transport cost as the auto-encoder objective function, while blending auto-encoders and GANs in a principled way. It also theoretically and experimentally explores the interesting relationships between WAEs, VAEs and adversarial auto-encoders.<br /> <br /> = Proposed Approach =<br /> ==Theory of Optimal Transport and Wasserstein Distance==<br /> Wasserstein Distance is a measure of the distance between two probability distributions. It is also called Earth Mover’s distance, short for EM distance, because informally it can be interpreted as moving piles of dirt that follow one probability distribution at a minimum cost to follow the other distribution. The cost is quantified by the amount of dirt moved times the moving distance. <br /> A simple case where the probability domain is discrete is presented below.<br /> <br /> <br /> [[File:em_distance.PNG|thumb|upright=1.4|center|Step-by-step plan of moving dirt between piles in ''P'' and ''Q'' to make them match (''W'' = 5).]]<br /> <br /> <br /> When dealing with the continuous probability domain, the EM distance or the minimum one among the costs of all dirt moving solutions becomes:<br /> \begin{align}<br /> \small W(p_r, p_g) = \underset{\gamma\sim\Pi(p_r, p_g)} {\inf}\pmb{\mathbb{E}}_{(x,y)\sim\gamma}[\parallel x-y\parallel]<br /> \end{align}<br /> <br /> Where &lt;math&gt;\Pi(p_r, p_g)&lt;/math&gt; is the set of all joint probability distributions with marginals &lt;math&gt;p_r&lt;/math&gt; and &lt;math&gt;p_g&lt;/math&gt;. Here the distribution &lt;math&gt;\gamma&lt;/math&gt; is called a transport plan because it's marginal structure gives some intuition that it represents the amount of probability mass to be moved from x to y. This intuition can be explained by looking at the following equation.<br /> <br /> \begin{align}<br /> \int\gamma(x, y)dx = p_g(y)<br /> \end{align}<br /> <br /> The Wasserstein distance or the cost of Optimal Transport (OT) provides a much weaker topology, which informally means that it makes it easier for a sequence of distribution to converge as compared to other ''f''-divergences. This is particularly important in applications where data is supported on low dimensional manifolds in the input space. As a result, stronger notions of distances such as KL-divergence, often max out, providing no useful gradients for training. In contrast, OT has a much nicer linear behaviour even upon saturation. It can be shown that the Wasserstein distance has guarantees of continuity and differentiability (Arjovsky et al., 2017). Moreover, Arjovsky et al. show there is a nice relationship between the magnitude of the Wasserstein distance and the distance between distributions; a smaller distance nicely corresponds to a smaller distance between the two distributions, and vice versa.<br /> <br /> ==Problem Formulation and Notation==<br /> In this paper, calligraphic letters, i.e. &lt;math&gt;\small {\mathcal{X}}&lt;/math&gt;, are used for sets, capital letters, i.e. &lt;math&gt;\small X&lt;/math&gt;, are used for random variables and lower case letters, i.e. &lt;math&gt;\small x&lt;/math&gt;, for their values. Probability distributions are denoted with capital letters, i.e. &lt;math&gt;\small P(X)&lt;/math&gt;, and corresponding densities with lower case letters, i.e. &lt;math&gt;\small p(x)&lt;/math&gt;.<br /> <br /> This work aims to minimize OT &lt;math&gt;\small W_c(P_X, P_G)&lt;/math&gt; between the true (but unknown) data distribution &lt;math&gt;\small P_X&lt;/math&gt; and a latent variable model &lt;math&gt;\small P_G&lt;/math&gt; specified by the prior distribution &lt;math&gt;\small P_Z&lt;/math&gt; of latent codes &lt;math&gt;\small Z \in \pmb{\mathbb{Z}}&lt;/math&gt; and the generative model &lt;math&gt;\small P_G(X|Z)&lt;/math&gt; of the data points &lt;math&gt;\small X \in \pmb{\mathbb{X}}&lt;/math&gt; given &lt;math&gt;\small Z&lt;/math&gt;. <br /> <br /> Kantorovich's formulation of the OT problem is given by:<br /> \begin{align}<br /> \small W_c(P_X, P_G) := \underset{\Gamma\sim {\mathcal{P}}(X \sim P_X, Y \sim P_G)}{\inf} {\pmb{\mathbb{E}}_{(X,Y)\sim\Gamma}[c(X,Y)]}<br /> \end{align}<br /> where &lt;math&gt;\small c(x,y)&lt;/math&gt; is any measurable cost function and &lt;math&gt;\small {\mathcal{P}(X \sim P_X,Y \sim P_G)}&lt;/math&gt; is a set of all joint distributions of &lt;math&gt;\small (X,Y)&lt;/math&gt; with marginals &lt;math&gt;\small P_X&lt;/math&gt; and &lt;math&gt;\small P_G&lt;/math&gt;. When &lt;math&gt;\small c(x,y)=d(x,y)&lt;/math&gt;, the following Kantorovich-Rubinstein duality holds for the &lt;math&gt;\small 1^{st}&lt;/math&gt; root of &lt;math&gt;\small W_c&lt;/math&gt;:<br /> \begin{align}<br /> \small W_1(P_X, P_G) := \underset{f \in {\mathcal{F_L}}} {\sup} {\pmb{\mathbb{E}}_{X \sim P_X}[f(X)]} -{\pmb{\mathbb{E}}_{Y \sim P_G}[f(Y)]}<br /> \end{align}<br /> where &lt;math&gt;\small {\mathcal{F_L}}&lt;/math&gt; is the class of all bounded [https://en.wikipedia.org/wiki/Lipschitz_continuity Lipschitz continuous functions]. A reference that provides an intuitive explanation for how the Kantorovich-Rubinstein duality was applied in this case is [https://vincentherrmann.github.io/blog/wasserstein/ here].<br /> <br /> ==Wasserstein Auto-Encoders==<br /> The proposed method focuses on latent variables &lt;math&gt;\small P_G &lt;/math&gt; defined by a two step procedure, where first a code &lt;math&gt;\small Z&lt;/math&gt; is sampled from a fixed prior distribution &lt;math&gt;\small P_Z&lt;/math&gt; on a latent space &lt;math&gt;\small {\mathcal{Z}}&lt;/math&gt; and then &lt;math&gt;\small Z&lt;/math&gt; is mapped to the image &lt;math&gt;\small X \in {\mathcal{X}}&lt;/math&gt; with a transformation. This results in a density of the form<br /> \begin{align}<br /> \small p_G(x) := \int_{{\mathcal{Z}}} p_G(x|z)p_z(z)dz, \forall x\in{\mathcal{X}}<br /> \end{align}<br /> assuming all the densities are properly defined. It turns out that if the focus is only on generative models deterministically mapping &lt;math&gt;\small Z &lt;/math&gt; to &lt;math&gt;\small X = G(Z) &lt;/math&gt;, then the OT cost takes a much simpler form as stated below by Theorem 1.<br /> <br /> '''Theorem 1''' For any function &lt;math&gt;\small G:{\mathcal{Z}} \rightarrow {\mathcal{X}}&lt;/math&gt;, where &lt;math&gt;\small Q(Z) &lt;/math&gt; is the marginal distribution of &lt;math&gt;\small Z &lt;/math&gt; when &lt;math&gt;\small X \in P_X &lt;/math&gt; and &lt;math&gt;\small Z \in Q(Z|X) &lt;/math&gt;,<br /> \begin{align}<br /> \small \underset{\Gamma\sim {\mathcal{P}}(X \sim P_X, Y \sim P_G)}{\inf} {\pmb{\mathbb{E}}_{(X,Y)\sim\Gamma}[c(X,Y)]} = \underset{Q : Q_z=P_z}{\inf} {{\pmb{\mathbb{E}}_{P_X}}{\pmb{\mathbb{E}}_{Q(Z|X)}}[c(X,G(Z))]}<br /> \end{align}<br /> This essentially means that instead of finding a coupling &lt;math&gt;\small \Gamma &lt;/math&gt; between two random variables living in the &lt;math&gt;\small {\mathcal{X}} &lt;/math&gt; space, one distributed according to &lt;math&gt;\small P_X &lt;/math&gt; and the other one according to &lt;math&gt;\small P_G &lt;/math&gt;, it is sufficient to find a conditional distribution &lt;math&gt;\small Q(Z|X) &lt;/math&gt; such that its &lt;math&gt;\small Z &lt;/math&gt; marginal &lt;math&gt;\small Q_Z(Z) := {\pmb{\mathbb{E}}_{X \sim P_X}[Q(Z|X)]} &lt;/math&gt; is identical to the prior distribution &lt;math&gt;\small P_Z &lt;/math&gt;. In order to implement a numerical solution to Theorem 1, the constraints on &lt;math&gt;\small Q(Z|X) &lt;/math&gt; and &lt;math&gt;\small P_Z &lt;/math&gt; are relaxed and a penalty function is added to the objective leading to the WAE objective function given by:<br /> <br /> \begin{align}<br /> \small D_{WAE}(P_X, P_G):= \underset{Q(Z|X) \in Q}{\inf} {{\pmb{\mathbb{E}}_{P_X}}{\pmb{\mathbb{E}}_{Q(Z|X)}}[c(X,G(Z))]} + {\lambda} {{\mathcal{D}}_Z(Q_Z,P_Z)}<br /> \end{align}<br /> where &lt;math&gt;\small Q &lt;/math&gt; is any non-parametric set of probabilistic encoders, &lt;math&gt;\small {\mathcal{D}}_Z &lt;/math&gt; is an arbitrary divergence between <br /> &lt;math&gt;\small Q_Z &lt;/math&gt; and &lt;math&gt;\small P_Z &lt;/math&gt;, and &lt;math&gt;\small \lambda &gt; 0 &lt;/math&gt; is a hyperparameter. The authors propose two different penalties &lt;math&gt;\small {\mathcal{D}}_Z(Q_Z,P_Z) &lt;/math&gt; based on adversarial training (GANs) and maximum mean discrepancy (MMD). The authors note that a numerical solution to the dual formulation of the problem has been tried by clipping the weights of the network (to satisfy the Lipschitz condition) and by penalizing the objective with &lt;math&gt;\small \lambda \mathbb{E}(\parallel \nabla f(X) \parallel - 1)^2 &lt;/math&gt;<br /> <br /> ===WAE-GAN: GAN-based===<br /> The first option is to choose &lt;math&gt;\small {\mathcal{D}}_Z(Q_Z,P_Z) = D_{JS}(Q_Z,P_Z)&lt;/math&gt;, where &lt;math&gt;\small D_{JS} &lt;/math&gt; is the Jensen-Shannon divergence metric, and use adversarial training to estimate it. Specifically a discriminator is introduced in the latent space &lt;math&gt;\small {\mathcal{Z}} &lt;/math&gt; trying to separate true points sampled from &lt;math&gt;\small P_Z &lt;/math&gt; from fake ones sampled from &lt;math&gt;\small Q_Z &lt;/math&gt;. This results in Algorithm 1. It is interesting that the min-max problem is moved from the input pixel space to the latent space.<br /> <br /> <br /> [[File:wae-gan.PNG|270px|center]]<br /> <br /> ===WAE-MMD: MMD-based===<br /> For a positive definite kernel &lt;math&gt;\small k: {\mathcal{Z}} \times {\mathcal{Z}} \rightarrow {\mathcal{R}}&lt;/math&gt;, the following expression is called the maximum mean discrepancy:<br /> \begin{align}<br /> \small {MMD}_k(P_Z,Q_Z) = \parallel \int_{{\mathcal{Z}}} k(z,\cdot)dP_z(z) - \int_{{\mathcal{Z}}} k(z,\cdot)dQ_z(z) \parallel_{\mathcal{H}_k},<br /> \end{align}<br /> <br /> where &lt;math&gt;\mathcal{H}_k&lt;/math&gt; is the reproducing kernel Hilbert space of real-valued functions mapping &lt;math&gt;\mathcal{Z}&lt;/math&gt; to &lt;math&gt;\mathcal{R}&lt;/math&gt;. This can be used as a divergence measure and the authors propose to use &lt;math&gt;\small {\mathcal{D}}_Z(Q_Z,P_Z) = MMD_k(P_Z,Q_Z) &lt;/math&gt;, which leads to Algorithm 2.<br /> <br /> <br /> [[File:wae-mmd.PNG|270px|center]]<br /> <br /> = Comparison with Related Work =<br /> ==Auto-Encoders, VAEs and WAEs==<br /> Classical unregularized encoders only minimized the reconstruction cost, and resulted in training points being chaotically scattered across the latent space with holes in between, where the decoder had never been trained. They were hard to sample from and did not provide a useful representation. VAEs circumvented this problem by maximizing a variational lower-bound term comprising of a reconstruction cost and a KL-divergence measure which captures how distinct each training example is from the prior &lt;math&gt;\small P_Z&lt;/math&gt;. This however does not guarantee that the overall encoded distribution &lt;math&gt;\small {{\pmb{\mathbb{E}}_{P_X}}}[Q(Z|X)]&lt;/math&gt; matches &lt;math&gt;\small P_Z&lt;/math&gt;. This is ensured by WAE however, is a direct consequence of our objective function derived from Theorem 1, and is visually represented in the figure below. It is also interesting to note that this also allows WAE to have deterministic encoder-decoder pairs.<br /> <br /> <br /> [[File:vae-wae.PNG|500px|thumb|center|WAE and VAE regularization]]<br /> <br /> <br /> It is also shown that if &lt;math&gt;\small c(x,y)={\parallel x-y \parallel}_2^2&lt;/math&gt;, WAE-GAN is equivalent to adversarial autoencoders (AAE). Thus the theory suggests that AAE minimize the 2-Wasserstein distance between &lt;math&gt;\small P_X&lt;/math&gt; and &lt;math&gt;\small P_G&lt;/math&gt;.<br /> <br /> ==OT, W-GAN and WAE==<br /> The Wasserstein GAN (W-GAN) minimizes the 1-Wasserstein distance &lt;math&gt;\small W_1(P_X,P_G)&lt;/math&gt; for generative modeling. The W-GAN formulation is approached from the dual form and thus it cannot be applied to another other cost &lt;math&gt;\small W_c&lt;/math&gt; as the neat form of the Kantorovich-Rubinstein duality holds only for &lt;math&gt;\small W_1&lt;/math&gt;. WAE approaches the same problem from the primal form, can be applied to any cost function &lt;math&gt;\small c&lt;/math&gt; and comes naturally with an encoder. The constraint on OT in Theorem 1, is relaxed in line with theory on unbalanced optimal transport by adding a penalty or additional divergences to the objective.<br /> <br /> ==GANs and WAEs==<br /> Many of the GAN variations including f-GAN and W-GAN come without an encoder. Often it may be desirable to reconstruct the latent codes and use the learned manifold in which case they won't be applicable. For works which try to blend adversarial auto-encoder structures, encoders and decoders do not have incentive to be reciprocal. WAE does not necessarily lead to a min-max game and has a clear theoretical foundation for using penalties for regularization.<br /> <br /> =Experimental Results=<br /> The authors empirically evaluate the proposed WAE generative model by specifically testing if data points are accurately reconstructed, if the latent manifold has reasonable geometry, and if random samples of good visual quality are generated. <br /> <br /> '''Experimental setup:'''<br /> Gaussian prior distribution &lt;math&gt; \small P_Z&lt;/math&gt; and squared cost function &lt;math&gt; \small c(x,y)&lt;/math&gt; are used for data-points. The encoder-decoder pairs were deterministic. The convolutional deep neural network for encoder mapping and decoder mapping are similar to DC-GAN with batch normalization. Real world datasets, MNIST with 70k images and CelebA with 203k images were used for training and testing. For interpolations a pair of of held out images, &lt;math&gt;(x,y)&lt;/math&gt; from the test set are Auto-encoded (separately), to produce &lt;math&gt;(z_x, z_y)&lt;/math&gt; in the latent space. The elements of the latent space are linearly interpolated and decoded to produce the images below. <br /> <br /> '''WAE-GAN and WAE-MMD:'''<br /> In WAE-GAN, the discriminator &lt;math&gt; \small D &lt;/math&gt; composed of several fully connected layers with ReLu activations. For WAE-MMD, the RBF kernel failed to penalize outliers and thus the authors resorted to using inverse multiquadratics kernel &lt;math&gt; \small k(x,y)=C/(C+\parallel{x-y}_2^2\parallel) &lt;/math&gt;. Trained models are presented in the figure below.<br /> As far as random sampled results are concerned, WAE-GAN seems to be highly unstable but do lead to better matching scores among WAE-GAN, WAE-MMD and VAE. WAE-MMD on the other hand has much more stable training and fairly good quality of sampled results.<br /> <br /> '''Qualitative assessment:'''<br /> In order to quantitatively assess the quality of the generated images, they use the Fréchet Inception Distance and report the results on CelebA (The Fréchet Inception Distance measures the similarity between two sets of images, by comparing the Fréchet distance of multivariate Gaussian distributions fitted to their feature representations. In more detail, let &lt;math&gt; (m,C) &lt;/math&gt; denote the mean vector and covariance matrix of the features of the inception network (Szegedy et al. 2017) applied to model samples. Let &lt;math&gt;(m_w,C_w) &lt;/math&gt; denote the mean vector and covariance matrix of the features of the inception network applied to real data. Then the Fréchet Inception Distance between the model samples and the real data is &lt;math&gt; ||m-m_w||^2 +\mathrm{tr}(C+C_w-2(CC_w)^{\frac{1}{2}} )\,&lt;/math&gt; (Heusel et al. 2017). ) These results confirm that the sampled images from WAE are of better quality than from VAE (score: 82), and WAE-GAN gets a slightly better score (score:42) than WAE-MMD (score:55), which correlates with visual inspection of the images.<br /> <br /> [[File:results.png|800px|thumb|center|Results on MNIST and Celeb-A dataset. In &quot;test reconstructions&quot; (middle row of images), odd rows correspond to the real test points.]]<br /> <br /> <br /> <br /> The authors also heuristically evaluate the sharpness of generated samples using the Laplace filter. The numbers, summarized in Table1, show that WAE-MMD has samples of slightly better quality than VAE, while WAE-GAN achieves the best results overall.<br /> [[File: paper17_Table.png|300px|thumb|right|Qualitative Assessment of Images]]<br /> <br /> = Commentary and Conclusion =<br /> This paper presents an interesting theoretical justification for a new family of auto-encoders called Wasserstein Auto-Encoders (WAE). The objective function minimizes the optimal transport cost in the form of the Wasserstein distance, but relaxes theoretical constraints to separate it into a reconstruction cost and a regularization penalty. The regularization penalizes divergences between a prior and the distribution of encoded latent space training data, and is estimated by means of adversarial training (WAE-GAN), or kernel-based techniques (WAE-MMD). They show that they achieve samples of better visual quality than VAEs, while achieving stable training at the same time. They also theoretically show that WAEs are a generalization of adversarial auto-encoders (AAEs).<br /> <br /> Although the paper mentions that encoder-decoder pairs can be deterministic, they do not show the geometry of the latent space that is obtained. It is necessary to study the effect of randomness of encoders on the quality of obtained samples. While this method is evaluated on MNIST and CelebA datasets, it is also important to see their performance on other real world data distributions. The authors do not provide a comprehensive evaluation of WAE-GAN regularization, thus making it hard to comment on whether moving an adversarial problem to the latent space results in less instability. Reasons for better sample quality of WAE-GAN over WAE-MMD also need to be inspected. In the future it would be interesting to investigate different ways to compute the divergences between the encoded distribution and the prior distribution.<br /> <br /> =Open Source Code=<br /> 1. https://github.com/tolstikhin/wae <br /> <br /> 2. https://github.com/maitek/waae-pytorch<br /> <br /> =Sources=<br /> 1. M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN, 2017<br /> <br /> 2. Martin Heusel et al. &quot;Gans trained by a two time-scale update rule converge to a local nash equilibrium.&quot; Advances in Neural Information Processing Systems. 2017.<br /> <br /> 3. Christian Szegedy et al. &quot;Inception-v4, inception-resnet and the impact of residual connections on learning.&quot; AAAI. Vol. 4. 2017.<br /> <br /> 4. Ilya Tolstikhin, Olivier Bousquet, Sylvain Gelly, Bernhard Scholkopf. Wasserstein Auto-Encoders, 2017<br /> <br /> 5. https://lilianweng.github.io/lil-log/2017/08/20/from-GAN-to-WGAN.html</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_Audio_Synthesis_of_Musical_Notes_with_WaveNet_autoencoders&diff=36189 Neural Audio Synthesis of Musical Notes with WaveNet autoencoders 2018-04-05T20:52:50Z <p>S3wen: /* WaveNet Autoencoder */</p> <hr /> <div>= Introduction =<br /> The authors of this paper have pointed out that the method in which most notes are created are hand-designed instruments modifying pitch, velocity and filter parameters to produce the required tone, timbre and dynamics of a sound. The authors suggest that this may be a problem and thus suggest a data-driven approach to audio synthesis. They demonstrate how to generate new types of expressive and realistic instrument sounds using a neural network model instead of using specific arrangements of oscillators or algorithms for sample playback. The model is capable of learning semantically meaningful hidden representations which can be used as control signals for manipulating tone, timbre, and dynamics during playback. To train such a data expensive model the authors highlight the need for a large dataset much like ImageNet for music. The motivation for this work stems from recent advances in autoregressive models like WaveNet  and SampleRNN . These models are effective at modeling short and medium scale (~500ms) signals, but rely on external conditioning for large-term dependencies; the proposed model removes the need for external conditioning.<br /> <br /> = Contributions =<br /> To solve the problem highlighted above the authors propose two main contributions of their paper: <br /> * Wavenet-style autoencoder that learn to encode temural data over a long term audio structures without requiring external conditioning<br /> * NSynth: a large dataset of musical notes inspired by the emerging of large image datasets<br /> <br /> <br /> = Models =<br /> <br /> [[File:paper26-figure1-models.png|center]]<br /> <br /> == WaveNet Autoencoder ==<br /> <br /> While the proposed autoencoder structure is very similar to that of WaveNet the authors argue that the algorithm is novel in two ways:<br /> * It is able to attain consistent long-term structure without any external conditioning <br /> * Creating meaningful embedding which can be interpolated between<br /> <br /> In the original WaveNet architecture the authors use a stack of dilated convolutions to predict the next sample of audio given a prior sample. This approach was prone to &quot;babbling&quot; since it did not take into account long-term structure of the audio. In this model the joint probability of generating audio &lt;math&gt;x&lt;/math&gt; is:<br /> <br /> \begin{align}<br /> p(x) = \prod_{i=1}^N\{x_i | x_1, … , x_N-1\}<br /> \end{align}<br /> <br /> They authors try to capture long-term structure by passing the raw audio through the encoder to produce an embedding &lt;math&gt;Z = f(x) &lt;/math&gt;, and then shifting the input and feeding it into the decoder which reproduces the input. The resulting probability distribution: <br /> <br /> \begin{align}<br /> p(x) = \prod_{i=1}^N\{x_i | x_1, … , x_N-1, f(x) \}<br /> \end{align}<br /> <br /> <br /> A detailed block diagram of the modified WaveNet structure can be seen in figure 1b. This diagram demonstrates the encoder as a 30 layer network in each each node is a ReLU nonlinearity followed by a non-causal dilated convolution. Dilated convolution (aka convolutions with holes) is a type of convolution in which the filter skips input values with a certain step (step size of 1 is equivalent to the standard convolution), effectively allowing the network to operate at a coarser scale compared to traditional convolutional layers and have very large receptive fields. The resulting convolution is 128 channels all feed into another ReLU nonlinearity which is feed into another 1x1 convolution before getting down sampled with average pooling to produce a 16 dimension &lt;math&gt;Z &lt;/math&gt; distribution. Each &lt;math&gt;Z &lt;/math&gt; encoding is for a specific temporal resolution which the authors of the paper tuned to 32ms. This means that there are 125, 16 dimension &lt;math&gt;Z &lt;/math&gt; encodings for each 4 second note present in the NSynth database (1984 embeddings). <br /> Before the &lt;math&gt;Z &lt;/math&gt; embedding enters the decoder it is first upsampled to the original audio rate using nearest neighbor interpolation. The embedding then passes through the decoder to recreate the original audio note. The input audio data is first quantized using 8-bit mu-law encoding into 256 possible values, and the output prediction is the softmax over the possible values.<br /> <br /> == Baseline: Spectral Autoencoder ==<br /> Being unable to find an alternative fully deep model which the authors could use to compare to there proposed WaveNet autoencoder to, the authors just made a strong baseline. The baseline algorithm that the authors developed is a spectral autoencoder. The block diagram of its architecture can be seen in figure 1a. The baseline network is 10 layer deep. Each layer has a 4x4 kernels with 2x2 strides followed by a leaky-ReLU (0.1) and batch normalization. The final hidden vector(Z) was set to 1984 to exactly match the hidden vector of the WaveNet autoencoder. <br /> <br /> Given the simple architecture, the authors first attempted to train the baseline on raw waveforms as input, with a mean-squared error cost. This did not work well and showed the problem of the independent Gaussian assumption. Spectral representations from FFT worked better, but had low perceptual quality despite having low MSE cost after training. Training on the log magnitude of the power spectra, normalized between 0 and 1, was found to be best correlated with perceptual distortion. The authors also explored several representations of phase, finding that estimating magnitude and using established iterative techniques to reconstruct phase to be most effective. (The technique to reconstruct the phase from the magnitude comes from (Griffin and Lim 1984). It can be summarized as follows. In each iteration, generate a Fourier signal z by taking the Short Time Fourier transform of the current estimate of the complete time-domain signal, and replacing its magnitude component with the known true magnitude. Then find the time-domain signal whose Short Time Fourier transform is closest to z in the least-squares sense. This is the estimate of the complete signal for the next iteration. ) A final heuristic that was used by the authors to increase the accuracy of the baseline was weighting the mean square error (MSE) loss starting at 10 for 0 HZ and decreasing linearly to 1 at 4000 Hz and above. This is valid as the fundamental frequency of most instrument are found at lower frequencies. <br /> <br /> == Training ==<br /> Both the modified WaveNet and the baseline autoencoder used stochastic gradient descent with an Adam optimizer. The authors trained the baseline autoencoder model asynchronously for 1800000 epocs with a batch size of 8 with a learning rate of 1e-4. Where as the WaveNet modules were trained synchronously for 250000 epocs with a batch size of 32 with a decaying learning rate ranging from 2e-4 to 6e-6.<br /> <br /> = The NSynth Dataset =<br /> To evaluate the WaveNet autoencoder model, the authors' wanted an audio dataset that let them explore the learned embeddings. Musical notes are an ideal setting for this study. Prior to this paper, the existing music datasets included the RWC music database (Goto et al., 2003) and the dataset from Romani Picas et al. However, the authors wanted to develop a larger dataset.<br /> <br /> The NSynth dataset has 306 043 unique musical notes (each have a unique pitch, timbre, envelope) all 4 seconds in length sampled at 16,000 Hz. The data set consists of 1006 different instruments playing on average of 65.4 different pitches across on average 4.75 different velocities. Average pitches and velocities are used as not all instruments, can reach all 88 MIDI frequencies, or the 5 velocities desired by the authors. The dataset has the following split: training set with 289,205 notes, validation set with 12,678 notes, and test set with 4,096 notes.<br /> <br /> Along with each note the authors also included the following annotations:<br /> * Source - The way each sound was produced. There were 3 classes ‘acoustic’, ‘electronic’ and ‘synthetic’.<br /> * Family - The family class of instruments that produced each note. There are 11 classes which include: {‘bass’, ‘brass’, ‘vocal’ ext.}<br /> * Qualities - Sonic qualities about each note<br /> <br /> The full dataset is publicly available here: https://magenta.tensorflow.org/datasets/nsynth as TFRecord files with training and holdout splits.<br /> <br /> [[File:nsynth_table.png | 400px|thumb|center|Full details of the NSynth dataset.]]<br /> <br /> = Evaluation =<br /> <br /> To fully analyze all aspects of WaveNet the authors proposed three evaluations:<br /> * Reconstruction - Both Quantitative and Qualitative analysis were considered<br /> * Interpolation in Timbre and Dynamics<br /> * Entanglement of Pitch and Timbre <br /> <br /> Sound is historically very difficult to quantify from a picture representation as it requires training and expertise to analyze. Even with expertise it can be difficult to complete a full analysis as two very different sounds can look quite similar in their respective pictorial representations. This is why the authors recommend all readers to listen to the created notes which can be found here: https://magenta.tensorflow.org/nsynth.<br /> <br /> However, even when taking this under consideration the authors do pictorially demonstrate differences in the two proposed algorithms along with the original note, as it is hard to publish a paper with sound included. To demonstrate the pictorial difference the authors demonstrate each note using constant-q transform (CQT) which is able to capture the dynamics of timbre along with representing the frequencies of the sound.<br /> <br /> == Reconstruction ==<br /> <br /> [[File:paper27-figure2-reconstruction.png|center]]<br /> <br /> The authors attempted to show magnitude and phase on the same plot above. Instantaneous frequency is the derivative of the phase and the intensity of solid lines is proportional to the log magnitude of the power spectrum. If fharm and an FFT bin are not the same, then there will be a constant phase shift: <br /> &lt;math&gt;<br /> \triangle \phi = (f_{bin} − f_{harm}) \dfrac{hopsize}{samplerate}<br /> &lt;/math&gt;.<br /> <br /> === Qualitative Comparison ===<br /> In Figure 2, CQT spectrograms are displayed from 3 different instruments, including the original note spectrograms and the model reconstruction spectrograms. For the model reconstruction spectrograms, a baseline is adopted to compare with WaveNet. Each note contains some noise, a fundamental frequency with a series of harmonics, and a decay. In the Glockenspiel the WaveNet autoencoder is able to reproduce the magnitude, phase of the fundamental frequency (A and C in figure 2), and the attack (B in figure 2) of the instrument; Whereas the Baseline autoencoder introduces non existing harmonics (D in figure 2). The flugelhorn on the other hand, presents the starkest difference between the WaveNet and baseline autoencoders. The WaveNet while not perfect is able to reproduce the verbarto (I and J in figure 2) across multiple frequencies, which results in a natural sounding note. The baseline not only fails to do this but also adds extra noise (K in figure 2). The authors do add that the WaveNet produces some strikes (L in figure 2) however they argue that they are inaudible.<br /> <br /> [[File:paper27-table1.png|center]]<br /> <br /> Mu-law encoding was used in the original WaveNet [https://arxiv.org/pdf/1609.03499.pdf paper] to make the problem &quot;more tractable&quot; compared to raw 16-bit integer values. In that paper, they note that &quot;especially for speech, this non-linear quantization produces a significantly better reconstruction&quot; compared to a linear scheme. This might be expected considering that the mu-law companding transformation was designed to [https://www.cisco.com/c/en/us/support/docs/voice/h323/8123-waveform-coding.html#t4 encode speech]. In this application though, using this encoding creates perceptible distortion that sounds similar to clipping.<br /> <br /> === Quantitative Comparison ===<br /> For a quantitative comparison the authors trained a separate multi-task classifier to classify a note using given pitch or quality of a note. The results of both the Baseline and the WaveNet where then inputted and attempted to be classified. As seen in table 1 WaveNet significantly outperformed the Baseline in both metrics posting a ~70% increase when only considering pitch.<br /> <br /> == Interpolation in Timbre and Dynamics ==<br /> <br /> [[File:paper27-figure3-interpolation.png|center]]<br /> <br /> For this evaluation the authors reconstructed from linear interpolations in Z space among different instruments and compared these to superimposed position of the original two instruments. Not surprisingly the model fuse aspects of both instruments during the recreation. The authors claim however, that WaveNet produces much more realistic sounding results. <br /> To support their claim the authors the authors point to WaveNet ability to create dynamic mixing of overtone in time, even jumping to higher harmonics (A in figure 3), capturing the timbre and dynamics of both the bass and flute. This can be once again seen in (B in figure 3) where Wavenet adds additional harmonics as well as a sub-harmonics to the original flute note. <br /> <br /> <br /> == Entanglement of Pitch and Timbre ==<br /> <br /> [[File:paper27-table2.png|center]]<br /> <br /> [[File:paper27-figure4-entanglement.png|center]]<br /> <br /> To study the entanglement between pitch and Z space the authors constructed a classifier which was expected to drop in accuracy if the representation of pitch and timbre is disentangled as it relies heavily on the pitch information. This is clearly demonstrated by the first two rows of table 2 where WaveNet relies more strongly on pitch then the baseline algorithm. The authors provide a more qualitative demonstrating in figure 4. They demonstrate a situation in which a classifier may be confused; a note with pitch of +12 is almost exactly the same as the original apart from an emergence of sub-harmonics.<br /> <br /> Further insight can be gained on the relationship between pitch and timbre by studying the trend amongst the network embeddings among the pitches for specific instruments. This is depicted in figure 5 for several instruments across their entire 88 note range at 127 velocity. It can be noted from the figure that the instruments have unique separation of two or more registers over which the embeddings of notes with different pitches are similar. This is expected since instrumental dynamics and timbre varies dramatically over the range of the instrument.<br /> <br /> = Conclusion &amp; Future Directions =<br /> <br /> This paper presents a Wavelet autoencoder model which is built on top of the WaveNet model and evaluate the model on NSynth dataset. The paper also introduces a new large scale dataset of musical notes: NSynth.<br /> <br /> One significant area which the authors claim great improvement is needed is the large memory constraints required by there algorithm. Due to the large memory requirement the current WaveNet must rely on down sampling thus being unable to fully capture the global context. This is an area where model compression techniques could be beneficial. That is, quantization and pruning could be effective: with 4-bit quantization during the entire process (weights, activations, gradients, error as in the work of Wu et al., 2016), memory requirement could be reduced by at least 8 times. The authors also claim that research using different input representations (instead of mu-law) to minimize distortion is ongoing.<br /> <br /> = Critique = <br /> * Authors have never conducted a human study determining sound similarity between the original, baseline, and WaveNet.<br /> * Architecture is not very novel.<br /> * In order to have a comparison, they set out to create a straight-forward baseline for the neural audio synthesis experiments.<br /> <br /> = Open Source Code =<br /> <br /> Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/nsynth<br /> <br /> = References =<br /> <br /> # Engel, J., Resnick, C., Roberts, A., Dieleman, S., Norouzi, M., Eck, D. &amp; Simonyan, K.. (2017). Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders. Proceedings of the 34th International Conference on Machine Learning, in PMLR 70:1068-1077<br /> # Griffin, Daniel, and Jae Lim. &quot;Signal estimation from modified short-time Fourier transform.&quot; IEEE Transactions on Acoustics, Speech, and Signal Processing 32.2 (1984): 236-243.<br /> # NSynth: Neural Audio Synthesis. (2017, April 06). Retrieved March 19, 2018, from https://magenta.tensorflow.org/nsynth <br /> # The NSynth Dataset. (2017, April 05). Retrieved March 19, 2018, from https://magenta.tensorflow.org/datasets/nsynth<br /> # Oord, Aaron van den, Nal Kalchbrenner, and Koray Kavukcuoglu. &quot;Pixel recurrent neural networks.&quot; arXiv preprint arXiv:1601.06759 (2016).<br /> # Mehri, Soroush, et al. &quot;SampleRNN: An unconditional end-to-end neural audio generation model.&quot; arXiv preprint arXiv:1612.07837 (2016).<br /> # Wu, S., Li, G., Chen, F., &amp; Shi, L. (2018). Training and Inference with Integers in Deep Neural Networks. arXiv preprint arXiv:1802.04680.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36188 Training And Inference with Integers in Deep Neural Networks 2018-04-05T20:31:57Z <p>S3wen: /* Bitwidth of Errors */</p> <hr /> <div>== Introduction ==<br /> <br /> Deep neural networks have enjoyed much success in all manners of tasks, but it is common for these networks to be complicated and have high memory requirements while performing many floating-point operations (FLOPs). As a result, running many of these models are very expensive in terms of energy use, and using state-of-the-art networks in applications where energy is limited can be very difficult. In order to overcome this and allow use of these networks in situations with low energy availability, the energy costs must be reduced while trying to maintain as high network performance as possible and/or practical.<br /> <br /> Most existing methods focus on reducing the energy requirements during inference rather than training. Since training with SGD requires accumulation, training usually has higher precision demand than inference. Most of the existing methods focus on how to compress a model for inference, rather than during training. This paper proposes a framework to reduce complexity both during training and inference through the use of integers instead of floats. The authors address how to quantize all operations and operands as well as examining the bitwidth requirement for SGD computation &amp; accumulation. Using integers instead of floats results in energy-savings because integer operations are more efficient than floating point (see the table below). Also, there already exists dedicated hardware for deep learning that uses integer operations (such as the 1st generation of Google TPU) so understanding the best way to use integers is well-motivated. A TPU is a Tensor Processing Unit developed by Google for Tensor operations. TPU is comparative to a GPU but produces higher IO per second for low precision computations.<br /> {| class=&quot;wikitable&quot;<br /> |+Rough Energy Costs in 45nm 0.9V <br /> !<br /> ! colspan=&quot;2&quot; |Energy(pJ)<br /> ! colspan=&quot;2&quot; |Area(&lt;math&gt;\mu m^2&lt;/math&gt;)<br /> |-<br /> !Operation<br /> !MUL<br /> !ADD<br /> !MUL<br /> !ADD<br /> |-<br /> |8-bit INT<br /> |0.2<br /> |0.03<br /> |282<br /> |36<br /> |-<br /> |16-bit FP<br /> |1.1<br /> |0.4<br /> |1640<br /> |1360<br /> |-<br /> |32-bit FP<br /> |3.7<br /> |0.9<br /> |7700<br /> |4184<br /> |}<br /> The authors call the framework WAGE because they consider how best to handle the '''W'''eights, '''A'''ctivations, '''G'''radients, and '''E'''rrors separately.<br /> <br /> == Related Work ==<br /> <br /> === Weight and Activation ===<br /> Existing works to train DNNs on binary weights and activations  add noise to weights and activations as a form of regularization. The use of high-precision accumulation is required for SGD optimization since real-valued gradients are obtained from real-valued variables. XNOR-Net  uses bitwise operations to approximate convolutions in a highly memory-efficient manner, and applies a filter-wise scaling factor for weights to improve performance. However, these floating-point factors are calculated simultaneously during training, which aggravates the training effort. Ternary weight networks (TWN)  and Trained ternary quantization (TTQ)  offer more expressive ability than binary weight networks by constraining the weights to be ternary-valued {-1,0,1} using two symmetric thresholds. Tang et al.  achieve impressive results by using a binarization scheme according to which floating-point activation vectors are approximated as linear combinations of binary vectors, where the weights in the linear combination are floating-point. Still other approaches rely on relative quantization ; however, an efficient implementation is difficult to apply in practice due to the requirements of persisting and applying a codebook.<br /> <br /> === Gradient Computation and Accumulation ===<br /> The DoReFa-Net quantizes gradients to low-bandwidth floating point numbers with discrete states in the backwards pass. In order to reduce the overhead of gradient synchronization in distributed training the TernGrad method quantizes the gradient updates to ternary values. In both works the weights are still stored and updated with float32, and the quantization of batch normalization and its derivative is ignored.<br /> <br /> == WAGE Quantization ==<br /> The core idea of the proposed method is to constrain the following to low-bitwidth integers on each layer:<br /> * '''W:''' weight in inference<br /> * '''a:''' activation in inference<br /> * '''e:''' error in backpropagation<br /> * '''g:''' gradient in backpropagation<br /> [[File:p32fig1.PNG|center|thumb|800px|Four operators QW (·), QA(·), QG(·), QE(·) added in WAGE computation dataflow to reduce precision, bitwidth of signed integers are below or on the right of arrows, activations are included in MAC for concision.]]<br /> The error and gradient are defined as:<br /> <br /> &lt;math&gt;e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}&lt;/math&gt;<br /> <br /> where L is the loss function.<br /> <br /> The precision in bits of the errors, activations, gradients, and weights are &lt;math&gt;k_E&lt;/math&gt;, &lt;math&gt;k_A&lt;/math&gt;, &lt;math&gt;k_G&lt;/math&gt;, and &lt;math&gt;k_W&lt;/math&gt; respectively. As shown in the above figure, each quantity also has a quantization operators to reduce bitwidth increases caused by multiply-accumulate (MAC) operations. Also, note that since this is a layer-by-layer approach, each layer may be followed or preceded by a layer with different precision, or even a layer using floating point math.<br /> <br /> === Shift-Based Linear Mapping and Stochastic Mapping ===<br /> The proposed method makes use of a linear mapping where continuous, unbounded values are discretized for each bitwidth &lt;math&gt;k&lt;/math&gt; with a uniform spacing of<br /> <br /> &lt;math&gt;\sigma(k) = 2^{1-k}, k \in Z_+ &lt;/math&gt;<br /> With this, the full quantization function is<br /> <br /> &lt;math&gt;Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}&lt;/math&gt;, <br /> <br /> where &lt;math&gt;round&lt;/math&gt; approximates continuous values to their nearest discrete state, and &lt;math&gt;Clip&lt;/math&gt; is the saturation function that clips unbounded values to &lt;math&gt;[-1 + \sigma, 1 - \sigma]&lt;/math&gt;. Note that this function is only using when simulating integer operations on floating-point hardware, on native integer hardware, this is done automatically. In addition to this quantization function, a distribution scaling factor is used in some quantization operators to preserve as much variance as possible when applying the quantization function above. The scaling factor is defined below.<br /> <br /> &lt;math&gt;Shift(x) = 2^{round(log_2(x))}&lt;/math&gt;<br /> <br /> Finally, stochastic rounding is substituted for small or real-valued updates during gradient accumulation.<br /> <br /> A visual representation of these operations is below.<br /> [[File:p32fig2.PNG|center|thumb|800px|Quantization methods used in WAGE. The notation &lt;math&gt;P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil&lt;/math&gt; denotes probability, vector, floor and ceil, respectively. &lt;math&gt;Shift(\cdot)&lt;/math&gt; refers to distribution shifting with a certain argument]]<br /> <br /> === Weight Initialization ===<br /> In this work, batch normalization is simplified to a constant scaling layer in order to sidestep the problem of normalizing outputs without floating point math, and to remove the extra memory requirement with batch normalization. As such, some care must be taken when initializing weights. The authors use a modified initialization method base on MSRA .<br /> <br /> &lt;math&gt;W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma&lt;/math&gt;<br /> <br /> &lt;math&gt;n_{in}&lt;/math&gt; is the layer fan-in number, &lt;math&gt;U&lt;/math&gt; denotes uniform distribution. The original initialization method for &lt;math&gt;\eta&lt;/math&gt; is modified by adding the condition that the distribution width should be at least &lt;math&gt;\beta \sigma&lt;/math&gt;, where &lt;math&gt;\beta&lt;/math&gt; is a constant greater than 1 and &lt;math&gt;\sigma&lt;/math&gt; is the minimum step size seen already. This prevents weights being initialised to all-zeros in the case where the bitwidth is low, or the fan-in number is high.<br /> <br /> === Quantization Details ===<br /> <br /> ==== Weight &lt;math&gt;Q_W(\cdot)&lt;/math&gt; ====<br /> &lt;math&gt;W_q = Q_W(W) = Q(W, k_W)&lt;/math&gt;<br /> <br /> The quantization operator is simply the quantization function previously introduced. <br /> <br /> ==== Activation &lt;math&gt;Q_A(\cdot)&lt;/math&gt; ====<br /> The authors say that the variance of the weights passed through this function will be scaled compared to the variance of the weights as initialized. To prevent this effect from blowing up the network outputs, they introduce a scaling factor &lt;math&gt;\alpha&lt;/math&gt;. Notice that it is constant for each layer.<br /> <br /> &lt;math&gt;\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}&lt;/math&gt;<br /> <br /> The quantization operator is then<br /> <br /> &lt;math&gt;a_q = Q_A(a) = Q(a/\alpha, k_A)&lt;/math&gt;<br /> <br /> The scaling factor approximates batch normalization.<br /> <br /> ==== Error &lt;math&gt;Q_E(\cdot)&lt;/math&gt; ====<br /> The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net ) solves the issue by using an affine transform to map the error to the range &lt;math&gt;[-1, 1]&lt;/math&gt;, apply quantization, and then applying the inverse transform. However, the authors claim that this approach still requires using float32, and that the magnitude of the error is unimportant: rather it is the orientation of the error. Thus, they only scale the error distribution to the range &lt;math&gt;\left [ -\sqrt2, \sqrt2 \right ]&lt;/math&gt; and quantise:<br /> <br /> &lt;math&gt;e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)&lt;/math&gt;<br /> <br /> Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br /> <br /> ==== Gradient &lt;math&gt;Q_G(\cdot)&lt;/math&gt; ====<br /> Similar to the activations and errors, the gradients are rescaled:<br /> <br /> &lt;math&gt;g_s = \eta \cdot g/Shift(max\{|g|\})&lt;/math&gt;<br /> <br /> &lt;math&gt; \eta &lt;/math&gt; is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes &lt;math&gt; \sigma(k) &lt;/math&gt;. When reducing the bitwidth of the gradients (remember that the gradients are coming out of a MAC operation, so the bitwidth may have increased) stochastic rounding is used as a substitute for small gradient accumulation.<br /> <br /> &lt;math&gt;\Delta W = Q_G(g) = \sigma(k_G) \cdot sgn(g_s) \cdot \left \{ \lfloor | g_s | \rfloor + Bernoulli(|g_s|<br /> - \lfloor | g_s | \rfloor) \right \}&lt;/math&gt;<br /> <br /> This randomly rounds the result of the MAC operation up or down to the nearest quantization for the given gradient bitwidth. The weights are updated with the resulting discrete increments:<br /> <br /> &lt;math&gt;W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}&lt;/math&gt;<br /> <br /> === Miscellaneous ===<br /> To train WAGE networks, the authors used pure SGD exclusively because more complicated techniques such as Momentum or RMSProp increase memory consumption and are complicated by the rescaling that happens within each quantization operator.<br /> <br /> The quantization and stochastic rounding are a form of regularization.<br /> <br /> The authors didn't use a traditional softmax with cross-entropy loss for the experiments because there does not yet exist a softmax layer for low-bit integers. Instead, they use a sum of squared error loss. This works for tasks with a small number of categories, but does not scale well.<br /> <br /> == Experiments ==<br /> For all experiments, the default layer bitwidth configuration is 2-8-8-8 for Weights, Activations, Gradients, and Error bits. The weight bitwidth is set to 2 because that results in ternary weights, and therefore no multiplication during inference. They authors argue that the bitwidth for activation and errors should be the same because the computation graph for each is similar and might use the same hardware. During training, the weight bitwidth is 8. For inference the weights are ternarized.<br /> <br /> === Implementation Details ===<br /> MNIST: Network is LeNet-5 variant  with 32C5-MP2-64C5-MP2-512FC-10SSE.<br /> <br /> SVHN &amp; CIFAR10: VGG variant  with 2×(128C3)-MP2-2×(256C3)-MP2-2×(512C3)-MP2-1024FC-10SSE. For CIFAR10 dataset, the data augmentation is followed in Lee et al. (2015)  for training.<br /> <br /> ImageNet: AlexNet variant  on ILSVRC12 dataset.<br /> {| class=&quot;wikitable&quot;<br /> |+Test or validation error rates (%) in previous works and WAGE on multiple datasets. Opt denotes gradient descent optimizer, withM means SGD with momentum, BN represents batch normalization, 32 bit refers to float32, and ImageNet top-k format: top1/top5.<br /> !Method<br /> !&lt;math&gt;k_W&lt;/math&gt;<br /> !&lt;math&gt;k_A&lt;/math&gt;<br /> !&lt;math&gt;k_G&lt;/math&gt;<br /> !&lt;math&gt;k_E&lt;/math&gt;<br /> !Opt<br /> !BN<br /> !MNIST<br /> !SVHN<br /> !CIFAR10<br /> !ImageNet<br /> |-<br /> |BC<br /> |1<br /> |32<br /> |32<br /> |32<br /> |Adam<br /> |yes<br /> |1.29<br /> |2.30<br /> |9.90<br /> |<br /> |-<br /> |BNN<br /> |1<br /> |1<br /> |32<br /> |32<br /> |Adam<br /> |yes <br /> |0.96<br /> |2.53<br /> |10.15<br /> |<br /> |-<br /> |BWN<br /> |1<br /> |32<br /> |32<br /> |32<br /> |withM<br /> |yes<br /> |<br /> |<br /> |<br /> |43.2/20.6<br /> |-<br /> |XNOR<br /> |1<br /> |1<br /> |32<br /> |32<br /> |Adam<br /> |yes<br /> |<br /> |<br /> |<br /> |55.8/30.8<br /> |-<br /> |TWN<br /> |2<br /> |32<br /> |32<br /> |32<br /> |withM<br /> |yes<br /> |0.65<br /> |<br /> |7.44<br /> |'''34.7/13.8'''<br /> |-<br /> |TTQ<br /> |2<br /> |32<br /> |32<br /> |32<br /> |Adam<br /> |yes<br /> |<br /> |<br /> |6.44<br /> |42.5/20.3<br /> |-<br /> |DoReFa<br /> |8<br /> |8<br /> |32<br /> |8<br /> |Adam<br /> |yes<br /> |<br /> |2.30<br /> |<br /> |47.0/<br /> |-<br /> |TernGrad<br /> |32<br /> |32<br /> |2<br /> |32<br /> |Adam<br /> |yes<br /> |<br /> |<br /> |14.36<br /> |42.4/19.5<br /> |-<br /> |WAGE<br /> |2<br /> |8<br /> |8<br /> |8<br /> |SGD<br /> |no<br /> |'''0.40'''<br /> |'''1.92'''<br /> |'''6.78'''<br /> |51.6/27.8<br /> |}<br /> <br /> === Training Curves and Regularization ===<br /> The authors compare the 2-8-8-8 WAGE configuration introduced above, a 2-8-f-f (meaning float32) configuration, and a completely floating point version on CIFAR10. The test error is plotted against epoch. For training these networks, the learning rate is divided by 8 at the 200th epoch and again at the 250th epoch.<br /> [[File:p32fig3.PNG|center|thumb|800px|Training curves of WAGE variations and a vanilla CNN on CIFAR10]]<br /> The convergence of the 2-8-8-8 has comparable convergence to the vanilla CNN and outperforms the 2-8-f-f variant. The authors speculate that this is because the extra discretization acts as a regularizer.<br /> <br /> === Bitwidth of Errors ===<br /> The CIFAR10 test accuracy is plotted against bitwidth below and the error density for a single layer is compared with the Vanilla network.<br /> [[File:p32fig4.PNG|center|thumb|520x522px|The 10 run accuracies of different &lt;math&gt;k_E&lt;/math&gt;]]<br /> <br /> [[File:32_error.png|center|thumb|520x522px|Histogram of errors for Vanilla network and Wage network. After being quantized and shifted each layer, the error is reshaped and so most orientation information is retained. ]]<br /> <br /> The table below shows the test error rates on CIFAR10 when left-shift upper boundary with factor γ. From this table we could see that large values play critical roles for backpropagation training even though they are infrequent while the majority with small values are just noise.<br /> <br /> [[File:testerror_rate.png|center]]<br /> <br /> === Bitwidth of Gradients ===<br /> <br /> The authors next investigated the choice of a proper &lt;math&gt;k_G&lt;/math&gt; for gradients using the CIFAR10 dataset. <br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Test error rates (%) on CIFAR10 with different &lt;math&gt;k_G&lt;/math&gt;<br /> !&lt;math&gt;k_G&lt;/math&gt;<br /> !2<br /> !3<br /> !4<br /> !5<br /> !6<br /> !7<br /> !8<br /> !9<br /> !10<br /> !11<br /> !12<br /> |-<br /> |error<br /> |54.22<br /> |51.57<br /> |28.22<br /> |18.01<br /> |11.48<br /> |7.61<br /> |6.78<br /> |6.63<br /> |6.43<br /> |6.55<br /> |6.57<br /> |}<br /> <br /> The results show similar bitwidth requirements as the last experiment for &lt;math&gt;k_E&lt;/math&gt;.<br /> <br /> The authors also examined the effect of bitwidth on the ImageNet implementation.<br /> <br /> Here, C denotes 12 bits (Hexidecimal) and BN refers to batch normalization being added. 7 models are used: 2888 from the first experiment, 288C for more accurate errors (12 bits), 28C8 for larger buffer space, 28f8 for non-quantization of gradients, 28ff for errors and gradients in float32, and 28ff with BN added. The baseline vanilla model refers to the original AlexNet architecture. <br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Top-5 error rates (%) on ImageNet with different &lt;math&gt;k_G&lt;/math&gt;and &lt;math&gt;k_E&lt;/math&gt;<br /> !Pattern<br /> !vanilla<br /> !28ff-BN<br /> !28ff<br /> !28f8<br /> !28C8<br /> !288C<br /> !2888<br /> |-<br /> |error<br /> |19.29<br /> |20.67<br /> |24.14<br /> |23.92<br /> |26.88<br /> |28.06<br /> |27.82<br /> |}<br /> <br /> The comparison between 28C8 and 288C shows that the model may perform better if it has more buffer space &lt;math&gt;k_G&lt;/math&gt; for gradient accumulation than if it has high-resolution orientation &lt;math&gt;k_E&lt;/math&gt;. The authors also noted that batch normalization and &lt;math&gt;k_G&lt;/math&gt; are more important for ImageNet because the training set samples are highly variant.<br /> <br /> == Discussion ==<br /> The authors have a few areas they believe this approach could be improved.<br /> <br /> '''MAC Operation:''' The 2-8-8-8 configuration was chosen because the low weight bitwidth means there aren't any multiplication during inference. However, this does not remove the requirement for multiplication during training. 2-2-8-8 configuration satisfies this requirement, but it is difficult to train and detrimental to the accuracy.<br /> <br /> '''Non-linear Quantization:''' The linear mapping used in this approach is simple, but there might be a more effective mapping. For example, a logarithmic mapping could be more effective if the weights and activations have a log-normal distribution.<br /> <br /> '''Normalization:''' Normalization layers (softmax, batch normalization) were not used in this paper. Quantized versions are an area of future work<br /> <br /> == Conclusion ==<br /> <br /> A framework for training and inference without the use of floating-point representation is presented. By quantizing all operations and operands of a network, the authors successfully reduce the energy costs of both training and inference with deep learning architectures. Future work may further improve compression and memory requirements.<br /> <br /> == Implementation ==<br /> The following repository provides the source code for the paper: https://github.com/boluoweifenda/WAGE. The repository provides the source code as written by the authors, in Tensorflow.<br /> <br /> <br /> <br /> == References ==<br /> <br /> # Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 &quot;Efficient Processing of Deep Neural Networks: A Tutorial and Survey&quot;]. arXiv:1703.09039 [cs].<br /> # Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 &quot;BinaryConnect: Training Deep Neural Networks with binary weights during propagations&quot;]. arXiv:1511.00363 [cs].<br /> # Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 &quot;Ternary Weight Networks&quot;]. arXiv:1605.04711 [cs].<br /> # He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 &quot;Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification&quot;]. arXiv:1502.01852 [cs].<br /> # Zhou, Shuchang; Wu, Yuxin; Ni, Zekun; Zhou, Xinyu; Wen, He; Zou, Yuheng (2016-06-20). [http://arxiv.org/abs/1606.06160 &quot;DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients&quot;]. arXiv:1606.06160 [cs].<br /> # Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true &quot;Gradient-based learning applied to document recognition&quot;]. Proceedings of the IEEE. 86 (11): 2278–2324. doi:10.1109/5.726791. ISSN 0018-9219.<br /> # Simonyan, Karen; Zisserman, Andrew (2014-09-04). [http://arxiv.org/abs/1409.1556 &quot;Very Deep Convolutional Networks for Large-Scale Image Recognition&quot;]. arXiv:1409.1556 [cs].<br /> # Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E (2012). Pereira, F.; Burges, C. J. C.; Bottou, L.; Weinberger, K. Q., eds. [http://papers.nips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf Advances in Neural Information Processing Systems 25 (PDF)]. Curran Associates, Inc. pp. 1097–1105.<br /> # Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016.<br /> # Chen-Yu Lee, Saining Xie, Patrick Gallagher, Zhengyou Zhang, and Zhuowen Tu. Deeplysupervisednets. In Artificial Intelligence and Statistics, pp. 562–570, 2015.<br /> # Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. In European Conference on Computer Vision, pp. 525–542. Springer, 2016.<br /> # “Boluoweifenda/WAGE.” GitHub, github.com/boluoweifenda/WAGE.<br /> # Han, S., Mao, H., &amp; Dally, W. J. (2015). Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149.<br /> # Tang, Wei, Gang Hua, and Liang Wang. &quot;How to train a compact binary neural network with high accuracy?.&quot; AAAI. 2017.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=35907 Multi-scale Dense Networks for Resource Efficient Image Classification 2018-03-30T00:41:02Z <p>S3wen: /* Anytime Prediction */</p> <hr /> <div>= Introduction = <br /> <br /> Multi-Scale Dense Networks, MSDNets, are designed to address the growing demand for efficient object recognition. The issue with existing recognition networks is that they are either:<br /> efficient networks, but don't do well on hard examples, or large networks that do well on all examples but require a large amount of resources.<br /> <br /> In order to be efficient on all difficulties MSDNets propose a structure that can accurately output classifications for varying levels of computational requirements. The two cases that are used to evaluate the network are:<br /> Anytime Prediction: What is the best prediction the network can provide when suddenly prompted.<br /> Budget Batch Predictions: Given a maximum amount of computational resources how well does the network do on the batch.<br /> <br /> = Related Networks =<br /> <br /> == Computationally Efficient Networks ==<br /> <br /> Much of the existing work on convolution networks that are computationally efficient at test time focus on reducing model size after training. Many existing methods for refining an accurate network to be more efficient include weight pruning [3,4,5], quantization of weights [6,7] (during or after training), and knowledge distillation [8,9], which trains smaller student networks to reproduce the output of a much larger teacher network. The proposed work differs from these approaches as it trains a single model which trades computation efficiency for accuracy at test time without re-training or finetuning.<br /> <br /> == Resource Efficient Networks == <br /> <br /> Unlike the above, resource efficient concepts consider limited resources as a part of the structure/loss.<br /> Examples of work in this area include: <br /> * Efficient variants to existing state of the art networks<br /> * Gradient boosted decision trees, which incorporate computational limitations into the training<br /> * Fractal nets<br /> * Adaptive computation time method<br /> <br /> == Related architectures ==<br /> <br /> MSDNets pull on concepts from a number of existing networks:<br /> * Neural fabrics and others, are used to quickly establish a low resolution feature map, which is integral for classification.<br /> * Deeply supervised nets, introduced the incorporation of multiple classifiers throughout the network<br /> * The feature concatenation method from DenseNets allows the later classifiers to not be disrupted by the weight updates from earlier classifiers.<br /> <br /> = Problem Setup =<br /> The authors consider two settings that impose computational constraints at prediction time.<br /> <br /> == Anytime Prediction ==<br /> In the anytime prediction setting (Grubb &amp; Bagnell, 2012), there is a finite computational budget &lt;math&gt;B &gt; 0&lt;/math&gt; available for each test example &lt;math&gt;x&lt;/math&gt;. Once the budget is exhausted, the prediction for the class is output using early exit. The budget is nondeterministic and varies per test instance.<br /> <br /> == Budgeted Batch Classification ==<br /> In the budgeted batch classification setting, the model needs to classify a set of examples &lt;math&gt;D_test = {x_1, . . . , x_M}&lt;/math&gt; within a finite computational budget &lt;math&gt;B &gt; 0&lt;/math&gt; that is known in advance.<br /> <br /> = Multi-Scale Dense Networks =<br /> <br /> == Integral Contributions ==<br /> <br /> The way MSDNets aims to provide efficient classification with varying computational costs is to create one network that outputs results at depths. While this may seem trivial, as intermediate classifiers can be inserted into any existing network, two major problems arise.<br /> <br /> === Coarse Level Features Needed For Classification ===<br /> <br /> [[File:paper29 fig3.png | 700px|thumb|center]]<br /> <br /> The term coarse level feature refers to a set of filters in a CNN with low resolution. There are several ways to create such features. These methods are typically refereed to as down sampling. Some example of layers that perform this function are: max pooling, average pooling and convolution with strides. In this architecture, convolution with strides will be used to create coarse features. <br /> <br /> Coarse level features are needed to gain context of scene. In typical CNN based networks, the features propagate from fine to coarse. Classifiers added to the early, fine featured, layers do not output accurate predictions due to the lack of context.<br /> <br /> Figure 3 depicts relative accuracies of the intermediate classifiers and shows that the accuracy of a classifier is highly correlated with its position in the network. It is easy to see, specifically with the case of ResNet, that the classifiers improve in a staircase pattern. All of the experiments were performed on Cifar-100 dataset and it can be seen that the intermediate classifiers perform worst than the final classifiers, thus highlighting the problem with the lack of coarse level features early on.<br /> <br /> To address this issue, MSDNets proposes an architecture in which uses multi scaled feature maps. The feature maps at a particular layer and scale are computed by concatenating results from up to two convolutions: a standard convolution is first applied to same-scale features from the previous layer to pass on high-resolution information that subsequent layers can use to construct better coarse features, and if possible, a strided convolution is also applied on the finer-scale feature map from the previous layer to produce coarser features amenable to classification. The network is quickly formed to contain a set number of scales ranging from fine to coarse. These scales are propagated throughout, so that for the length of the network there are always coarse level features for classification and fine features for learning more difficult representations.<br /> <br /> === Training of Early Classifiers Interferes with Later Classifiers ===<br /> <br /> When training a network containing intermediate classifiers, the training of early classifiers will cause the early layers to focus on features for that classifier. These learned features may not be as useful to the later classifiers and degrade their accuracy.<br /> <br /> MSDNets use dense connectivity to avoid this issue. By concatenating all prior layers to learn future layers, the gradient propagation is spread throughout the available features. This allows later layers to not be reliant on any single prior, providing opportunities to learn new features that priors have ignored.<br /> <br /> == Architecture ==<br /> <br /> [[File:MSDNet_arch.png | 700px|thumb|center|Left: the MSDNet architecture. Right: example calculations for each output given 3 scales and 4 layers.]]<br /> <br /> The architecture of MSDNet is a structure of convolutions with a set number of layers and a set number of scales. Layers allow the network to build on the previous information to generate more accurate predictions, while the scales allow the network to maintain coarse level features throughout.<br /> <br /> The first layer is a special, mini-CNN-network, that quickly fills all required scales with features. The following layers are generated through the convolutions of the previous layers and scales.<br /> <br /> Each output at a given s scale is given by the convolution of all prior outputs of the same scale, and the strided-convolution of all prior outputs from the previous scale. <br /> <br /> The classifiers are run on the concatenation of all of the coarsest outputs from the preceding layers.<br /> <br /> === Loss Function ===<br /> <br /> The loss is calculated as a weighted sum of each classifier's logistic loss: <br /> <br /> &lt;math&gt;\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) &lt;/math&gt;<br /> <br /> Here &lt;math&gt;w_i&lt;/math&gt; represents the weights and &lt;math&gt;L(f_k)&lt;/math&gt; represents the logistic loss of each classifier. The weighted loss is taken as an average over a set of training samples. The weights can be determined from a budget of computational power, but results also show that setting all to 1 is also acceptable.<br /> <br /> === Computational Limit Inclusion ===<br /> <br /> When running in a budgeted batch scenario, the network attempts to provide the best overall accuracy. To do this with a set limit on computational resources, it works to use less of the budget on easy detections in order to allow more time to be spent on hard ones. <br /> In order to facilitate this, the classifiers are designed to exit when the confidence of the classification exceeds a preset threshold. To determine the threshold for each classifier, &lt;math&gt;|D_{test}|\sum_{k}(q_k C_k) \leq B &lt;/math&gt; must be true. Where &lt;math&gt;|D_{test}|&lt;/math&gt; is the total number of test samples, &lt;math&gt;C_k&lt;/math&gt; is the computational requirement to get an output from the &lt;math&gt;k&lt;/math&gt;th classifier, and &lt;math&gt;q_k &lt;/math&gt; is the probability that a sample exits at the &lt;math&gt;k&lt;/math&gt;th classifier. Assuming that all classifiers have the same base probability, &lt;math&gt;q&lt;/math&gt;, then &lt;math&gt;q_k&lt;/math&gt; can be used to find the threshold.<br /> <br /> === Network Reduction and Lazy Evaluation ===<br /> There are two ways to reduce the computational needs of MSDNets:<br /> <br /> # Reduce the size of the network by splitting it into &lt;math&gt;S&lt;/math&gt; blocks along the depth dimension and keeping the &lt;math&gt;(S-i+1)&lt;/math&gt; scales in the &lt;math&gt;i^{\text{th}}&lt;/math&gt; block.<br /> # Remove unnecessary computations: Group the computation in &quot;diagonal blocks&quot;; this propagates the example along paths that are required for the evaluation of the next classifier.<br /> <br /> = Experiments = <br /> <br /> When evaluating on CIFAR-10 and CIFAR-100 ensembles and multi-classifier versions of ResNets and DenseNets, as well as FractalNet are used to compare with MSDNet. <br /> <br /> When evaluating on ImageNet ensembles and individual versions of ResNets and DenseNets are compared with MSDNets.<br /> <br /> == Anytime Prediction ==<br /> <br /> In anytime prediction MSDNets are shown to have highly accurate with very little budget, and continue to remain above the alternate methods as the budget increases. The authors attributed this to the fact that MSDNets are able to produce low-resolution feature maps well-suited for classification after just a few layers, in contrast to the high-resolution feature maps in early layers of ResNets or DenseNets. Ensemble networks need to repeat computations of similar low-level features repeatedly when new models need to be evaluated, so their accuracy results do not increase as fast when computational budget increases. <br /> <br /> [[File:MSDNet_anytime.png | 700px|thumb|center|Accuracy of the anytime classification models.]] [[File:cifar10msdnet.png | 700px|thumb|center|CIFAR-10 results.]]<br /> <br /> == Budget Batch ==<br /> <br /> For budget batch 3 MSDNets are designed with classifiers set-up for varying ranges of budget constraints. On both dataset options the MSDNets exceed all alternate methods with a fraction of the budget required.<br /> <br /> [[File:MSDNet_budgetbatch.png | 700px|thumb|center|Accuracy of the budget batch classification models.]]<br /> <br /> The following figure shows examples of what was deemed &quot;easy&quot; and &quot;hard&quot; examples by the network. The top row contains images of either red wine or volcanos that were easily classified, thus exiting the network early and reducing required computations. The bottom row contains examples of &quot;hard&quot; images that were incorrectly classified by the first classifier but were correctly classified by the last layer.<br /> <br /> [[File:MSDNet_visualizingearlyclassifying.png | 700px|thumb|center|Examples of &quot;hard&quot;/&quot;easy&quot; classification]]<br /> <br /> = Ablation study =<br /> Additional experiments were performed to shed light on multi-scale feature maps, dense connectivity, and intermediate classifiers. This experiment started with an MSDNet with six intermediate classifiers and each of these components were removed, one at a time. To make our comparisons fair, the computational costs of the full networks were kept similar by adapting the network width. After removing all the three components, a VGG-like convolutional network is obtained. The classification accuracy of all classifiers is shown in the image below.<br /> <br /> [[File:Screenshot_from_2018-03-29_14-58-03.png]]<br /> <br /> = Critique = <br /> <br /> The problem formulation and scenario evaluation were very well formulated, and according to independent reviews, the results were reproducible. Where the paper could improve is on explaining how to implement the threshold; it isn't very well explained how the use of the validation set can be used to set the threshold value.<br /> <br /> = Implementation =<br /> The following repository provides the source code for the paper, written by the authors: https://github.com/gaohuang/MSDNet<br /> <br /> = Sources =<br /> # Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., &amp; Weinberger, K. Q. (n.d.). Multi-Scale Dense Networks for Resource Efficient Image Classification. ICLR 2018. doi:1703.09844 <br /> # Huang, G. (n.d.). Gaohuang/MSDNet. Retrieved March 25, 2018, from https://github.com/gaohuang/MSDNet<br /> # LeCun, Yann, John S. Denker, and Sara A. Solla. &quot;Optimal brain damage.&quot; Advances in neural information processing systems. 1990.<br /> # Hassibi, Babak, David G. Stork, and Gregory J. Wolff. &quot;Optimal brain surgeon and general network pruning.&quot; Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br /> # Li, Hao, et al. &quot;Pruning filters for efficient convnets.&quot; arXiv preprint arXiv:1608.08710 (2016).<br /> # Hubara, Itay, et al. &quot;Binarized neural networks.&quot; Advances in neural information processing systems. 2016.<br /> # Rastegari, Mohammad, et al. &quot;Xnor-net: Imagenet classification using binary convolutional neural networks.&quot; European Conference on Computer Vision. Springer, Cham, 2016.<br /> # Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ACM SIGKDD, pp. 535–541. ACM, 2006.<br /> # Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2014.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:cifar10msdnet.png&diff=35906 File:cifar10msdnet.png 2018-03-30T00:39:56Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=35905 Multi-scale Dense Networks for Resource Efficient Image Classification 2018-03-30T00:36:46Z <p>S3wen: /* Loss Function */</p> <hr /> <div>= Introduction = <br /> <br /> Multi-Scale Dense Networks, MSDNets, are designed to address the growing demand for efficient object recognition. The issue with existing recognition networks is that they are either:<br /> efficient networks, but don't do well on hard examples, or large networks that do well on all examples but require a large amount of resources.<br /> <br /> In order to be efficient on all difficulties MSDNets propose a structure that can accurately output classifications for varying levels of computational requirements. The two cases that are used to evaluate the network are:<br /> Anytime Prediction: What is the best prediction the network can provide when suddenly prompted.<br /> Budget Batch Predictions: Given a maximum amount of computational resources how well does the network do on the batch.<br /> <br /> = Related Networks =<br /> <br /> == Computationally Efficient Networks ==<br /> <br /> Much of the existing work on convolution networks that are computationally efficient at test time focus on reducing model size after training. Many existing methods for refining an accurate network to be more efficient include weight pruning [3,4,5], quantization of weights [6,7] (during or after training), and knowledge distillation [8,9], which trains smaller student networks to reproduce the output of a much larger teacher network. The proposed work differs from these approaches as it trains a single model which trades computation efficiency for accuracy at test time without re-training or finetuning.<br /> <br /> == Resource Efficient Networks == <br /> <br /> Unlike the above, resource efficient concepts consider limited resources as a part of the structure/loss.<br /> Examples of work in this area include: <br /> * Efficient variants to existing state of the art networks<br /> * Gradient boosted decision trees, which incorporate computational limitations into the training<br /> * Fractal nets<br /> * Adaptive computation time method<br /> <br /> == Related architectures ==<br /> <br /> MSDNets pull on concepts from a number of existing networks:<br /> * Neural fabrics and others, are used to quickly establish a low resolution feature map, which is integral for classification.<br /> * Deeply supervised nets, introduced the incorporation of multiple classifiers throughout the network<br /> * The feature concatenation method from DenseNets allows the later classifiers to not be disrupted by the weight updates from earlier classifiers.<br /> <br /> = Problem Setup =<br /> The authors consider two settings that impose computational constraints at prediction time.<br /> <br /> == Anytime Prediction ==<br /> In the anytime prediction setting (Grubb &amp; Bagnell, 2012), there is a finite computational budget &lt;math&gt;B &gt; 0&lt;/math&gt; available for each test example &lt;math&gt;x&lt;/math&gt;. Once the budget is exhausted, the prediction for the class is output using early exit. The budget is nondeterministic and varies per test instance.<br /> <br /> == Budgeted Batch Classification ==<br /> In the budgeted batch classification setting, the model needs to classify a set of examples &lt;math&gt;D_test = {x_1, . . . , x_M}&lt;/math&gt; within a finite computational budget &lt;math&gt;B &gt; 0&lt;/math&gt; that is known in advance.<br /> <br /> = Multi-Scale Dense Networks =<br /> <br /> == Integral Contributions ==<br /> <br /> The way MSDNets aims to provide efficient classification with varying computational costs is to create one network that outputs results at depths. While this may seem trivial, as intermediate classifiers can be inserted into any existing network, two major problems arise.<br /> <br /> === Coarse Level Features Needed For Classification ===<br /> <br /> [[File:paper29 fig3.png | 700px|thumb|center]]<br /> <br /> The term coarse level feature refers to a set of filters in a CNN with low resolution. There are several ways to create such features. These methods are typically refereed to as down sampling. Some example of layers that perform this function are: max pooling, average pooling and convolution with strides. In this architecture, convolution with strides will be used to create coarse features. <br /> <br /> Coarse level features are needed to gain context of scene. In typical CNN based networks, the features propagate from fine to coarse. Classifiers added to the early, fine featured, layers do not output accurate predictions due to the lack of context.<br /> <br /> Figure 3 depicts relative accuracies of the intermediate classifiers and shows that the accuracy of a classifier is highly correlated with its position in the network. It is easy to see, specifically with the case of ResNet, that the classifiers improve in a staircase pattern. All of the experiments were performed on Cifar-100 dataset and it can be seen that the intermediate classifiers perform worst than the final classifiers, thus highlighting the problem with the lack of coarse level features early on.<br /> <br /> To address this issue, MSDNets proposes an architecture in which uses multi scaled feature maps. The feature maps at a particular layer and scale are computed by concatenating results from up to two convolutions: a standard convolution is first applied to same-scale features from the previous layer to pass on high-resolution information that subsequent layers can use to construct better coarse features, and if possible, a strided convolution is also applied on the finer-scale feature map from the previous layer to produce coarser features amenable to classification. The network is quickly formed to contain a set number of scales ranging from fine to coarse. These scales are propagated throughout, so that for the length of the network there are always coarse level features for classification and fine features for learning more difficult representations.<br /> <br /> === Training of Early Classifiers Interferes with Later Classifiers ===<br /> <br /> When training a network containing intermediate classifiers, the training of early classifiers will cause the early layers to focus on features for that classifier. These learned features may not be as useful to the later classifiers and degrade their accuracy.<br /> <br /> MSDNets use dense connectivity to avoid this issue. By concatenating all prior layers to learn future layers, the gradient propagation is spread throughout the available features. This allows later layers to not be reliant on any single prior, providing opportunities to learn new features that priors have ignored.<br /> <br /> == Architecture ==<br /> <br /> [[File:MSDNet_arch.png | 700px|thumb|center|Left: the MSDNet architecture. Right: example calculations for each output given 3 scales and 4 layers.]]<br /> <br /> The architecture of MSDNet is a structure of convolutions with a set number of layers and a set number of scales. Layers allow the network to build on the previous information to generate more accurate predictions, while the scales allow the network to maintain coarse level features throughout.<br /> <br /> The first layer is a special, mini-CNN-network, that quickly fills all required scales with features. The following layers are generated through the convolutions of the previous layers and scales.<br /> <br /> Each output at a given s scale is given by the convolution of all prior outputs of the same scale, and the strided-convolution of all prior outputs from the previous scale. <br /> <br /> The classifiers are run on the concatenation of all of the coarsest outputs from the preceding layers.<br /> <br /> === Loss Function ===<br /> <br /> The loss is calculated as a weighted sum of each classifier's logistic loss: <br /> <br /> &lt;math&gt;\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) &lt;/math&gt;<br /> <br /> Here &lt;math&gt;w_i&lt;/math&gt; represents the weights and &lt;math&gt;L(f_k)&lt;/math&gt; represents the logistic loss of each classifier. The weighted loss is taken as an average over a set of training samples. The weights can be determined from a budget of computational power, but results also show that setting all to 1 is also acceptable.<br /> <br /> === Computational Limit Inclusion ===<br /> <br /> When running in a budgeted batch scenario, the network attempts to provide the best overall accuracy. To do this with a set limit on computational resources, it works to use less of the budget on easy detections in order to allow more time to be spent on hard ones. <br /> In order to facilitate this, the classifiers are designed to exit when the confidence of the classification exceeds a preset threshold. To determine the threshold for each classifier, &lt;math&gt;|D_{test}|\sum_{k}(q_k C_k) \leq B &lt;/math&gt; must be true. Where &lt;math&gt;|D_{test}|&lt;/math&gt; is the total number of test samples, &lt;math&gt;C_k&lt;/math&gt; is the computational requirement to get an output from the &lt;math&gt;k&lt;/math&gt;th classifier, and &lt;math&gt;q_k &lt;/math&gt; is the probability that a sample exits at the &lt;math&gt;k&lt;/math&gt;th classifier. Assuming that all classifiers have the same base probability, &lt;math&gt;q&lt;/math&gt;, then &lt;math&gt;q_k&lt;/math&gt; can be used to find the threshold.<br /> <br /> === Network Reduction and Lazy Evaluation ===<br /> There are two ways to reduce the computational needs of MSDNets:<br /> <br /> # Reduce the size of the network by splitting it into &lt;math&gt;S&lt;/math&gt; blocks along the depth dimension and keeping the &lt;math&gt;(S-i+1)&lt;/math&gt; scales in the &lt;math&gt;i^{\text{th}}&lt;/math&gt; block.<br /> # Remove unnecessary computations: Group the computation in &quot;diagonal blocks&quot;; this propagates the example along paths that are required for the evaluation of the next classifier.<br /> <br /> = Experiments = <br /> <br /> When evaluating on CIFAR-10 and CIFAR-100 ensembles and multi-classifier versions of ResNets and DenseNets, as well as FractalNet are used to compare with MSDNet. <br /> <br /> When evaluating on ImageNet ensembles and individual versions of ResNets and DenseNets are compared with MSDNets.<br /> <br /> == Anytime Prediction ==<br /> <br /> In anytime prediction MSDNets are shown to have highly accurate with very little budget, and continue to remain above the alternate methods as the budget increases. The authors attributed this to the fact that MSDNets are able to produce low-resolution feature maps well-suited for classification after just a few layers, in contrast to the high-resolution feature maps in early layers of ResNets or DenseNets. Ensemble networks need to repeat computations of similar low-level features repeatedly when new models need to be evaluated, so their accuracy results do not increase as fast when computational budget increases. <br /> <br /> [[File:MSDNet_anytime.png | 700px|thumb|center|Accuracy of the anytime classification models.]]<br /> <br /> == Budget Batch ==<br /> <br /> For budget batch 3 MSDNets are designed with classifiers set-up for varying ranges of budget constraints. On both dataset options the MSDNets exceed all alternate methods with a fraction of the budget required.<br /> <br /> [[File:MSDNet_budgetbatch.png | 700px|thumb|center|Accuracy of the budget batch classification models.]]<br /> <br /> The following figure shows examples of what was deemed &quot;easy&quot; and &quot;hard&quot; examples by the network. The top row contains images of either red wine or volcanos that were easily classified, thus exiting the network early and reducing required computations. The bottom row contains examples of &quot;hard&quot; images that were incorrectly classified by the first classifier but were correctly classified by the last layer.<br /> <br /> [[File:MSDNet_visualizingearlyclassifying.png | 700px|thumb|center|Examples of &quot;hard&quot;/&quot;easy&quot; classification]]<br /> <br /> = Ablation study =<br /> Additional experiments were performed to shed light on multi-scale feature maps, dense connectivity, and intermediate classifiers. This experiment started with an MSDNet with six intermediate classifiers and each of these components were removed, one at a time. To make our comparisons fair, the computational costs of the full networks were kept similar by adapting the network width. After removing all the three components, a VGG-like convolutional network is obtained. The classification accuracy of all classifiers is shown in the image below.<br /> <br /> [[File:Screenshot_from_2018-03-29_14-58-03.png]]<br /> <br /> = Critique = <br /> <br /> The problem formulation and scenario evaluation were very well formulated, and according to independent reviews, the results were reproducible. Where the paper could improve is on explaining how to implement the threshold; it isn't very well explained how the use of the validation set can be used to set the threshold value.<br /> <br /> = Implementation =<br /> The following repository provides the source code for the paper, written by the authors: https://github.com/gaohuang/MSDNet<br /> <br /> = Sources =<br /> # Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., &amp; Weinberger, K. Q. (n.d.). Multi-Scale Dense Networks for Resource Efficient Image Classification. ICLR 2018. doi:1703.09844 <br /> # Huang, G. (n.d.). Gaohuang/MSDNet. Retrieved March 25, 2018, from https://github.com/gaohuang/MSDNet<br /> # LeCun, Yann, John S. Denker, and Sara A. Solla. &quot;Optimal brain damage.&quot; Advances in neural information processing systems. 1990.<br /> # Hassibi, Babak, David G. Stork, and Gregory J. Wolff. &quot;Optimal brain surgeon and general network pruning.&quot; Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br /> # Li, Hao, et al. &quot;Pruning filters for efficient convnets.&quot; arXiv preprint arXiv:1608.08710 (2016).<br /> # Hubara, Itay, et al. &quot;Binarized neural networks.&quot; Advances in neural information processing systems. 2016.<br /> # Rastegari, Mohammad, et al. &quot;Xnor-net: Imagenet classification using binary convolutional neural networks.&quot; European Conference on Computer Vision. Springer, Cham, 2016.<br /> # Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ACM SIGKDD, pp. 535–541. ACM, 2006.<br /> # Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2014.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35904 MarrNet: 3D Shape Reconstruction via 2.5D Sketches 2018-03-29T23:48:13Z <p>S3wen: </p> <hr /> <div>= Introduction =<br /> Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br /> <br /> [[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br /> <br /> In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images and also enforce the re projection consistency between the 3D shape and the estimated sketch. 2.5D is the construction of 3D environment using the 2D retina projection along with depth perception obtained from the image. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth maps (contains information related to the distance of surfaces from a viewpoint) and surface normal maps (technique for adding the illusion of depth details to surfaces using an image's RGB information), the authors design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape. MarrNet uses an encoder-decoder structure for the sub-components of the framework. <br /> <br /> The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br /> <br /> Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br /> <br /> The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br /> <br /> = Related Work =<br /> <br /> == 2.5D Sketch Recovery ==<br /> Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br /> <br /> [[File:2-5d_example.PNG|700px|thumb|center|Results from the paper: Learning Non-Lambertian Object Intrinsics across ShapeNet Categories. The results show that neural networks can be trained to recover 2.5D information from an image. The top row predicts the albedo and the bottom row predicts the shading. It can be observed that the results are still blurry and the fine details are not fully recovered.]]<br /> <br /> == Single Image 3D Reconstruction ==<br /> The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. A voxel is an abbreviation for volume element, the three-dimensional version of a pixel. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br /> <br /> == 2D-3D Consistency ==<br /> Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, and has been widely used in 3D shape completion with the use of depths and silhouettes. A few recent papers [5,6,7,8] discussed enforcing differentiable 2D-3D constraints between shape and silhouettes to enable joint training of deep networks for the task of 3D reconstruction. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br /> <br /> = Approach =<br /> The 3D structure is recovered from a single RGB view using three steps, shown in the figure below. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br /> <br /> [[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br /> <br /> == 2.5D Sketch Estimation ==<br /> The first step takes a 2D RGB image and predicts the 2.5 sketch with surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information such as texture and lighting. An encoder-decoder architecture is used. The encoder is a A ResNet-18 network, which takes a 256 x 256 RGB image and produces 512 feature maps of size 8 x 8. The decoder is four sets of 5 x 5 fully convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br /> <br /> == 3D Shape Estimation ==<br /> The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem since it only takes in surface normal and depth images as input. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 fully convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br /> <br /> == Re-projection Consistency ==<br /> The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, &lt;math&gt;v_{x, y, z}&lt;/math&gt; represents the value at position &lt;math&gt;(x, y, z)&lt;/math&gt; in a 3D voxel grid, with &lt;math&gt;v_{x, y, z} \in [0, 1] ∀ x, y, z&lt;/math&gt;. &lt;math&gt;d_{x, y}&lt;/math&gt; denotes the estimated depth at position &lt;math&gt;(x, y)&lt;/math&gt;, &lt;math&gt;n_{x, y} = (n_a, n_b, n_c)&lt;/math&gt; denotes the estimated surface normal. Orthographic projection is used.<br /> <br /> [[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br /> <br /> === Depths ===<br /> The voxel with depth &lt;math&gt;v_{x, y}, d_{x, y}&lt;/math&gt; should be 1, while all voxels in front of it should be 0. This ensures the estimated 3D shape matches the estimated depth values. The projected depth loss and its gradient are defined as follows:<br /> <br /> &lt;math&gt;<br /> L_{depth}(x, y, z)=<br /> \left\{<br /> \begin{array}{ll}<br /> v^2_{x, y, z}, &amp; z &lt; d_{x, y} \\<br /> (1 - v_{x, y, z})^2, &amp; z = d_{x, y} \\<br /> 0, &amp; z &gt; d_{x, y} \\<br /> \end{array}<br /> \right.<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br /> \left\{<br /> \begin{array}{ll}<br /> 2v{x, y, z}, &amp; z &lt; d_{x, y} \\<br /> 2(v_{x, y, z} - 1), &amp; z = d_{x, y} \\<br /> 0, &amp; z &gt; d_{x, y} \\<br /> \end{array}<br /> \right.<br /> &lt;/math&gt;<br /> <br /> When &lt;math&gt;d_{x, y} = \infty&lt;/math&gt;, all voxels in front of it should be 0.<br /> <br /> === Surface Normals ===<br /> Since vectors &lt;math&gt;n_{x} = (0, −n_{c}, n_{b})&lt;/math&gt; and &lt;math&gt;n_{y} = (−n_{c}, 0, n_{a})&lt;/math&gt; are orthogonal to the normal vector &lt;math&gt;n_{x, y} = (n_{a}, n_{b}, n_{c})&lt;/math&gt;, they can be normalized to obtain &lt;math&gt;n’_{x} = (0, −1, n_{b}/n_{c})&lt;/math&gt; and &lt;math&gt;n’_{y} = (−1, 0, n_{a}/n_{c})&lt;/math&gt; on the estimated surface plane at &lt;math&gt;(x, y, z)&lt;/math&gt;. The projected surface normal tried to guarantee voxels at &lt;math&gt;(x, y, z) ± n’_{x}&lt;/math&gt; and &lt;math&gt;(x, y, z) ± n’_{y}&lt;/math&gt; should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br /> <br /> The projected surface normal loss is defined as follows, with &lt;math&gt;z = d_{x, y}&lt;/math&gt;:<br /> <br /> &lt;math&gt;<br /> L_{normal}(x, y, z) =<br /> (1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br /> (1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br /> &lt;/math&gt;<br /> <br /> Gradients along x are:<br /> <br /> &lt;math&gt;<br /> \frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br /> &lt;/math&gt;<br /> and<br /> &lt;math&gt;<br /> \frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br /> &lt;/math&gt;<br /> <br /> Gradients along y are similar to x.<br /> <br /> = Training =<br /> The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br /> <br /> For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br /> <br /> Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br /> <br /> Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br /> <br /> = Evaluation =<br /> Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets; ShapeNet, PASCAL 3D+, and IKEA. Intersection-over-Union (IoU) is the main measurement of comparison between the models. However the authors note that models which focus on the IoU metric fail to capture the details of the object they are trying to model, disregarding details to focus on the overall shape. To counter this drawback they poll people on which reconstruction is preferred. IoU is also computationally inefficient since it has to check over all possible scales.<br /> <br /> == ShapeNet ==<br /> Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br /> <br /> === Method ===<br /> MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br /> <br /> === Results ===<br /> The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. The estimated normal and depth images are able to extract intrinsic information about object shape while leaving behind non-essential information such as textures from the original images. Quantitatively, the full model also achieves 0.57 integer over union score (which compares the overlap of the predicted model and ground truth), which is higher than the direct prediction baseline.<br /> <br /> [[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br /> <br /> == PASCAL 3D+ ==<br /> Rough 3D models are provided from real-life images.<br /> <br /> === Method ===<br /> Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br /> <br /> === Results ===<br /> The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br /> <br /> [[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br /> <br /> Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. Since PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth. This shows how MarrNet produces nice shapes and also highlights the fact that ground truth shapes are not very good.<br /> <br /> [[File:human_studies.png|400px|thumb|center|Human preferences on chairs in PASCAL 3D+ (Xiang et al. 2014). The numbers show the percentage of how often humans prefered the 3D shape from DRC (state-of-the-art), MarrNet, or GT.]]<br /> <br /> <br /> [[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br /> <br /> Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br /> <br /> [[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br /> <br /> == IKEA ==<br /> This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br /> <br /> === Results ===<br /> Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br /> <br /> [[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br /> <br /> == Other Data ==<br /> MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br /> <br /> [[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br /> <br /> MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br /> <br /> [[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br /> <br /> = Commentary =<br /> Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br /> <br /> As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br /> <br /> As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br /> <br /> = Conclusion =<br /> The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br /> <br /> = Implementation =<br /> The following repository provides the source code for the paper. The repository provides the source code as written by the authors: https://github.com/jiajunwu/marrnet<br /> <br /> = References =<br /> # Jiajun Wu, Yifan Wang, Tianfan Xue, Xingyuan Sun, William T. Freeman, Joshua B. Tenenbaum. MarrNet: 3D Shape Reconstruction via 2.5D Sketches, 2017<br /> # David Marr. Vision: A computational investigation into the human representation and processing of visual information. W. H. Freeman and Company, 1982.<br /> # Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br /> # JiajunWu, Chengkai Zhang, Tianfan Xue,William T Freeman, and Joshua B Tenenbaum. Learning a Proba- bilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling. In NIPS, 2016b.<br /> # Wu, J. (n.d.). Jiajunwu/marrnet. Retrieved March 25, 2018, from https://github.com/jiajunwu/marrnet<br /> # Jiajun Wu, Tianfan Xue, Joseph J Lim, Yuandong Tian, Joshua B Tenenbaum, Antonio Torralba, and William T Freeman. Single image 3d interpreter network. In ECCV, 2016a.<br /> # Xinchen Yan, Jimei Yang, Ersin Yumer, Yijie Guo, and Honglak Lee. Perspective transformer nets: Learning single-view 3d object reconstruction without 3d supervision. In NIPS, 2016.<br /> # Danilo Jimenez Rezende, SM Ali Eslami, Shakir Mohamed, Peter Battaglia, Max Jaderberg, and Nicolas Heess. Unsupervised learning of 3d structure from images. In NIPS, 2016.<br /> # Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=34609 stat946w18/Spectral normalization for generative adversial network 2018-03-18T05:56:14Z <p>S3wen: </p> <hr /> <div>= Presented by =<br /> <br /> 1. liu, wenqing<br /> <br /> = Introduction =<br /> Generative adversarial networks (GANs) (Goodfellow et al., 2014) have been enjoying considerable success as a framework of generative models in recent years. The concept is to consecutively train the model distribution and the discriminator in turn, with the goal of reducing the difference between the model distribution and the target distribution measured by the best discriminator possible at each step of the training.<br /> <br /> A persisting challenge in the training of GANs is the performance control of the discriminator. When the support of the model distribution and the support of target distribution are disjoint, there exists a discriminator that can perfectly distinguish the model distribution from the target (Arjovsky &amp; Bottou, 2017). One such discriminator is produced in this situation, the training of the generator comes to complete stop, because the derivative of the so-produced discriminator with respect to the input turns out to be 0. This motivates us to introduce some form of restriction to the choice of discriminator.<br /> <br /> In this paper, we propose a novel weight normalization method called ''spectral normalization'' that can stabilize the training of discriminator networks. Our normalization enjoys following favorable properties:<br /> <br /> * The only hyper-parameter that needs to be tuned is the Lipschitz constant, and the algorithm is not too sensitive to this constant's value<br /> * The additional computational needed to implement spectral normalization is small <br /> <br /> In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br /> <br /> = Model =<br /> <br /> Let us consider a simple discriminator made of a neural network of the following form, with the input x:<br /> <br /> $f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots))))$<br /> <br /> where &lt;math&gt; \theta:=W^1,\cdots,W^L, W^{L+1} &lt;/math&gt; is the learning parameters set, &lt;math&gt;W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} &lt;/math&gt;, and &lt;math&gt;a_l &lt;/math&gt; is an element-wise non-linear activation function.The final output of the discriminator function is given by &lt;math&gt;D(x,\theta) = A(f(x,\theta)) &lt;/math&gt;. The standard formulation of GANs is given by &lt;math&gt;\min_{G}\max_{D}V(G,D)&lt;/math&gt; where min and max of G and D are taken over the set of generator and discriminator functions, respectively. <br /> <br /> The conventional form of &lt;math&gt;V(G,D) &lt;/math&gt; is given by:<br /> <br /> $E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x'))]$<br /> <br /> where &lt;math&gt;q_{data}&lt;/math&gt; is the data distribution and &lt;math&gt;p_G(x)&lt;/math&gt; is the model generator distribution to be learned through the adversarial min-max optimization. It is known that, for a fixed generator G, the optimal discriminator for this form of &lt;math&gt;V(G,D) &lt;/math&gt; is given by <br /> <br /> $D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))$<br /> <br /> We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <br /> <br /> $\arg\max_{||f||_{Lip}\le k}V(G,D)$<br /> <br /> where we mean by &lt;math&gt; ||f||_{lip}&lt;/math&gt; the smallest value M such that &lt;math&gt; ||f(x)-f(x')||/||x-x'||\le M &lt;/math&gt; for any x,x', with the norm being the &lt;math&gt; l_2 &lt;/math&gt; norm.<br /> <br /> Our spectral normalization controls the Lipschitz constant of the discriminator function &lt;math&gt; f &lt;/math&gt; by literally constraining the spectral norm of each layer &lt;math&gt; g: h_{in}\rightarrow h_{out}&lt;/math&gt;. By definition, Lipschitz norm &lt;math&gt; ||g||_{Lip} &lt;/math&gt; is equal to &lt;math&gt; \sup_h\sigma(\nabla g(h)) &lt;/math&gt;, where &lt;math&gt; \sigma(A) &lt;/math&gt; is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer &lt;math&gt; g(h)=Wh &lt;/math&gt;, the norm is given by &lt;math&gt; ||g||_{Lip}=\sigma(W) &lt;/math&gt;. Observing the following bound:<br /> <br /> $||f||_{Lip}\le ||(h_L\rightarrow W^{L+1}h_{L})||_{Lip}*||a_{L}||_{Lip}*||(h_{L-1}\rightarrow W^{L}h_{L-1})||_{Lip}\cdots ||a_1||_{Lip}*||(h_0\rightarrow W^1h_0)||_{Lip}=\prod_{l=1}^{L+1}\sigma(W^l) *\prod_{l=1}^{L} ||a_l||_{Lip}$<br /> <br /> Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint &lt;math&gt; \sigma(W)=1 &lt;/math&gt;:<br /> <br /> $\bar{W_{SN}}:= W/\sigma(W)$<br /> <br /> In summary, just like what weight normalization does, we reparameterize weight matrix &lt;math&gt; \bar{W_{SN}} &lt;/math&gt; as &lt;math&gt; W/\sigma(W) &lt;/math&gt; to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br /> <br /> $\frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W}$<br /> <br /> $\frac{\partial \bar{W_{SN}}}{\partial W_{ij}} = \frac{1}{\sigma(W)}E_{ij}-\frac{1}{\sigma(W)^2}*\frac{\partial \sigma(W)}{\partial(W_{ij})}W=\frac{1}{\sigma(W)}E_{ij}-\frac{[u_1v_1^T]_{ij}}{\sigma(W)^2}W=\frac{1}{\sigma(W)}(E_{ij}-[u_1v_1^T]_{ij}\bar{W_{SN}})$<br /> <br /> where &lt;math&gt; E_{ij} &lt;/math&gt; is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and &lt;math&gt; u_1, v_1&lt;/math&gt; are respectively the first left and right singular vectors of W.<br /> <br /> To understand the above computation in more detail, note that <br /> \begin{align}<br /> \sigma(W)= \sup_{||u||=1, ||v||=1} \langle Wv, u \rangle = \sup_{||u||=1, ||v||=1} \text{trace} ( (uv^T)^T W).<br /> \end{align}<br /> By Theorem 4.4.2 in Lemaréchal and Hiriart-Urruty (1996), the sub-differential of a convex function defined as the the maximum of a set of differentiable convex functions over a compact index set is the convex hull of the gradients of the maximizing functions. Thus we have the sub-differential:<br /> <br /> \begin{align}<br /> \partial \sigma = \text{convex hull} \{ u v^T: u,v \text{ are left/right singular vectors associated with } \sigma(W) \}.<br /> \end{align}<br /> <br /> However, the authors assume that the maximum singular value of W has only one left and one right normalized singular vector. Thus &lt;math&gt; \sigma &lt;/math&gt; is differentiable and <br /> \begin{align}<br /> \nabla_W \sigma(W) =u_1v_1^T,<br /> \end{align}<br /> which explains the above computation.<br /> <br /> = Spectral Normalization VS Other Regularization Techniques =<br /> <br /> The weight normalization introduced by Salimans &amp; Kingma (2016) is a method that normalizes the &lt;math&gt; l_2 &lt;/math&gt; norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization &lt;math&gt; \bar{W_{WN}} &lt;/math&gt;:<br /> <br /> &lt;math&gt; \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) &lt;/math&gt; where &lt;math&gt; \sigma_t(A) &lt;/math&gt; is a t-th singular value of matrix A. <br /> <br /> Note, if &lt;math&gt; \bar{W_{WN}} &lt;/math&gt; is the weight normalized matrix of dimension &lt;math&gt; d_i*d_0 &lt;/math&gt;, the norm &lt;math&gt; ||\bar{W_{WN}}h||_2 &lt;/math&gt; for a fixed unit vector &lt;math&gt; h &lt;/math&gt; is maximized at &lt;math&gt; ||\bar{W_{WN}}h||_2 \text{ when } \sigma_1(\bar{W_{WN}})=\sqrt{d_0} \text{ and } \sigma_t(\bar{W_{WN}})=0, t=2, \cdots, T &lt;/math&gt; which means that &lt;math&gt; \bar{W_{WN}} &lt;/math&gt; is of rank one. In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of &lt;math&gt; \bar{W_{WN}}h &lt;/math&gt; large. For weight normalization, however, this comes at the cost of reducing the rank and hence the number of features to be used for the discriminator. Thus, there is a conflict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution. The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators. Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features. <br /> <br /> Brock et al. (2016) introduced orthonormal regularization on each weight to stabilize the training of GANs. In their work, Brock et al. (2016) augmented the adversarial objective function by adding the following term:<br /> <br /> &lt;math&gt; ||W^TW-I||^2_F &lt;/math&gt;<br /> <br /> While this seems to serve the same purpose as spectral normalization, orthonormal regularization is mathematically quite different from our spectral normalization because the orthonormal regularization destroys the information about the spectrum by setting all the singular values to one. On the other hand, spectral normalization only scales the spectrum so that its maximum will be one. <br /> <br /> Gulrajani et al. (2017) used gradient penalty method in combination with WGAN. In their work, they placed K-Lipschitz constant on the discriminator by augmenting the objective function with the regularizer that rewards the function for having local 1-Lipschitz constant(i.e &lt;math&gt; ||\nabla_{\hat{x}} f ||_2 = 1 &lt;/math&gt;) at discrete sets of points of the form &lt;math&gt; \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x &lt;/math&gt; generated by interpolating a sample &lt;math&gt; \tilde{x} &lt;/math&gt; from generative distribution and a sample &lt;math&gt; x &lt;/math&gt; from the data distribution. This approach has an obvious weakness of being heavily dependent on the support of the current generative distribution. Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of &lt;math&gt; ||\nabla_{\hat{x}} f ||_2 &lt;/math&gt; requires one whole round of forward and backward propagation.<br /> <br /> = Experimental settings and results = <br /> == Objective function ==<br /> For all methods other than WGAN-GP, we use <br /> &lt;math&gt; V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]&lt;/math&gt;<br /> to update D, for the updates of G, use &lt;math&gt; -E_{z\sim p(z)}[\log(D(G(z)))] &lt;/math&gt;. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br /> &lt;math&gt; V_D(\hat{G},D)= E_{x\sim q_{data}(x)}[\min(0,-1+D(x))] + E_{z\sim p(z)}[\min(0,-1-D(\hat{G}(z)))] &lt;/math&gt;, &lt;math&gt; V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] &lt;/math&gt;<br /> <br /> For WGAN-GP, we choose <br /> &lt;math&gt; V(G,D):=E_{x\sim q_{data}}[D(x)]-E_{z\sim p(z)}[D(G(z))]- \lambda E_{\hat{x}\sim p(\hat{x})}[(||\nabla_{\hat{x}}D(\hat{x}||-1)^2)]&lt;/math&gt;<br /> <br /> == Optimization ==<br /> Adam optimizer: 6 settings in total, related to <br /> * &lt;math&gt; n_{dis} &lt;/math&gt;, the number of updates of the discriminator per one update of Adam. <br /> * learning rate &lt;math&gt; \alpha &lt;/math&gt;<br /> * the first and second momentum parameters &lt;math&gt; \beta_1, \beta_2 &lt;/math&gt; of Adam<br /> <br /> [[File:inception score.png]]<br /> <br /> [[File:FID score.png]]<br /> <br /> The above image show the inception core and FID score of with settings A-F, and table show the inception scores of the different methods with optimal settings on CIFAR-10 and STL-10 dataset.<br /> <br /> == Singular values analysis on the weights of the discriminator D ==<br /> [[File:singular value.png]]<br /> <br /> In above figure, we show the squared singular values of the weight matrices in the final discriminator D produced by each method using the parameter that yielded the best inception score. As we predicted before, the singular values of the first fifth layers trained with weight clipping and weight normalization concentrate on a few components. On the other hand, the singular values of the weight matrices in those layers trained with spectral normalization is more broadly distributed.<br /> <br /> == Training time ==<br /> On CIFAR-10, SN-GANs is slightly slower than weight normalization, but significantly faster than WGAN-GP. As we mentioned in section 3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm. For STL-10, the computational time of SN-GANs is almost the same as vanilla GANs.<br /> <br /> == Comparison between GN-GANs and orthonormal regularization ==<br /> [[File:comparison.png]]<br /> Above we explained in Section 3, orthonormal regularization is different from our method in that it destroys the spectral information and puts equal emphasis on all feature dimensions, including the ones that shall be weeded out in the training process. To see the extent of its possibly detrimental effect, we experimented by increasing the dimension of the feature space, especially at the final layer for which the training with our spectral normalization prefers relatively small feature space. Above figure shows the result of our experiments. As we predicted, the performance of the orthonormal regularization deteriorates as we increase the dimension of the feature maps at the final layer. SN-GANs, on the other hand, does not falter with this modification of the architecture.<br /> <br /> We also applied our method to the training of class conditional GANs on ILSVRC2012 dataset with 1000 classes, each consisting of approximately 1300 images, which we compressed to 128*128 pixels. GAN without normalization and GAN with layer normalization collapsed in the beginning of training and failed to produce any meaningful images. Above picture shows that the inception score of the orthonormal normalization plateaued around 20k iterations, while SN kept improving even afterward.<br /> <br /> [[File:sngan.jpg]]<br /> <br /> Samples generated by various networks trained on CIFAR10. Momentum and training rates are increasing to the right. We can see that for high learning rates and momentum the Wasserstein-GAN does not generate good images, while weight and spectral normalization generate good samples.<br /> <br /> = Algorithm of spectral normalization =<br /> To calculate the largest singular value of matrix &lt;math&gt; W &lt;/math&gt; to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br /> <br /> * Initialize &lt;math&gt;\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L &lt;/math&gt; with a random vector (sampled from isotropic distribution) <br /> * For each update and each layer l:<br /> * Apply power iteration method to a unnormalized weight &lt;math&gt; W^l &lt;/math&gt;:<br /> <br /> &lt;math&gt; \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 &lt;/math&gt;<br /> <br /> &lt;math&gt;\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| &lt;/math&gt;<br /> <br /> * Calculate &lt;math&gt; \bar{W_{SN}} &lt;/math&gt; with the spectral norm :<br /> <br /> &lt;math&gt; \bar{W_{SN}}(W^l)=W^l/\sigma(W^l)&lt;/math&gt;, where &lt;math&gt;\sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} &lt;/math&gt;<br /> <br /> * Update &lt;math&gt; W^l &lt;/math&gt; with SGD on mini-batch dataset &lt;math&gt; D_M &lt;/math&gt; with a learning rate &lt;math&gt; \alpha &lt;/math&gt;<br /> <br /> &lt;math&gt; W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) &lt;/math&gt;<br /> <br /> == Conclusions ==<br /> This paper proposes spectral normalization as a stabilizer of training of GANs. When we apply spectral normalization to the GANs on image generation tasks, the generated examples are more diverse than the conventional weight normalization and achieve better or comparative inception scores relative to previous studies. The method imposes global regularization on the discriminator as opposed to local regularization introduced by WGAN-GP, and can possibly used in combinations. In the future work, we would like to further investigate where our methods stand amongest other methods on more theoretical basis, and experiment our algorithm on larger and more complex datasets.<br /> <br /> == Critique(to be edited) ==<br /> <br /> == References ==<br /> # Lemaréchal, Claude, and J. B. Hiriart-Urruty. &quot;Convex analysis and minimization algorithms I.&quot; Grundlehren der mathematischen Wissenschaften 305 (1996).</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:sngan.jpg&diff=34608 File:sngan.jpg 2018-03-18T05:48:07Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=On_The_Convergence_Of_ADAM_And_Beyond&diff=34240 On The Convergence Of ADAM And Beyond 2018-03-15T07:16:23Z <p>S3wen: </p> <hr /> <div>= Introduction =<br /> Somewhat different to the presentation I gave in class, this paper focuses strictly on the pitfalls in convergance of the ADAM training algorithm for neural networks from a theoretical standpoint and proposes a novel improvement to ADAM called AMSGrad. The paper introduces the idea that it is possible for ADAM to get &quot;stuck&quot; in it's weighted average history, preventing it from converging to an optimal solution. An example is that in an experiment there may be a large spike in the gradient during some minibatches, but since ADAM weighs the current update by the exponential moving averages of squared past<br /> gradients, the effect of the large spike in gradient is lost. This can be prevented through novel adjustments to the ADAM optimization algorithm, which can improve convergence.<br /> <br /> == Notation ==<br /> The paper presents the following framework that generalizes training algorithms to allow us to define a specific variant such as AMSGrad or SGD entirely within it:<br /> <br /> <br /> [[File:training_algo_framework.png]]<br /> <br /> Where we have &lt;math&gt; x_t &lt;/math&gt; as our network parameters defined within a vector space &lt;math&gt; \mathcal{F} &lt;/math&gt;. &lt;math&gt; \prod_{\mathcal{F}} (y) = &lt;/math&gt; the projection of &lt;math&gt; y &lt;/math&gt; on to the set &lt;math&gt; \mathcal{F} &lt;/math&gt;.<br /> &lt;math&gt; \psi_t &lt;/math&gt; and &lt;math&gt; \phi_t &lt;/math&gt; correspond to arbitrary functions we will provide later, The former maps from the history of gradients to &lt;math&gt; \mathbb{R}^d &lt;/math&gt; and the latter maps from the history of the gradients to positive semi definite matrices. And finally &lt;math&gt; f_t &lt;/math&gt; is our loss function at some time &lt;math&gt; t &lt;/math&gt;, the rest should be pretty self explanatory. Using this framework and defining different &lt;math&gt; \psi_t &lt;/math&gt; , &lt;math&gt; \phi_t &lt;/math&gt; will allow us to recover all different kinds of training algorithms under this one roof.<br /> <br /> <br /> === SGD As An Example ===<br /> To recover SGD using this framework we simply select &lt;math&gt; \phi_t (g_1, \dotsc, g_t) = g_t&lt;/math&gt;, &lt;math&gt; \psi_t (g_1, \dotsc, g_t) = I &lt;/math&gt; and &lt;math&gt;\alpha_t = \alpha / \sqrt{t}&lt;/math&gt;. It's easy to see that no transformations are ultimately applied to any of the parameters based on any gradient history other than the most recent from &lt;math&gt; \phi_t &lt;/math&gt; and that &lt;math&gt; \psi_t &lt;/math&gt; in no way transforms any of the parameters by any specific amount as &lt;math&gt; V_t = I &lt;/math&gt; has no impact later on.<br /> <br /> <br /> === ADAM As Another Example ===<br /> Once you can convince yourself that SGD is correct, you should understand the framework enough to see why the following setup for ADAM will allow us to recover the behavior we want. ADAM has the ability to define a &quot;learning rate&quot; for every parameter based on how much that parameter moves over time (a.k.a it's momentum) supposedly to help with the learning process.<br /> <br /> In order to do this we will choose &lt;math&gt; \phi_t (g_1, \dotsc, g_t) = (1 - \beta_1) \sum_{i=0}^{t} {\beta_1}^{t - i} g_t &lt;/math&gt;, psi to be &lt;math&gt; \psi_t (g_1, \dotsc, g_t) = (1 - \beta_2)&lt;/math&gt;diag&lt;math&gt;( \sum_{i=0}^{t} {\beta_2}^{t - i} {g_t}^2) &lt;/math&gt;, and keep &lt;math&gt;\alpha_t = \alpha / \sqrt{t}&lt;/math&gt;. This set up is equivalent to choosing a learning rate decay of &lt;math&gt;\alpha / \sqrt{\sum_i g_{i,j}}&lt;/math&gt; for &lt;math&gt;j \in [d]&lt;/math&gt;.<br /> <br /> From this we can now see that &lt;math&gt;m_t &lt;/math&gt; gets filled up with the exponentially weighted average of the history of our gradients that we've come across so far in the algorithm. And that as we proceed to update we scale each one of our paramaters by dividing out &lt;math&gt; V_t &lt;/math&gt; (in the case of diagonal it's just 1/the diagonal entry) which contains the exponentially weighted average of each parameters momentum (&lt;math&gt; {g_t}^2 &lt;/math&gt;) across our training so far in the algorithm. Thus giving each parameter it's own unique scaling by it's second moment or momentum. Intuitively from a physical perspective if each parameter is a ball rolling around in the optimization landscape what we are now doing is instead of having the ball changed positions on the landscape at a fixed velocity (i.e. momentum of 0) the ball now has the ability to accelerate and speed up or slow down if it's on a steep hill or flat trough in the landscape (i.e. a momentum that can change with time).<br /> <br /> = &lt;math&gt; \Gamma_t &lt;/math&gt; An Interesting Quantity =<br /> Now that we have an idea of what ADAM looks like in this framework, let us now investigate the following:<br /> <br /> &lt;math&gt; \Gamma_{t + 1} = \frac{\sqrt{V_{t+1}}}{\alpha_{t+1}} - \frac{\sqrt{V_t}}{\alpha_t} &lt;/math&gt;<br /> <br /> <br /> Which essentially measure the change of the &quot;Inverse of the learning rate&quot; across time (since we are using alpha's as step sizes). Looking back to our example of SGD it's not hard to see that this quantity is strictly positive, which leads to &quot;non-increasing&quot; learning rates a desired property. However that is not the case with ADAM, and can pose a problem in a theoretical and applied setting. The problem ADAM can face is that &lt;math&gt; \Gamma_t &lt;/math&gt; can be indefinite, which the original proof assumed it could not be. The math for this proof is VERY long so instead we will opt for an example to showcase why this could be an issue.<br /> <br /> Consider the loss function &lt;math&gt; f_t(x) = \begin{cases} <br /> Cx &amp; \text{for } t \text{ mod 3} = 1 \\<br /> -x &amp; \text{otherwise}<br /> \end{cases} &lt;/math&gt;<br /> <br /> Where we have &lt;math&gt; C &gt; 2 &lt;/math&gt; and &lt;math&gt; \mathcal{F} &lt;/math&gt; is &lt;math&gt; [-1,1] &lt;/math&gt;<br /> Additionally we choose &lt;math&gt; \beta_1 = 0 &lt;/math&gt; and &lt;math&gt; \beta_2 = 1/(1+C^2) &lt;/math&gt;. We then proceed to plug this into our framework from before. <br /> This function is periodic and it's easy to see that it has the gradient of C once and then a gradient of -1 twice every period. It has an optimal solution of &lt;math&gt; x = -1 &lt;/math&gt; (from a regret standpoint), but using ADAM we would eventually converge at &lt;math&gt; x = 1 &lt;/math&gt; since &lt;math&gt; C &lt;/math&gt; would &quot;overpower&quot; the -1's.<br /> <br /> = AMSGrad as an improvement to ADAM =<br /> <br /> There is a very simple intuitive fix to ADAM to handle this problem. We simply scale our historical weighted average by the maximum we have seen so far to avoid the negative sign problem. There is a very simple one liner adaptation of ADAM to get to AMSGRAD:<br /> [[File:AMSGrad_algo.png]]<br /> <br /> Below are some simple plots comparing ADAM and AMSGrad, the first are from the paper and the second are from another individual who attempted to recreate the experiments. The two plots somewhat disagree with one another so take this heuristic improvement with a grain of salt.<br /> [[File:AMSGrad_vs_adam.png]]<br /> <br /> Here is another example of a one-dimensional convex optimization problem where ADAM fails to converge<br /> <br /> [[File:AMSGrad_vs_adam3.png | 1000px]]<br /> <br /> [[File:AMSGrad_vs_adam2.png]]<br /> <br /> = Conclusion =<br /> We have introduced a framework for which we could view several different training algorithms. From there we used it to recover SGD as well as ADAM. In our recovery of ADAM we investigated the change of the inverse of the learning rate over time to discover in certain cases there were convergence issues. We proposed a new heuristic AMSGrad to help deal with this problem and presented some empirical results that show it may have helped ADAM slightly. Thanks for your time.<br /> <br /> = Source =<br /> 1. Sashank J. Reddi and Satyen Kale and Sanjiv Kumar. &quot;On the Convergence of Adam and Beyond.&quot; International Conference on Learning Representations. 2018</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_Image_Motion_with_Group_Representations&diff=34211 Understanding Image Motion with Group Representations 2018-03-15T06:45:52Z <p>S3wen: </p> <hr /> <div>== Introduction ==<br /> Motion perception is a key component of computer vision. It is critical to problems such as optical flow and visual odometry, where a sequence of images are used to calculate either the pixel level (local) motion or the motion of the entire scene (global). The smooth image transformation caused by camera motion is a subspace of all position image transformations. Here, we are interested in realistic transformation caused by motion, therefore unrealistic motion caused by say, face swapping, are not considered. <br /> <br /> Supervised learning of 3D motion is challenging since explicit motion labels are no trivial to obtain. The proposed learning method does not need label data. Instead, the method constrains learning by using the properties of motion space. The paper presents a general model of visual motion, and how the motion space properties of associativity and invertibility can be used to constrain the learning of a deep neural network. The results show evidence that the learned model captions motion in both 2D and 3D settings.<br /> <br /> [[File:paper13_fig1.png|650px|center|]]<br /> <br /> == Related Work ==<br /> The most common global representations of motion are from structure from motion (SfM) and simultaneous localization and mapping (SLAM), which represent poses in special Euclidean group &lt;math&gt; SE(3) &lt;/math&gt; to represent a sequence of motions. However, these cannot be used to represent non-rigid or independent motions. Another approach to representing motion is spatiotemporal features (STFs), which are flexible enough to represent non-rigid motions. However these approaches are restricted to fixed windows of representation. <br /> <br /> There are also works using CNN’s to learn optical flow using brightness constancy assumptions, and/or photometric local constraints. Works on stereo depth estimation using learning has also shown results. Regarding image sequences, there are works on shuffling the order of images to learn representations of its contents, as well as learning representations equivariant to the egomotion of the camera.<br /> <br /> == Approach ==<br /> The proposed method is based on the observation that 3D motions, equipped with composition, form a group. By learning the underlying mapping that captures the motion transformations, we are approximating latent motion of the scene. The method is designed to capture group associativity and invertibility.<br /> <br /> Consider a latent structure space &lt;math&gt;S&lt;/math&gt;, element of the structure space generates images via projection &lt;math&gt;\pi:S\rightarrow I&lt;/math&gt;, latent motion space &lt;math&gt;M&lt;/math&gt; which is some closed subgroup of the set of homeomorphism on &lt;math&gt;S&lt;/math&gt;. For &lt;math&gt;s \in S&lt;/math&gt;, a continuous motion sequence &lt;math&gt; \{m_t \in M | t \geq 0\} &lt;/math&gt; generates continous image sequence &lt;math&gt; \{i_t \in I | t \geq 0\} &lt;/math&gt; where &lt;math&gt; i_t=\pi(m_t(s)) &lt;/math&gt;. Writing this as a hidden Markov model gives &lt;math&gt; i_t=\pi(m_{\Delta t}(s_{t-1}))) &lt;/math&gt; where the current state is based on the change from the previous. Since &lt;math&gt; M &lt;/math&gt; is a closed group on &lt;math&gt; S &lt;/math&gt;, it is associative, has inverse, and contains idenity. &lt;math&gt; SE(3) &lt;/math&gt; is an exmaple of this. To be more specific, the latent structure of a scene from rigid image motion could be modelled by a point cloud with a motion space &lt;math&gt;M=SE(3)&lt;/math&gt;, where rigid image motion can be produced by a camera translating and rotating through a rigid scene in 3D. When a scene has N rigid bodies, the motion space can be represented as &lt;math&gt;M=[SE(3)]^N&lt;/math&gt;.<br /> <br /> === Learning Motion by Group Properties ===<br /> The goal is to learn a function &lt;math&gt; \Phi : I \times I \rightarrow \overline{M} &lt;/math&gt;, &lt;math&gt; \overline{M} &lt;/math&gt; indicating mapping of image pairs from &lt;math&gt; M &lt;/math&gt; to its representation, as well as the composition operator &lt;math&gt; \diamond : \overline{M} \rightarrow \overline{M} &lt;/math&gt; that emulates the composition of these elements in &lt;math&gt; M &lt;/math&gt;. For all sequences, it is assumed that for all times &lt;math&gt; t_0 &lt; t_1 &lt; t_2 ... &lt;/math&gt;, the sequence representation should have the following properties: <br /> # Associativity: &lt;math&gt; \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_2}, I_{t_3}) = (\Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_2})) \diamond \Phi(I_{t_2}, I_{t_3}) = \Phi(I_{t_0}, I_{t_1}) \diamond (\Phi(I_{t_1}, I_{t_2}) \diamond \Phi(I_{t_2}, I_{t_3})) = \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_3})&lt;/math&gt;, which means that the motion of differently composed subsequences of a sequence are equivalent<br /> # Has Identity: &lt;math&gt; \Phi(I_{t_0}, I_{t_1}) \diamond e = \Phi(I_{t_0}, I_{t_1}) = e \diamond \Phi(I_{t_0}, I_{t_1}) &lt;/math&gt; and &lt;math&gt; e=\Phi(I_{t}, I_{t}) \forall t &lt;/math&gt;, where &lt;math&gt;e&lt;/math&gt; is the null image motion and the unique identity in the latent space<br /> # Invertibility: &lt;math&gt; \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_0}) = e &lt;/math&gt;, so the inverse of the motion of an image sequence is the motion of that image sequence reversed<br /> An embedding loss is used to approximately enforce associativity and invertibility among subsequences sampled from image sequence. Associativity is encouraged by pushing sequences with the same final motion but different transitions to the same representation. Invertibility is encouraged by pushing sequences corresponding to the same motion with but in opposite directions away from each other, as well as pushing all loops to the same representation. Uniqueness of the identity is encouraged by pushing loops away from non-identity representations. Loops from different sequences are also pushed to the same representation (the identity).<br /> <br /> These constraints are true to any type of transformation resulting from image motion. This puts little restriction on the learning problems and allows all features relevant to the motion structure to be captured. On the other hand, optical flow assumes unchanging brightness between frames of the same projected scene, and motion estimates would degrade when that assumption does not hold.<br /> <br /> Also with this method, it is possible multiple representations &lt;math&gt; \overline{M} &lt;/math&gt; can be learned from a single &lt;math&gt; M &lt;/math&gt;, thus the learned representation is not necessary unique. In addition, the scenes are not expected to have rapid changing content, scene cuts, or long-term occlusions.<br /> <br /> === Sequence Learning with Neural Networks ===<br /> The functions &lt;math&gt; \Phi &lt;/math&gt; and &lt;math&gt; \diamond &lt;/math&gt; are approximated by CNN and RNN, respectively. LSTM is used for RNN. The input to the network is a sequence of images &lt;math&gt; I_t = \{I_1,...,I_t\} &lt;/math&gt;. The CNN processes pairs of images and generates intermediate representations, and the LSTM operates over the sequence of CNN outputs to produce an embedding sequence &lt;math&gt; R_t = \{R_{1,2},...,R_{t-1,t}\} &lt;/math&gt;. Only the embedding at the final time step is used for loss. The network is trained to minimize a hinge loss with respect to embeddings to pairs of sequences as defined below:<br /> <br /> &lt;center&gt;&lt;math&gt;L(R^1,R^2) = \begin{cases} d(R^1,R^2), &amp; \text{if positive pair} \\ max(0, m - d(R^1,R^2)), &amp; \text{if negative pair} \end{cases}&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt; d_{cosine}(R^1,R^2)=1-\frac{\langle R^1,R^2 \rangle}{\lVert R^1 \rVert \lVert R^2 \rVert} &lt;/math&gt;&lt;/center&gt;<br /> <br /> where &lt;math&gt;d(R^1,R^2)&lt;/math&gt; measure the distance between the embeddings of two sequences used for training selected to be cosine distance, &lt;math&gt; m &lt;/math&gt; is a fixed scalar margin selected to be 0.5. Positive pairs are training examples where two sequences have the same final motion, negative pairs are training examples where two sequences have the exact opposite final motion. Using L2 distances yields similar results as cosine distances.<br /> <br /> Each training sequence is recomposed into 6 subsequences: two forward, two backward, and two identity. To prevent the network from only looking at static differences, subsequence pairs are sampled such that they have the same start and end frames but different motions in between. Sequences of varying lengths are also used to generalize motion on different temporal scales. Training the network with only one input images per time step was also tried, but consistently yielded work results than image pairs.<br /> <br /> [[File:paper13_fig2.png|650px|center|]]<br /> <br /> Overall, training with image pairs resulted in lower error than training with just single images. This is demonstrated in the below table.<br /> <br /> <br /> [[File:table.png|700px|center|]]<br /> <br /> == Experimentation ==<br /> Trained network using rotated and translated MNIST dataset as well as KITTI dataset. <br /> * Used Torch<br /> * Used Adam for optimization, decay schedule of 30 epochs, learning rate chosen by random serach<br /> * 50-60 batch size for MNIST, 25-30 batch size for KITTI<br /> * Dilated convolution with Relu and batch normalization<br /> * Two LSTM cell per layer 256 hidden units each<br /> * Sequence length of 3-5 images<br /> * MINIST networks with up to 12 images <br /> <br /> === Rigid Motion in 2D ===<br /> * MNIST data rotated &lt;math&gt;[0, 360)&lt;/math&gt; degrees and translated &lt;math&gt;[-10, 10] &lt;/math&gt; pixels, i.e. &lt;math&gt;SE(2)&lt;/math&gt; transformations<br /> * Visualized the representation using t-SNE<br /> ** Clear clustering by translation and rotation but not object classes<br /> ** Suggests the representation captures the motion properties in the dataset, but is independent of image contents<br /> * Visualized the image-conditioned saliency maps<br /> **In Figure 3, the red represents the positive gradients of the activation function with respect to the input image, and the negative gradients are represented in blue.<br /> **If we consider a saliency map as a first-order Taylor expansion, then the map could show the relationship between pixel and the representation.<br /> ** Take derivative of the network output respect to the map<br /> ** The area that has the highest gradient means that part contributes the most to the output<br /> ** The resulting salient map strongly resembles spatiotemporal energy filters of classical motion processing<br /> ** Suggests the network is learning the right motion structure<br /> <br /> [[File:paper13_fig3.png|700px|center|]]<br /> <br /> === Real World Motion in 3D ===<br /> * Uses KITTI dataset collected on a car driving through roads in Germany<br /> * On a separate dataset with ground truth camera pose, linearly regress the representation to the ground truth<br /> ** The result is compared against self supervised flow algorithm Yu et al.(2016) after the output from the flow algorithm is downsampled, then feed through PCA, then regressed against the camera motion<br /> ** The data shows it performs not as well as the supervised algorithm, but consistently better than chance (guessing the mean value)<br /> ** Shows the method is able to capture dominant motion structure<br /> * Test performance on interpolation task<br /> ** Check &lt;math&gt;R([I_1,I_T])&lt;/math&gt; against &lt;math&gt;R([I_1, I_m, I_T])&lt;/math&gt;, &lt;math&gt;R([I_1, I_{IN}, I_T])&lt;/math&gt;, and &lt;math&gt;R([I_1, I_{OUT}, I_T])&lt;/math&gt;<br /> ** Test how sensitive the network is to deviations from unnatural motion<br /> ** High errors &lt;math&gt;\gg 1&lt;/math&gt; means the network can distinguish between realistic and unrealistic motion<br /> ** In order to do this, the distance between the embeddings of the frame sequences of the first and last frame &lt;math&gt;R([I_1,I_T])&lt;/math&gt; and of the first, middle, and last frame &lt;math&gt;R([I_1, I_m, I_T])&lt;/math&gt; is computed. This distance is compared with the distance when the middle frame of the second embedding is changed to a frame that is visually similar (inside sequence): &lt;math&gt;R([I_1, I_{IN}, I_T])&lt;/math&gt; and one that is visually dissimilar (outside sequence): &lt;math&gt;R([I_1, I_{OUT}, I_T])&lt;/math&gt;. The results are shown in Table 3. The embedding distance method is compared to the Euclidean distance which is defined as the mean pixel distance between the test frame and &lt;math&gt;{I_1,I_T}&lt;/math&gt;, whichever is smaller. It can be seen from the results that the embedding distance of the true frame is significantly lower than other frames. This means that the embedding distance used in the network is more sensitive to any atypical motions of the scenes. <br /> * Visualized saliency maps<br /> ** Highlights objects moving in the background, and motion of the car in the foreground<br /> ** Suggests the method can be used for tracking as well<br /> <br /> [[File:paper13_tab2.png|700px|center|]]<br /> <br /> [[File:paper13_fig4.png|700px|center|]]<br /> <br /> [[File:paper13_fig5.png|700px|center|]]<br /> <br /> [[File:table3_motion.PNG|700px|center|]]<br /> <br /> * Figure 7 displays graphs comparing the mean squared error of the method presented in this paper to the baseline chance method and the supervised Flow PCA method.<br /> <br /> [[File:paper13_fig6.PNG|700px|center|]]<br /> <br /> == Conclusion ==<br /> The author presented a new model of motion and a method for learning motion representations. It is shown that by enforcing group properties we can learn motion representations that are able to generalize between scenes with disparate content. The results can be useful for navigation, prediction, and other behavioral tasks relying on motion. Due to the fact that this method does not require labelled data, it can be applied to a large variety of tasks.<br /> <br /> == Criticism ==<br /> Although this method does not require any labelled data, it is still learning by supervision through defined constraints. The idea of training using unlabelled data is interesting and it does have meaningful practical application. Unfortunately, the author did not provide convincing experimental results. Results from motion estimation problems are typically compared against ground truth data for their accuracy. The author performed experiments on transformed MNIST data and KITTI data. The MNIST data is transformed by the author, thus the ground truth is readily available. However the author only claimed the validity of the results through indirect means of using t-SNE and saliency map visualization. For the KITTI dataset, the author regressed the representations against ground truth for some mapping from the network output to some physical motion representation. Again, the results were compared only indirectly against ground truth. Such experimentation made the method hardly convincing and applicable to real world applications. In addition, the network does not output motion representations with physical meanings, make the proposed method useless for any real world applications.<br /> <br /> Another criticism is that the group-properties constraint the authors impose is too weak. Any set consisting of functions, their inverses, and the identity forms a group. While physical motions are one example of such a group, there are many valid groups that do not represent any coherent physical motions.<br /> <br /> Since the network has to learn both the group elements, &lt;math&gt;\overline{M}&lt;/math&gt;, and the composition function, &lt;math&gt;\diamond&lt;/math&gt;, associated with the group it is difficult to tell how each of them are performing.<br /> == References ==<br /> Jaegle, A. (2018). Understanding image motion with group representations . ICLR. Retrieved from https://openreview.net/pdf?id=SJLlmG-AZ.<br /> <br /> Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. In ''International Conference on Learning Representations (ICLR) Workshop'', 2013.<br /> <br /> Jason J Yu, Adam W Harley, and Konstantinos G Derpanis. Back to basics: Unsupervised learning of optical flow via brightness constancy and motion smoothness. ''European Conference on Computer Vision (ECCV) Workshops'', 2016.</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32391 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-28T05:13:46Z <p>S3wen: </p> <hr /> <div>= Introduction =<br /> Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, from here on referred to as measurements of the true data. If know the measurement function and we would like to train a generative model for the true data there are several ways to continue with varying degrees of success. We will use noisy MNIST data as an illustrative example. Suppose we only see MNIST data that have been run through a Gaussian kernel (blurred) with some noise from a &lt;math&gt;N(0, 0.5^2)&lt;/math&gt; distribution added to each pixel:<br /> <br /> &lt;gallery mode=&quot;packed&quot;&gt;<br /> File:mnist.png| True Data (Unobserved)<br /> File:mnistmeasured.png| Measured Data (Observed)<br /> &lt;/gallery&gt;<br /> <br /> <br /> === Ignore the problem ===<br /> [[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br /> <br /> Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br /> <br /> <br /> === Try to recover the information lost ===<br /> [[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br /> <br /> Works better than ignoring the problem but depends on how easily the measurement function can be inversed. <br /> <br /> === AmbientGAN ===<br /> [[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br /> <br /> Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. <br /> <br /> AmbientGAN works by training a generator which attempts to have the measruements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. <br /> <br /> = Model =<br /> For the following variables superscript &lt;math&gt;r&lt;/math&gt; represents the true distributions while superscript &lt;math&gt;g&lt;/math&gt; represents the generated distributions. Let &lt;math&gt;x&lt;/math&gt;, represent the underlying space and &lt;math&gt;y&lt;/math&gt; for the measurement.<br /> <br /> Thus &lt;math&gt;p_x^r&lt;/math&gt; is the real underlying distribution over &lt;math&gt;\mathbb{R}^n&lt;/math&gt; that we are interested in. However if we assume our (known) measurement functions, &lt;math&gt;f_\theta: \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; are parameterized by &lt;math&gt;\theta \sim p_\theta&lt;/math&gt;, we can only observe &lt;math&gt;y = f_\theta(x)&lt;/math&gt;.<br /> <br /> Mirroring the standard GAN setup we let &lt;math&gt;Z \in \mathbb{R}^k, Z \sim p_z&lt;/math&gt; and &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt; be random variables coming from a distribution that is easy to sample. <br /> <br /> If we have a generator &lt;math&gt;G: \mathbb{R}^k \to \mathbb{R}^n&lt;/math&gt; then we can generate &lt;math&gt;X^g = G(Z)&lt;/math&gt; which has distribution &lt;math&gt;p_x^g&lt;/math&gt; a measurement &lt;math&gt;Y^g = f_\Theta(G(Z))&lt;/math&gt; which has distribution &lt;math&gt;p_y^g&lt;/math&gt;. <br /> <br /> Unfortunately we do not observe any &lt;math&gt;X^g \sim p_x&lt;/math&gt; so we can use the discriminator directly on &lt;math&gt;G(Z)&lt;/math&gt; to train the generator. Instead we will use the discriminator to distinguish between the &lt;math&gt;Y^g -<br /> f_\Theta(G(Z))&lt;/math&gt; and &lt;math&gt;Y^r&lt;/math&gt;. That is we train the discriminator, &lt;math&gt;D: \mathbb{R}^m \to \mathbb{R}&lt;/math&gt; to detect if a measurement came from &lt;math&gt;p_y^r&lt;/math&gt; or &lt;math&gt;p_y^g&lt;/math&gt;.<br /> <br /> AmbientGAN has the objective function:<br /> <br /> &lt;math&gt;\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))] &lt;/math&gt;<br /> <br /> where &lt;math&gt;q(.)&lt;/math&gt; is the quality function; for the standard GAN &lt;math&gt;q(x) = log(x)&lt;/math&gt; and for Wasserstein GAN &lt;math&gt;q(x) = x&lt;/math&gt;.<br /> <br /> As a technical limitation we require &lt;math&gt;f_\theta&lt;/math&gt; to be differentiable with the respect each input for all values of &lt;math&gt;\theta&lt;/math&gt;.<br /> <br /> With this set up we sample &lt;math&gt;Z \sim p_z&lt;/math&gt;, &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt;, and &lt;math&gt;Y^r \sim U\{y_1, \cdots, y_s\}&lt;/math&gt; each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating &lt;math&gt;G&lt;/math&gt; and updating &lt;math&gt;D&lt;/math&gt;. <br /> <br /> = Empirical Results =<br /> <br /> The paper continues to present results of AmbientGAN under various measurement functions when compared to baseline models. We have already seen one example in the introduction: a comparison of AmbientGAN in the Convolve + Noise Measurement case compared to the ignore-baseline, and the unmeasure-baseline. <br /> <br /> === Block-Pixels ===<br /> With the block-pixels measurement function each pixel is independently set to 0 with probability &lt;math&gt;p&lt;/math&gt;.<br /> <br /> [[File:block-pixels.png]]<br /> <br /> Measurements from the celebA dataset with &lt;math&gt;p=0.95&lt;/math&gt; (left). Images generated from GAN trained on unmeasured (via blurring) data (middle). Results generated from AmbientGAN (right).<br /> <br /> === Block-Patch ===<br /> <br /> [[File:block-patch.png]]<br /> <br /> A random 14x14 patch is set to zero (left). Unmeasured using-navier-stoke inpainting (middle). AmbientGAN (right). <br /> <br /> === Pad-Rotate-Project-&lt;math&gt;\theta&lt;/math&gt; ===<br /> <br /> [[File:pad-rotate-project-theta.png]]<br /> <br /> Results generated by AmbientGAN where the measurement function 0 pads the images, rotates it by &lt;math&gt;\theta&lt;/math&gt;, and projects it on to the x axis. For each measurement the value of &lt;math&gt;\theta&lt;/math&gt; is known. <br /> <br /> The generated images only have the basic features of a face and is referred to as a failure case in the paper. However the measurement function performs relatively well given how lossy the measurement function is. <br /> <br /> === MNIST Inception ===<br /> <br /> [[File:MNIST-inception.png]]<br /> <br /> AmbientGAN is more robust than all other baseline models. Further AmbientGAN retains high inception scores as measurements become more and more lossy. <br /> <br /> === CIFAR-10 Inception ===<br /> <br /> [[File:CIFAR-inception.png]]<br /> <br /> AmbientGAN is faster to train and more robust even on more complex distributions such as CIFAR-10. <br /> <br /> = Theoretical Results =<br /> <br /> The theoretical results in the paper prove the true underlying distribution of &lt;math&gt;p_x^r&lt;/math&gt; can be recovered when we have data that comes from the Gaussian-Projection measurement, Fourier transform measurement and the block-pixels measurement. The do this by showing the distribution of the measurements &lt;math&gt;p_y^r&lt;/math&gt; corresponds to a unique distribution &lt;math&gt;p_x^r&lt;/math&gt;. Thus even when the measurement itself is non-invertible the effect of the measurement on the distribution &lt;math&gt;p_x^r&lt;/math&gt; is invertible. Lemma 5.1 ensures this is sufficient to provide the AmbientGAN training process with a consistency guarantee. For full proofs of the results please see appendix A. <br /> <br /> === Lemma 5.1 === <br /> Let &lt;math&gt;p_x^r&lt;/math&gt; be the true data distribution, and &lt;math&gt;p_\theta&lt;/math&gt; be the distributions over the parameters of the measurement function. Let &lt;math&gt;p_y^r&lt;/math&gt; be the induced measurement distribution. <br /> <br /> Assume for &lt;math&gt;p_\theta&lt;/math&gt; there is a unique probability distribution &lt;math&gt;p_x^r&lt;/math&gt; that induces &lt;math&gt;p_y^r&lt;/math&gt;. <br /> <br /> Then for the standard GAN model if the Discriminator is optimal, then a generator &lt;math&gt;G&lt;/math&gt; is optimal if and only if &lt;math&gt;p_x^g = p_x^r&lt;/math&gt;. <br /> <br /> === Theorems 5.2-5.4 ===<br /> When the measurement is a Gaussian-Projection (&lt;math&gt;f_\Theta(x) = \langle \Theta, \langle \Theta, x \rangle \rangle &lt;/math&gt;), the Convolution + Noise measurement, or the block-pixels measurement then there is a unique underlying distribution that induces the observed measurement distribution. <br /> <br /> = Future Research =<br /> <br /> One critical weakness of AmbientGAN is the assumption that the measurement model is known. It would be nice to be able to train an AmbientGAN model when we have an unknown measurement model but also a small sample of unmeasured data. <br /> <br /> = References =<br /> https://openreview.net/forum?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:CIFAR-inception.png&diff=32390 File:CIFAR-inception.png 2018-02-28T05:10:09Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:MNIST-inception.png&diff=32389 File:MNIST-inception.png 2018-02-28T05:06:29Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32388 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-28T05:00:09Z <p>S3wen: Theoretical Results</p> <hr /> <div>= Introduction =<br /> Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, from here on referred to as measurements of the true data. If know the measurement function and we would like to train a generative model for the true data there are several ways to continue with varying degrees of success. We will use noisy MNIST data as an illustrative example. Suppose we only see MNIST data that have been run through a Gaussian kernel (blurred) with some noise from a &lt;math&gt;N(0, 0.5^2)&lt;/math&gt; distribution added to each pixel:<br /> <br /> &lt;gallery mode=&quot;packed&quot;&gt;<br /> File:mnist.png| True Data (Unobserved)<br /> File:mnistmeasured.png| Measured Data (Observed)<br /> &lt;/gallery&gt;<br /> <br /> <br /> === Ignore the problem ===<br /> [[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br /> <br /> Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br /> <br /> <br /> === Try to recover the information lost ===<br /> [[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br /> <br /> Works better than ignoring the problem but depends on how easily the measurement function can be inversed. <br /> <br /> === AmbientGAN ===<br /> [[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br /> <br /> Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. <br /> <br /> AmbientGAN works by training a generator which attempts to have the measruements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. <br /> <br /> = Model =<br /> For the following variables superscript &lt;math&gt;r&lt;/math&gt; represents the true distributions while superscript &lt;math&gt;g&lt;/math&gt; represents the generated distributions. Let &lt;math&gt;x&lt;/math&gt;, represent the underlying space and &lt;math&gt;y&lt;/math&gt; for the measurement.<br /> <br /> Thus &lt;math&gt;p_x^r&lt;/math&gt; is the real underlying distribution over &lt;math&gt;\mathbb{R}^n&lt;/math&gt; that we are interested in. However if we assume our (known) measurement functions, &lt;math&gt;f_\theta: \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; are parameterized by &lt;math&gt;\theta \sim p_\theta&lt;/math&gt;, we can only observe &lt;math&gt;y = f_\theta(x)&lt;/math&gt;.<br /> <br /> Mirroring the standard GAN setup we let &lt;math&gt;Z \in \mathbb{R}^k, Z \sim p_z&lt;/math&gt; and &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt; be random variables coming from a distribution that is easy to sample. <br /> <br /> If we have a generator &lt;math&gt;G: \mathbb{R}^k \to \mathbb{R}^n&lt;/math&gt; then we can generate &lt;math&gt;X^g = G(Z)&lt;/math&gt; which has distribution &lt;math&gt;p_x^g&lt;/math&gt; a measurement &lt;math&gt;Y^g = f_\Theta(G(Z))&lt;/math&gt; which has distribution &lt;math&gt;p_y^g&lt;/math&gt;. <br /> <br /> Unfortunately we do not observe any &lt;math&gt;X^g \sim p_x&lt;/math&gt; so we can use the discriminator directly on &lt;math&gt;G(Z)&lt;/math&gt; to train the generator. Instead we will use the discriminator to distinguish between the &lt;math&gt;Y^g -<br /> f_\Theta(G(Z))&lt;/math&gt; and &lt;math&gt;Y^r&lt;/math&gt;. That is we train the discriminator, &lt;math&gt;D: \mathbb{R}^m \to \mathbb{R}&lt;/math&gt; to detect if a measurement came from &lt;math&gt;p_y^r&lt;/math&gt; or &lt;math&gt;p_y^g&lt;/math&gt;.<br /> <br /> AmbientGAN has the objective function:<br /> <br /> &lt;math&gt;\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))] &lt;/math&gt;<br /> <br /> where &lt;math&gt;q(.)&lt;/math&gt; is the quality function; for the standard GAN &lt;math&gt;q(x) = log(x)&lt;/math&gt; and for Wasserstein GAN &lt;math&gt;q(x) = x&lt;/math&gt;.<br /> <br /> As a technical limitation we require &lt;math&gt;f_\theta&lt;/math&gt; to be differentiable with the respect each input for all values of &lt;math&gt;\theta&lt;/math&gt;.<br /> <br /> With this set up we sample &lt;math&gt;Z \sim p_z&lt;/math&gt;, &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt;, and &lt;math&gt;Y^r \sim U\{y_1, \cdots, y_s\}&lt;/math&gt; each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating &lt;math&gt;G&lt;/math&gt; and updating &lt;math&gt;D&lt;/math&gt;. <br /> <br /> = Empirical Results =<br /> <br /> The paper continues to present results of AmbientGAN under various measurement functions when compared to baseline models. We have already seen one example in the introduction: a comparison of AmbientGAN in the Convolve + Noise Measurement case compared to the ignore-baseline, and the unmeasure-baseline. <br /> <br /> === Block-Pixels ===<br /> With the block-pixels measurement function each pixel is independently set to 0 with probability &lt;math&gt;p&lt;/math&gt;.<br /> <br /> [[File:block-pixels.png]]<br /> <br /> Measurements from the celebA dataset with &lt;math&gt;p=0.95&lt;/math&gt; (left). Images generated from GAN trained on unmeasured (via blurring) data (middle). Results generated from AmbientGAN (right).<br /> <br /> === Block-Patch ===<br /> <br /> [[File:block-patch.png]]<br /> <br /> A random 14x14 patch is set to zero (left). Unmeasured using-navier-stoke inpainting (middle). AmbientGAN (right). <br /> <br /> === Pad-Rotate-Project-&lt;math&gt;\theta&lt;/math&gt; ===<br /> <br /> [[File:pad-rotate-project-theta.png]]<br /> <br /> Results generated by AmbientGAN where the measurement function 0 pads the images, rotates it by &lt;math&gt;\theta&lt;/math&gt;, and projects it on to the x axis. For each measurement the value of &lt;math&gt;\theta&lt;/math&gt; is known. <br /> <br /> The generated images only have the basic features of a face and is referred to as a failure case in the paper. However the measurement function performs relatively well given how lossy the measurement function is. <br /> <br /> === Gaussian-Projection ===<br /> <br /> === Fourier transform ===<br /> <br /> = Theoretical Results =<br /> <br /> The theoretical results in the paper prove the true underlying distribution of &lt;math&gt;p_x^r&lt;/math&gt; can be recovered when we have data that comes from the Gaussian-Projection measurement, Fourier transform measurement and the block-pixels measurement. The do this by showing the distribution of the measurements &lt;math&gt;p_y^r&lt;/math&gt; corresponds to a unique distribution &lt;math&gt;p_x^r&lt;/math&gt;. Thus even when the measurement itself is non-invertible the effect of the measurement on the distribution &lt;math&gt;p_x^r&lt;/math&gt; is invertible. Lemma 5.1 ensures this is sufficient to provide the AmbientGAN training process with a consistency guarantee. For full proofs of the results please see appendix A. <br /> <br /> === Lemma 5.1 === <br /> Let &lt;math&gt;p_x^r&lt;/math&gt; be the true data distribution, and &lt;math&gt;p_\theta&lt;/math&gt; be the distributions over the parameters of the measurement function. Let &lt;math&gt;p_y^r&lt;/math&gt; be the induced measurement distribution. <br /> <br /> Assume for &lt;math&gt;p_\theta&lt;/math&gt; there is a unique probability distribution &lt;math&gt;p_x^r&lt;/math&gt; that induces &lt;math&gt;p_y^r&lt;/math&gt;. <br /> <br /> Then for the standard GAN model if the Discriminator is optimal, then a generator &lt;math&gt;G&lt;/math&gt; is optimal if and only if &lt;math&gt;p_x^g = p_x^r&lt;/math&gt;. <br /> <br /> === Theorems 5.2-5.4 ===<br /> When the measurement is a Gaussian-Projection (&lt;math&gt;f_\Theta(x) = \langle \Theta, \langle \Theta, x \rangle \rangle &lt;/math&gt;), the Convolution + Noise measurement, or the block-pixels measurement then there is a unique underlying distribution that induces the observed measurement distribution. <br /> <br /> = Criticisms =<br /> <br /> = References =<br /> https://openreview.net/forum?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:pad-rotate-project-theta.png&diff=32385 File:pad-rotate-project-theta.png 2018-02-28T02:33:01Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:block-patch.png&diff=32384 File:block-patch.png 2018-02-28T02:17:49Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:block-pixels.png&diff=32383 File:block-pixels.png 2018-02-28T01:30:18Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32382 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-28T00:58:23Z <p>S3wen: Model Specification</p> <hr /> <div>= Introduction =<br /> Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, from here on referred to as measurements of the true data. If know the measurement function and we would like to train a generative model for the true data there are several ways to continue with varying degrees of success. We will use noisy MNIST data as an illustrative example. Suppose we only see MNIST data that have been run through a Gaussian kernel (blurred) with some noise from a &lt;math&gt;N(0, 0.5^2)&lt;/math&gt; distribution added to each pixel:<br /> <br /> &lt;gallery mode=&quot;packed&quot;&gt;<br /> File:mnist.png| True Data (Unobserved)<br /> File:mnistmeasured.png| Measured Data (Observed)<br /> &lt;/gallery&gt;<br /> <br /> <br /> === Ignore the problem ===<br /> [[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br /> <br /> Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br /> <br /> <br /> === Try to recover the information lost ===<br /> [[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br /> <br /> Works better than ignoring the problem but depends on how easily the measurement function can be inversed. <br /> <br /> === AmbientGAN ===<br /> [[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br /> <br /> Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. <br /> <br /> AmbientGAN works by training a generator which attempts to have the measruements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. <br /> <br /> = Model =<br /> For the following variables superscript &lt;math&gt;r&lt;/math&gt; represents the true distributions while superscript &lt;math&gt;g&lt;/math&gt; represents the generated distributions. Let &lt;math&gt;x&lt;/math&gt;, represent the underlying space and &lt;math&gt;y&lt;/math&gt; for the measurement.<br /> <br /> Thus &lt;math&gt;p_x^r&lt;/math&gt; is the real underlying distribution over &lt;math&gt;\mathbb{R}^n&lt;/math&gt; that we are interested in. However if we assume our (known) measurement functions, &lt;math&gt;f_\theta: \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; are parameterized by &lt;math&gt;\theta \sim p_\theta&lt;/math&gt;, we can only observe &lt;math&gt;y = f_\theta(x)&lt;/math&gt;.<br /> <br /> Mirroring the standard GAN setup we let &lt;math&gt;Z \in \mathbb{R}^k, Z \sim p_z&lt;/math&gt; and &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt; be random variables coming from a distribution that is easy to sample. <br /> <br /> If we have a generator &lt;math&gt;G: \mathbb{R}^k \to \mathbb{R}^n&lt;/math&gt; then we can generate &lt;math&gt;X^g = G(Z)&lt;/math&gt; which has distribution &lt;math&gt;p_x^g&lt;/math&gt; a measurement &lt;math&gt;Y^g = f_\Theta(G(Z))&lt;/math&gt; which has distribution &lt;math&gt;p_y^g&lt;/math&gt;. <br /> <br /> Unfortunately we do not observe any &lt;math&gt;X^g \sim p_x&lt;/math&gt; so we can use the discriminator directly on &lt;math&gt;G(Z)&lt;/math&gt; to train the generator. Instead we will use the discriminator to distinguish between the &lt;math&gt;Y^g -<br /> f_\Theta(G(Z))&lt;/math&gt; and &lt;math&gt;Y^r&lt;/math&gt;. That is we train the discriminator, &lt;math&gt;D: \mathbb{R}^m \to \mathbb{R}&lt;/math&gt; to detect if a measurement came from &lt;math&gt;p_y^r&lt;/math&gt; or &lt;math&gt;p_y^g&lt;/math&gt;.<br /> <br /> AmbientGAN has the objective function:<br /> <br /> &lt;math&gt;\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))] &lt;/math&gt;<br /> <br /> where &lt;math&gt;q(.)&lt;/math&gt; is the quality function; for the standard GAN &lt;math&gt;q(x) = log(x)&lt;/math&gt; and for Wasserstein GAN &lt;math&gt;q(x) = x&lt;/math&gt;.<br /> <br /> As a technical limitation we require &lt;math&gt;f_\theta&lt;/math&gt; to be differentiable with the respect each input for all values of &lt;math&gt;\theta&lt;/math&gt;.<br /> <br /> With this set up we sample &lt;math&gt;Z \sim p_z&lt;/math&gt;, &lt;math&gt;\Theta \sim p_\theta&lt;/math&gt;, and &lt;math&gt;Y^r \sim U\{y_1, \cdots, y_s\}&lt;/math&gt; each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating &lt;math&gt;G&lt;/math&gt; and updating &lt;math&gt;D&lt;math/&gt;. <br /> <br /> = Empirical Results =<br /> <br /> = Theoretical Results =<br /> <br /> = Criticisms =<br /> <br /> = References =<br /> https://openreview.net/forum?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32378 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-27T09:29:14Z <p>S3wen: Introduction into the basic idea of AmbientGAN</p> <hr /> <div>= Introduction =<br /> Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, from here on referred to as measurements of the true data. If know the measurement function and we would like to train a generative model for the true data there are several ways to continue with varying degrees of success. We will use noisy MNIST data as an illustrative example. Suppose we only see MNIST data that have been run through a Gaussian kernel (blurred) with some noise from a &lt;math&gt;N(0, 0.5^2)&lt;/math&gt; distribution added to each pixel:<br /> <br /> &lt;gallery mode=&quot;packed&quot;&gt;<br /> File:mnist.png| True Data (Unobserved)<br /> File:mnistmeasured.png| Measured Data (Observed)<br /> &lt;/gallery&gt;<br /> <br /> <br /> === Ignore the problem ===<br /> [[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br /> <br /> Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br /> <br /> <br /> === Try to recover the information lost ===<br /> [[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br /> <br /> Works better than ignoring the problem but depends on how easily the measurement function can be inversed. <br /> <br /> === AmbientGAN ===<br /> [[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br /> <br /> Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. <br /> <br /> AmbientGAN works by training a generator which attempts to have the measruements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. <br /> <br /> = Model =<br /> For the following variables superscript &lt;math&gt;r&lt;/math&gt; represents the true distributions while superscript &lt;math&gt;g&lt;/math&gt; represents the generated distributions. Let &lt;math&gt;x&lt;/math&gt;, represent the underlying space and &lt;math&gt;y&lt;/math&gt; for the measurement.<br /> <br /> Thus &lt;math&gt;p_x^r&lt;/math&gt; is the real underlying distribution over &lt;math&gt;\mathbb{R}^n&lt;/math&gt; that we are interested in. However if we assume our (known) measurement functions, &lt;math&gt;f_\theta: \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; are parameterized by &lt;math&gt;\theta \sim p_\theta&lt;/math&gt;, we can only observe &lt;math&gt;y = f_\theta(x)&lt;/math&gt;.<br /> <br /> <br /> <br /> AmbientGAN has the objective function:<br /> <br /> &lt;math&gt;\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))] &lt;/math&gt;<br /> <br /> = Empirical Results =<br /> <br /> = Theoretical Results =<br /> <br /> = Criticisms =<br /> <br /> = References =<br /> https://openreview.net/forum?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:mnistambient.png&diff=32377 File:mnistambient.png 2018-02-27T09:19:26Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:mnistrecover.png&diff=32376 File:mnistrecover.png 2018-02-27T09:19:11Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:mnistignore.png&diff=32375 File:mnistignore.png 2018-02-27T09:18:27Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:GANambient.png&diff=32374 File:GANambient.png 2018-02-27T09:17:01Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:GANrecovery.png&diff=32373 File:GANrecovery.png 2018-02-27T09:16:51Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:GANignore.png&diff=32372 File:GANignore.png 2018-02-27T09:15:45Z <p>S3wen: </p> <hr /> <div></div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:mnistmeasured.png&diff=32371 File:mnistmeasured.png 2018-02-27T08:59:48Z <p>S3wen: Convolve + Noise measurement of MNIST</p> <hr /> <div>Convolve + Noise measurement of MNIST</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:mnist.png&diff=32370 File:mnist.png 2018-02-27T08:58:42Z <p>S3wen: Sample MNIST data</p> <hr /> <div>Sample MNIST data</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18&diff=32369 stat946w18 2018-02-26T07:10:05Z <p>S3wen: </p> <hr /> <div>=[https://piazza.com/uwaterloo.ca/fall2017/stat946/resources List of Papers]=<br /> <br /> = Record your contributions here [https://docs.google.com/spreadsheets/d/1fU746Cld_mSqQBCD5qadvkXZW1g-j-kHvmHQ6AMeuqU/edit?usp=sharing]=<br /> <br /> Use the following notations:<br /> <br /> P: You have written a summary/critique on the paper.<br /> <br /> T: You had a technical contribution on a paper (excluding the paper that you present).<br /> <br /> E: You had an editorial contribution on a paper (excluding the paper that you present).<br /> <br /> [https://docs.google.com/forms/d/1HrpW_lnn4jpFmoYKJBRAkm-GYa8djv9iZXcESeVB7Ts/prefill Your feedback on presentations]<br /> <br /> =Paper presentation=<br /> {| class=&quot;wikitable&quot;<br /> <br /> {| border=&quot;1&quot; cellpadding=&quot;3&quot;<br /> |-<br /> |width=&quot;60pt&quot;|Date<br /> |width=&quot;100pt&quot;|Name <br /> |width=&quot;30pt&quot;|Paper number <br /> |width=&quot;700pt&quot;|Title<br /> |width=&quot;30pt&quot;|Link to the paper<br /> |width=&quot;30pt&quot;|Link to the summary<br /> |-<br /> |Feb 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] || [http://wikicoursenote.com/wiki/Stat946f15/Sequence_to_sequence_learning_with_neural_networks#Long_Short-Term_Memory_Recurrent_Neural_Network Summary]<br /> |-<br /> |Feb 27 || || 1|| || || <br /> |-<br /> |Feb 27 || || 2|| || || <br /> |-<br /> |Feb 27 || || 3|| || || <br /> |-<br /> |Mar 1 || Peter Forsyth || 4|| Unsupervised Machine Translation Using Monolingual Corpora Only || [https://arxiv.org/pdf/1711.00043.pdf Paper] || [[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only Summary]]<br /> |-<br /> |Mar 1 || wenqing liu || 5|| Spectral Normalization for Generative Adversarial Networks || [https://openreview.net/pdf?id=B1QRgziT- Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network Summary]<br /> |-<br /> |Mar 1 || Ilia Sucholutsky || 6|| One-Shot Imitation Learning || [https://papers.nips.cc/paper/6709-one-shot-imitation-learning.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning#Performance_Evaluation Summary]<br /> |-<br /> |Mar 6 || George (Shiyang) Wen || 7|| AmbientGAN: Generative models from lossy measurements || [https://openreview.net/pdf?id=Hy7fDog0b Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements Summary]<br /> |-<br /> |Mar 6 || Raphael Tang || 8|| Snapshot Ensembles: Train 1, Get M for Free || [https://arxiv.org/abs/1704.00109 Paper] || <br /> |-<br /> |Mar 6 ||Fan Xia || 9|| Word translation without parallel data ||[https://arxiv.org/pdf/1710.04087.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data Summary]<br /> |-<br /> |Mar 8 || Alex (Xian) Wang || 10 || Self-Normalizing Neural Networks || [http://papers.nips.cc/paper/6698-self-normalizing-neural-networks.pdf Paper] || <br /> |-<br /> |Mar 8 || Guillaume Verdon || 11|| || || <br /> |-<br /> |Mar 8 || Wei Tao Chen || 12|| Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data || [https://openreview.net/forum?id=ryBnUWb0b] || <br /> |-<br /> |Mar 13 || Chunshang Li || 13 || UNDERSTANDING IMAGE MOTION WITH GROUP REPRESENTATIONS || [https://openreview.net/pdf?id=SJLlmG-AZ Paper] || <br /> |-<br /> |Mar 13 || Saifuddin Hitawala || 14 || Thinking Fast and Slow with Deep Learning and Tree Search || [https://papers.nips.cc/paper/7120-thinking-fast-and-slow-with-deep-learning-and-tree-search.pdf Paper] || <br /> |-<br /> |Mar 13 || Taylor Denouden || 15|| A neural representation of sketch drawings || [https://arxiv.org/pdf/1704.03477.pdf Paper] || [Summary]<br /> |-<br /> |Mar 15 || Zehao xu || 16|| Synthetic and natural noise both break neural machine translation || || <br /> |-<br /> |Mar 15 || Prarthana Bhattacharyya || 17|| Semi-Supervised Learning for Optical Flow with Generative Adversarial Networks || [http://papers.nips.cc/paper/6639-semi-supervised-learning-for-optical-flow-with-generative-adversarial-networks.pdf] || [Summary] <br /> |-<br /> |Mar 15 || Changjian Li || 18|| Imagination-Augmented Agents for Deep Reinforcement Learning || [https://papers.nips.cc/paper/7152-imagination-augmented-agents-for-deep-reinforcement-learning.pdf Paper] || <br /> |-<br /> |Mar 20 || Travis Dunn || 19|| Continuous Adaptation via Meta-Learning in Nonstationary and Competitive Environments || [https://openreview.net/pdf?id=Sk2u1g-0- Paper] || [Summary]<br /> |-<br /> |Mar 20 || Sushrut Bhalla || 20|| MaskRNN: Instance Level Video Object Segmentation || [https://papers.nips.cc/paper/6636-maskrnn-instance-level-video-object-segmentation.pdf Paper] || [Summary]<br /> |-<br /> |Mar 20 || Hamid Tahir || 21|| Wavelet Pooling for Convolution Neural Networks || [https://openreview.net/pdf?id=rkhlb8lCZ Paper] || Summary<br /> |-<br /> |Mar 22 || Dongyang Yang|| 22|| Don't Decay the Learning Rate, Increase the Batch Size || [https://arxiv.org/pdf/1711.00489.pdf Paper] || <br /> |-<br /> |Mar 22 || Yao Li || 23||Toward Multimodal Image-to-Image Translation || [http://papers.nips.cc/paper/6650-toward-multimodal-image-to-image-translation.pdf Paper] || <br /> |-<br /> |Mar 22 || Sahil Pereira || 24||End-to-End Differentiable Adversarial Imitation Learning|| [http://proceedings.mlr.press/v70/baram17a/baram17a.pdf Paper] || [http://proceedings.mlr.press/v70/baram17a/baram17a.pdf Summary]<br /> |-<br /> |Mar 27 || Jaspreet Singh Sambee || 25|| Gated Recurrent Convolution Neural Network for OCR || [http://papers.nips.cc/paper/6637-gated-recurrent-convolution-neural-network-for-ocr.pdf Paper] || <br /> |-<br /> |Mar 27 || Braden Hurl || 26|| Spherical CNNs || [https://openreview.net/pdf?id=Hkbd5xZRb Paper] || <br /> |-<br /> |Mar 27 || Marko Ilievski || 27|| Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders || [http://proceedings.mlr.press/v70/engel17a/engel17a.pdf Paper] || <br /> |-<br /> |Mar 29 || Alex Pon || 28||Wasserstein GAN || [https://arxiv.org/pdf/1701.07875.pdf Paper] ||<br /> |-<br /> |Mar 29 || Sean Walsh || 29||Improved Training of Wasserstein GANs || [https://arxiv.org/pdf/1704.00028.pdf Paper] ||<br /> |-<br /> |Mar 29 || Jason Ku || 30||MarrNet: 3D Shape Reconstruction via 2.5D Sketches ||[https://arxiv.org/pdf/1711.03129.pdf Paper] ||<br /> |-<br /> |Apr 3 || Tong Yang || 31|| Dynamic Routing Between Capsules. || [http://papers.nips.cc/paper/6975-dynamic-routing-between-capsules.pdf Paper] || <br /> |-<br /> |Apr 3 || Benjamin Skikos || 32|| Training and Inference with Integers in Deep Neural Networks || [https://openreview.net/pdf?id=HJGXzmspb Paper] || <br /> |-<br /> |Apr 3 || Weishi Chen || 33|| Progressive Growing of GANs for Improved Quality, Stability, and Variation || [https://openreview.net/pdf?id=Hk99zCeAb Paper] || <br /> |-</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32368 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-26T07:08:42Z <p>S3wen: Basic information</p> <hr /> <div>= Introduction =<br /> Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from lossy data under certain conditions. Even when these conditions are not met the results from AmbientGAN still produce high quality results. <br /> <br /> = Problem Specification =<br /> <br /> = Model =<br /> AmbientGAN compared to GAN (AmbientGAN passes the generator output through a measurement function) <br /> <br /> = Results =<br /> <br /> = Open questions =<br /> <br /> = Criticisms =<br /> <br /> = References =<br /> https://openreview.net/forum?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32200 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-24T21:15:00Z <p>S3wen: S3wen moved page AmbientGAN: Generative Models from Lossy Measurements to stat946w18/AmbientGAN: Generative Models from Lossy Measurements</p> <hr /> <div>https://openreview.net/pdf?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32201 AmbientGAN: Generative Models from Lossy Measurements 2018-02-24T21:15:00Z <p>S3wen: S3wen moved page AmbientGAN: Generative Models from Lossy Measurements to stat946w18/AmbientGAN: Generative Models from Lossy Measurements</p> <hr /> <div>#REDIRECT [[stat946w18/AmbientGAN: Generative Models from Lossy Measurements]]</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32199 stat946w18/AmbientGAN: Generative Models from Lossy Measurements 2018-02-24T21:12:04Z <p>S3wen: Created page with &quot;https://openreview.net/pdf?id=Hy7fDog0b&quot;</p> <hr /> <div>https://openreview.net/pdf?id=Hy7fDog0b</div> S3wen http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18&diff=31881 stat946w18 2018-02-14T04:24:16Z <p>S3wen: /* Paper presentation */</p> <hr /> <div>=[https://piazza.com/uwaterloo.ca/fall2017/stat946/resources List of Papers]=<br /> <br /> = Record your contributions here [https://docs.google.com/spreadsheets/d/1fU746Cld_mSqQBCD5qadvkXZW1g-j-kHvmHQ6AMeuqU/edit?usp=sharing]=<br /> <br /> Use the following notations:<br /> <br /> P: You have written a summary/critique on the paper.<br /> <br /> T: You had a technical contribution on a paper (excluding the paper that you present).<br /> <br /> E: You had an editorial contribution on a paper (excluding the paper that you present).<br /> <br /> [https://docs.google.com/forms/d/1HrpW_lnn4jpFmoYKJBRAkm-GYa8djv9iZXcESeVB7Ts/prefill Your feedback on presentations]<br /> <br /> =Paper presentation=<br /> {| class=&quot;wikitable&quot;<br /> <br /> {| border=&quot;1&quot; cellpadding=&quot;3&quot;<br /> |-<br /> |width=&quot;60pt&quot;|Date<br /> |width=&quot;100pt&quot;|Name <br /> |width=&quot;30pt&quot;|Paper number <br /> |width=&quot;700pt&quot;|Title<br /> |width=&quot;30pt&quot;|Link to the paper<br /> |width=&quot;30pt&quot;|Link to the summary<br /> |-<br /> |Feb 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] || [http://wikicoursenote.com/wiki/Stat946f15/Sequence_to_sequence_learning_with_neural_networks#Long_Short-Term_Memory_Recurrent_Neural_Network Summary]<br /> |-<br /> |Feb 27 || Peter Forsyth || 1|| Unsupervised Machine Translation Using Monolingual Corpora Only || [https://arxiv.org/pdf/1711.00043.pdf Paper] || <br /> |-<br /> |Feb 27 || Ilia Sucholutsky || 2|| || || <br /> |-<br /> |Feb 27 || || 3|| || || <br /> |-<br /> |Mar 1 || || 4|| || || <br /> |-<br /> |Mar 1 || wenqing liu || 5|| || || <br /> |-<br /> |Mar 1 || || 6|| || || <br /> |-<br /> |Mar 6 || George (Shiyang) Wen || 7|| AmbientGAN: Generative models from lossy measurements || [https://openreview.net/pdf?id=Hy7fDog0b Paper] || <br /> |-<br /> |Mar 6 || || 8|| || || <br /> |-<br /> |Mar 6 || || 9|| || || <br /> |-<br /> |Mar 8 || Alex (Xian) Wang || 10 || Self-Normalizing Neural Networks || [http://papers.nips.cc/paper/6698-self-normalizing-neural-networks.pdf Paper] || <br /> |-<br /> |Mar 8 || || 11|| || || <br /> |-<br /> |Mar 8 || Wei Tao Chen || 12|| || || <br /> |-<br /> |Mar 13 || Chunshang Li || 13 || UNDERSTANDING IMAGE MOTION WITH GROUP REPRESENTATIONS || [https://openreview.net/pdf?id=SJLlmG-AZ Paper] || <br /> |-<br /> |Mar 13 || Saifuddin Hitawala || 14 || Thinking Fast and Slow with Deep Learning and Tree Search || [https://papers.nips.cc/paper/7120-thinking-fast-and-slow-with-deep-learning-and-tree-search.pdf Paper] || <br /> |-<br /> |Mar 13 || Taylor Denouden || 15|| A neural representation of sketch drawings || [https://arxiv.org/pdf/1704.03477.pdf Paper] || [Summary]<br /> |-<br /> |Mar 15 || Zehao xu || 16|| Synthetic and natural noise both break neural machine translation || || <br /> |-<br /> |Mar 15 || Prarthana Bhattacharyya || 17|| || || <br /> |-<br /> |Mar 15 || Changjian Li || 18|| Imagination-Augmented Agents for Deep Reinforcement Learning || [https://papers.nips.cc/paper/7152-imagination-augmented-agents-for-deep-reinforcement-learning.pdf Paper] || <br /> |-<br /> |Mar 20 || Travis Dunn || 19|| Continuous Adaptation via Meta-Learning in Nonstationary and Competitive Environments || [https://openreview.net/pdf?id=Sk2u1g-0- Paper] || [Summary]<br /> |-<br /> |Mar 20 || Sushrut Bhalla || 20|| MaskRNN: Instance Level Video Object Segmentation || [https://papers.nips.cc/paper/6636-maskrnn-instance-level-video-object-segmentation.pdf Paper] || [Summary]<br /> |-<br /> |Mar 20 || Hamid Tahir || 21|| Wavelet Pooling for Convolution Neural Networks || [https://openreview.net/pdf?id=rkhlb8lCZ Paper] || Summary<br /> |-<br /> |Mar 22 || Dongyang Yang|| 22|| Don't Decay the Learning Rate, Increase the Batch Size || [https://arxiv.org/pdf/1711.00489.pdf Paper] || <br /> |-<br /> |Mar 22 || Yao Li || 23||Toward Multimodal Image-to-Image Translation || [http://papers.nips.cc/paper/6650-toward-multimodal-image-to-image-translation.pdf Paper] || <br /> |-<br /> |Mar 22 || Sahil Pereira || 24||End-to-End Differentiable Adversarial Imitation Learning|| [http://proceedings.mlr.press/v70/baram17a/baram17a.pdf Paper] || [http://proceedings.mlr.press/v70/baram17a/baram17a.pdf Summary]<br /> |-<br /> |Mar 27 || Jaspreet Singh Sambee || 25|| Gated Recurrent Convolution Neural Network for OCR || [http://papers.nips.cc/paper/6637-gated-recurrent-convolution-neural-network-for-ocr.pdf Paper] || <br /> |-<br /> |Mar 27 || Braden Hurl || 26|| Spherical CNNs || [https://openreview.net/pdf?id=Hkbd5xZRb Paper] || <br /> |-<br /> |Mar 27 || Marko Ilievski || 27|| Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders || [http://proceedings.mlr.press/v70/engel17a/engel17a.pdf Paper] || <br /> |-<br /> |Mar 29 || Alex Pon || 28||Wasserstein GAN || [https://arxiv.org/pdf/1701.07875.pdf Paper] ||<br /> |-<br /> |Mar 29 || Sean Walsh || 29||Improved Training of Wasserstein GANs || [https://arxiv.org/pdf/1704.00028.pdf Paper] ||<br /> |-<br /> |Mar 29 || Jason Ku || 30||MarrNet: 3D Shape Reconstruction via 2.5D Sketches ||[https://arxiv.org/pdf/1711.03129.pdf Paper] ||<br /> |-<br /> |Apr 3 || Tong Yang || 31|| Dynamic Routing Between Capsules. || [http://papers.nips.cc/paper/6975-dynamic-routing-between-capsules.pdf Paper] || <br /> |-<br /> |Apr 3 || Benjamin Skikos || 32|| Training and Inference with Integers in Deep Neural Networks || [https://openreview.net/pdf?id=HJGXzmspb Paper] || <br /> |-<br /> |Apr 3 || Weishi Chen || 33|| || || <br /> |-</div> S3wen