http://wiki.math.uwaterloo.ca/statwiki/api.php?action=feedcontributions&user=W285liu&feedformat=atomstatwiki - User contributions [US]2021-08-02T16:37:43ZUser contributionsMediaWiki 1.28.3http://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/IMPROVING_GANS_USING_OPTIMAL_TRANSPORT&diff=36439stat946w18/IMPROVING GANS USING OPTIMAL TRANSPORT2018-04-21T03:08:58Z<p>W285liu: </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 <math> \prod (p,g) </math> is the set of all joint distributions <math> \gamma (x,y) </math> with marginals <math> p(x) </math> (real data), <math> g(y) </math> (generated data). <math> c(x,y) </math> 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 <math> g(y) </math> and <math> p(x) </math> such that the generator distribution <math> g(y) </math> is similar to the real data distribution <math> p(x) </math>.<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 (<math> \beta </math>) to the joint distribution <math> \prod_{\beta} (p,g) </math>. This distance could be generalized to approximate the mini-batches of data <math> X ,Y</math> with <math> K </math> vectors of <math> x, y</math>. The <math> i, j </math> th entry of the cost matrix <math> C </math> can be interpreted as the cost it needs to transport the <math> x_i </math> in mini-batch X to the <math> y_i </math> in mini-batch <math>Y </math>. The resulting distance will be:<br />
<br />
[[File: equation5.png|550px]]<br />
<br />
where <math> M </math> is a <math> K \times K </math> matrix, each row of <math> M </math> is a joint distribution of <math> \gamma (x,y) </math> with positive entries. The summmation of rows or columns of <math> M </math> 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 <math> \mathcal{W}_{c} </math> 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 <math> x, x' </math> and <math> y, y'</math> are independent samples from data distribution <math> p </math> and generator distribution <math> g </math>, 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 <math> g(X), p(X) </math>. 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, <math> X, X' </math> and <math> Y, Y'</math> can be the independent samples from data distribution <math> p </math> and the generator distribution <math> g </math>, respectively. While in Generalized engergy distance, <math> X, X' </math> and <math> Y, Y'</math> can also be valid for mini-batches. The <math> D_{GED}(p,g) </math> is a metric when having <math> d </math> as a metric. Thus, taking the triangle inequality of <math> d </math> into account, <math> D(p,g) \geq 0,</math> and <math> D(p,g)=0 </math> when <math> p=g </math>.<br />
<br />
===Mini-Batch Energy Distance===<br />
As <math> d </math> is free to choose, authors proposed Mini-batch Energy Distance by using entropy-regularized Wasserstein distance as <math> d </math>. <br />
<br />
[[File: equation8.png|650px]]<br />
<br />
where <math> X, X' </math> and <math> Y, Y'</math> are independent sampled mini-batches from the data distribution <math> p </math> and the generator distribution <math> g </math>, respectively. This distance metric combines the energy distance with primal form of optimal transport over mini-batch distributions <math> g(Y) </math> and <math> p(X) </math>. Inside the generalized energy distance, the Sinkhorn distance is a valid metric between each mini-batches. By adding the <math> - \mathcal{W}_c (Y,Y')</math> and <math> \mathcal{W}_c (X,Y)</math> 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 <math>c(x,y)</math>. 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 <math>g</math> in higher dimensions for which the mini-batch energy distance is zero such that it is difficult to tell <math>p</math> and <math>g</math> 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 <math>g</math> and thereby become more discriminative. <br />
<br />
In practice, in order to secure the statistical efficiency (i.e. being able to tell <math>p</math> and <math>g</math> apart without requiring an enormous sample size when their distance is close to zero), authors suggested using cosine distance between vectors <math> v_\eta (x) </math> and <math> v_\eta (y) </math> 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 <math> v_\eta </math> 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 "inception score" 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 <math> 0.0003, \beta_1 = 0.5, \beta_2 = 0.999 </math> . 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 <math> 1 / \lambda </math> 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 <math>s</math> 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 <math> \mathcal{W}_c </math>. 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 />
<br />
= Discussion =<br />
We have presented OT-GAN, a new variant of GANs where the generator is trained to minimize<br />
a novel distance metric over probability distributions. This metric, which we call mini-batch energy<br />
distance, combines optimal transport in primal form with an energy distance defined in an<br />
adversarially learned feature space, resulting in a highly discriminative distance function with unbiased<br />
mini-batch gradients. OT-GAN was shown to be uniquely stable when trained with large<br />
mini-batches and to achieve state-of-the-art results on several common benchmarks.<br />
One downside of OT-GAN, as currently proposed, is that it requires large amounts of computation<br />
and memory. We achieve the best results when using very large mini-batches, which increases the<br />
time required for each update of the parameters. All experiments in this paper, except for the mixture<br />
of Gaussians toy example, were performed using 8 GPUs and trained for several days. In future work<br />
we hope to make the method more computationally efficient, as well as to scale up our approach to<br />
multi-machine training to enable generation of even more challenging and high resolution image<br />
data sets.<br />
A unique property of OT-GAN is that the mini-batch energy distance remains a valid training objective<br />
even when we stop training the critic. Our implementation of OT-GAN updates the generative<br />
model more often than the critic, where GANs typically do this the other way around (see e.g. Gulrajani<br />
et al., 2017). As a result we learn a relatively stable transport cost function c(x, y), describing<br />
how (dis)similar two images are, as well as an image embedding function vη(x) capturing the geometry<br />
of the training data. Preliminary experiments suggest these learned functions can be used<br />
successfully for unsupervised learning and other applications, which we plan to investigate further<br />
in future work.<br />
<br />
==Reference==<br />
Salimans, Tim, Han Zhang, Alec Radford, and Dimitris Metaxas. "Improving GANs using optimal transport." (2018).</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Implicit_Causal_Models_for_Genome-wide_Association_Studies&diff=36437stat946w18/Implicit Causal Models for Genome-wide Association Studies2018-04-21T03:04:43Z<p>W285liu: /* Causal Inference with a Latent Confounder */</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 <math>f</math> and a noise <math>n</math>. For working simplicity, we usually assume <math>f</math> 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 <math>\epsilon</math>, representing unknown background quantities which are jointly independent and global variable <math>\beta</math>, some function of this noise, where<br />
<br />
[[File: eq1.1.png|800px|center]]<br />
<br />
Each <math>\beta</math> and <math>x</math> is a function of noise; <math>y</math> is a function of noise and <math>x</math>，<br />
<br />
[[File: eqt1.png|800px|center]]<br />
<br />
The target is the causal mechanism <math>f_y</math> so that the causal effect <math>p(y|do(X=x),\beta)</math> can be calculated. <math>do(X=x)</math> means that we specify a value of <math>X</math> under the fixed structure <math>\beta</math>. By other paper’s work, it is assumed that <math>p(y|do(x),\beta) = p(y|x, \beta)</math>.<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 />
<math>f(.)</math> is usually a linear function or spline functions for nonlinearities. <math>\epsilon</math> is assumed to be standard normal, as well as <math>y</math>. Thus the posterior <math>p(\theta | x, y, \beta)</math> can be represented as <br />
<br />
[[File: eqt2.png|800px|center]]<br />
<br />
where <math>p(\theta)</math> 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 using an additive noise term, implicit causal models directly take noise <math>\epsilon</math> as input and outputs <math>x</math> given parameter <math>\theta</math>.<br />
<br />
<math><br />
x=g(\epsilon | \theta), \epsilon \tilde s(\cdot)<br />
</math><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 />
==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 />
Similar to before, the interest is the causal effect <math>p(y|do(x_m), x_{-m})</math>. Here, the SNPs other than <math>x_m</math> is also under consideration. However, it is confounded by the unobserved confounder <math>z_n</math>. 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 <math>n=1,…,N</math> and each SNP <math>m=1,…,M</math>,<br />
<br />
[[File: eqt4.png|800px|center]]<br />
<br />
<br />
The mechanism for latent confounder <math>z_n</math> 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 <math>\theta</math> is needed to be calculate in order to estimate the mechanism <math>g_y</math> as well as the causal effect <math>p(y|do(x_m), x_{-m})</math>, so that it can be explained how changes to each SNP <math>X_m</math> cause changes to the trait <math>Y</math>.<br />
<br />
[[File: eqt5.png|800px|center]]<br />
<br />
Note that the latent structure <math>p(z|x, y)</math> 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 <math>X_m → Y</math>. 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 <math>p(θ | x, y)<br />
</math> provides a consistent estimator of the causal mechanism <math>f_y</math>.<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 <math>z_n</math>, and thus we can estimate the causal mechanism given each data point’s confounder <math>z_n</math>. As more data points arrive (“N → ∞, M fixed”), we can estimate the causal mechanism given any confounder <math>z_n</math> 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 <math>z_n</math>.====<br />
The distribution of confounders is set as standard normal. <math>z_n \in R^K</math> , where <math>K</math> is the dimension of <math>z_n</math> and <math>K</math> should make the latent space as close as possible to the true population structural. <br />
<br />
====Generative Process of SNPs <math>x_{nm}</math>.====<br />
Given SNP is coded for,<br />
<br />
[[File: SNP.png|300px|center]]<br />
<br />
The authors defined a <math>Binomial(2,\pi_{nm})</math> distribution on <math>x_{nm}</math>. 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 <math>N*M</math> matrix due the the variables <math>w_m</math>, which act as principal component in PCA. Here, <math>\phi</math> has a standard normal prior distribution. The weights <math>w</math> and biases <math>\phi</math> are shared over the <math>m</math> SNPs and <math>n</math> individuals, which makes it possible to learn nonlinear interactions between <math>z_n</math> and <math>w_m</math>.<br />
<br />
====Generative Process of Traits <math>y_n</math>.====<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 <math>\theta</math> 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 <math>w_m</math> and <math>z_n</math> 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 <math>a</math> 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 "uncorrected" (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>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Implicit_Causal_Models_for_Genome-wide_Association_Studies&diff=36436stat946w18/Implicit Causal Models for Genome-wide Association Studies2018-04-21T03:03:21Z<p>W285liu: /* Probabilistic Causal Models */</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 <math>f</math> and a noise <math>n</math>. For working simplicity, we usually assume <math>f</math> 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 <math>\epsilon</math>, representing unknown background quantities which are jointly independent and global variable <math>\beta</math>, some function of this noise, where<br />
<br />
[[File: eq1.1.png|800px|center]]<br />
<br />
Each <math>\beta</math> and <math>x</math> is a function of noise; <math>y</math> is a function of noise and <math>x</math>，<br />
<br />
[[File: eqt1.png|800px|center]]<br />
<br />
The target is the causal mechanism <math>f_y</math> so that the causal effect <math>p(y|do(X=x),\beta)</math> can be calculated. <math>do(X=x)</math> means that we specify a value of <math>X</math> under the fixed structure <math>\beta</math>. By other paper’s work, it is assumed that <math>p(y|do(x),\beta) = p(y|x, \beta)</math>.<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 />
<math>f(.)</math> is usually a linear function or spline functions for nonlinearities. <math>\epsilon</math> is assumed to be standard normal, as well as <math>y</math>. Thus the posterior <math>p(\theta | x, y, \beta)</math> can be represented as <br />
<br />
[[File: eqt2.png|800px|center]]<br />
<br />
where <math>p(\theta)</math> 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 using an additive noise term, implicit causal models directly take noise <math>\epsilon</math> as input and outputs <math>x</math> given parameter <math>\theta</math>.<br />
<br />
<math><br />
x=g(\epsilon | \theta), \epsilon \tilde s(\cdot)<br />
</math><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 />
==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 <math>p(y|do(x_m), x_{-m})</math>. Here, the SNPs other than <math>x_m</math> is also under consideration. However, it is confounded by the unobserved confounder <math>z_n</math>. 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 <math>n=1,…,N</math> and each SNP <math>m=1,…,M</math>,<br />
<br />
[[File: eqt4.png|800px|center]]<br />
<br />
<br />
The mechanism for latent confounder <math>z_n</math> 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 <math>\theta</math> is needed to be calculate in order to estimate the mechanism <math>g_y</math> as well as the causal effect <math>p(y|do(x_m), x_{-m})</math>, so that it can be explained how changes to each SNP <math>X_m</math> cause changes to the trait <math>Y</math>.<br />
<br />
[[File: eqt5.png|800px|center]]<br />
<br />
Note that the latent structure <math>p(z|x, y)</math> 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 <math>X_m → Y</math>. 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 <math>p(θ | x, y)<br />
</math> provides a consistent estimator of the causal mechanism <math>f_y</math>.<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 <math>z_n</math>, and thus we can estimate the causal mechanism given each data point’s confounder <math>z_n</math>. As more data points arrive (“N → ∞, M fixed”), we can estimate the causal mechanism given any confounder <math>z_n</math> 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 <math>z_n</math>.====<br />
The distribution of confounders is set as standard normal. <math>z_n \in R^K</math> , where <math>K</math> is the dimension of <math>z_n</math> and <math>K</math> should make the latent space as close as possible to the true population structural. <br />
<br />
====Generative Process of SNPs <math>x_{nm}</math>.====<br />
Given SNP is coded for,<br />
<br />
[[File: SNP.png|300px|center]]<br />
<br />
The authors defined a <math>Binomial(2,\pi_{nm})</math> distribution on <math>x_{nm}</math>. 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 <math>N*M</math> matrix due the the variables <math>w_m</math>, which act as principal component in PCA. Here, <math>\phi</math> has a standard normal prior distribution. The weights <math>w</math> and biases <math>\phi</math> are shared over the <math>m</math> SNPs and <math>n</math> individuals, which makes it possible to learn nonlinear interactions between <math>z_n</math> and <math>w_m</math>.<br />
<br />
====Generative Process of Traits <math>y_n</math>.====<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 <math>\theta</math> 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 <math>w_m</math> and <math>z_n</math> 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 <math>a</math> 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 "uncorrected" (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>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Implicit_Causal_Models_for_Genome-wide_Association_Studies&diff=36435stat946w18/Implicit Causal Models for Genome-wide Association Studies2018-04-21T03:03:11Z<p>W285liu: /* Probabilistic Causal Models */</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 <math>f</math> and a noise <math>n</math>. For working simplicity, we usually assume <math>f</math> 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 <math>\epsilon</math>, representing unknown background quantities which are jointly independent and global variable <math>\beta</math>, some function of this noise, where<br />
<br />
[[File: eq1.1.png|800px|center]]<br />
<br />
Each <math>\beta</math> and <math>x</math> is a function of noise; <math>y</math> is a function of noise and <math>x</math>，<br />
<br />
[[File: eqt1.png|800px|center]]<br />
<br />
The target is the causal mechanism <math>f_y</math> so that the causal effect <math>p(y|do(X=x),\beta)</math> can be calculated. <math>do(X=x)</math> means that we specify a value of <math>X</math> under the fixed structure <math>\beta</math>. By other paper’s work, it is assumed that <math>p(y|do(x),\beta) = p(y|x, \beta)</math>.<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 />
<math>f(.)</math> is usually a linear function or spline functions for nonlinearities. <math>\epsilon</math> is assumed to be standard normal, as well as <math>y</math>. Thus the posterior <math>p(\theta | x, y, \beta)</math> can be represented as <br />
<br />
[[File: eqt2.png|800px|center]]<br />
<br />
where <math>p(\theta)</math> is the prior which is known.<br />
<br />
===Implicit Causal Models===<br />
The difference between implicit causal models and probabilistic causal models is the noise variable. Instead of using an additive noise term, implicit causal models directly take noise <math>\epsilon</math> as input and outputs <math>x</math> given parameter <math>\theta</math>.<br />
<br />
<math><br />
x=g(\epsilon | \theta), \epsilon \tilde s(\cdot)<br />
</math><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 />
==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 <math>p(y|do(x_m), x_{-m})</math>. Here, the SNPs other than <math>x_m</math> is also under consideration. However, it is confounded by the unobserved confounder <math>z_n</math>. 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 <math>n=1,…,N</math> and each SNP <math>m=1,…,M</math>,<br />
<br />
[[File: eqt4.png|800px|center]]<br />
<br />
<br />
The mechanism for latent confounder <math>z_n</math> 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 <math>\theta</math> is needed to be calculate in order to estimate the mechanism <math>g_y</math> as well as the causal effect <math>p(y|do(x_m), x_{-m})</math>, so that it can be explained how changes to each SNP <math>X_m</math> cause changes to the trait <math>Y</math>.<br />
<br />
[[File: eqt5.png|800px|center]]<br />
<br />
Note that the latent structure <math>p(z|x, y)</math> 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 <math>X_m → Y</math>. 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 <math>p(θ | x, y)<br />
</math> provides a consistent estimator of the causal mechanism <math>f_y</math>.<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 <math>z_n</math>, and thus we can estimate the causal mechanism given each data point’s confounder <math>z_n</math>. As more data points arrive (“N → ∞, M fixed”), we can estimate the causal mechanism given any confounder <math>z_n</math> 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 <math>z_n</math>.====<br />
The distribution of confounders is set as standard normal. <math>z_n \in R^K</math> , where <math>K</math> is the dimension of <math>z_n</math> and <math>K</math> should make the latent space as close as possible to the true population structural. <br />
<br />
====Generative Process of SNPs <math>x_{nm}</math>.====<br />
Given SNP is coded for,<br />
<br />
[[File: SNP.png|300px|center]]<br />
<br />
The authors defined a <math>Binomial(2,\pi_{nm})</math> distribution on <math>x_{nm}</math>. 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 <math>N*M</math> matrix due the the variables <math>w_m</math>, which act as principal component in PCA. Here, <math>\phi</math> has a standard normal prior distribution. The weights <math>w</math> and biases <math>\phi</math> are shared over the <math>m</math> SNPs and <math>n</math> individuals, which makes it possible to learn nonlinear interactions between <math>z_n</math> and <math>w_m</math>.<br />
<br />
====Generative Process of Traits <math>y_n</math>.====<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 <math>\theta</math> 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 <math>w_m</math> and <math>z_n</math> 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 <math>a</math> 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 "uncorrected" (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>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=End-to-End_Differentiable_Adversarial_Imitation_Learning&diff=36434End-to-End Differentiable Adversarial Imitation Learning2018-04-21T03:00:30Z<p>W285liu: /* Continuous Action Distributions */</p>
<hr />
<div>= Introduction =<br />
The ability to imitate an expert policy is very beneficial in the case of automating human demonstrated tasks. Assuming that a sequence of state action pairs (trajectories) of an expert policy are available, a new policy can be trained that imitates the expert without having access to the original reward signal used by the expert. There are two main approaches to solve the problem of imitating a policy; they are Behavioural Cloning (BC) and Inverse Reinforcement Learning (IRL). BC directly learns the conditional distribution of actions over states in a supervised fashion by training on single time-step state-action pairs. The disadvantage of BC is that the training requires large amounts of expert data, which is hard to obtain. In addition, an agent trained using BC is unaware of how its action can affect future state distribution. The second method using IRL involves recovering a reward signal under which the expert is uniquely optimal; the main disadvantage is that it’s an ill-posed problem.<br />
<br />
To address the problem of imitating an expert policy, techniques based on Generative Adversarial Networks (GANs) have been proposed in recent years. GANs use a discriminator to guide the generative model towards producing patterns like those of the expert. The generator is guided as it tries to produce samples on the correct side of the discriminators decision boundary hyper-plane, as seen in Figure 1. This idea was used by (Ho & Ermon, 2016)<sup>[[#References|[2]]]</sup> in their work titled Generative Adversarial Imitation Learning (GAIL) to imitate an expert policy in a model-free setup. A model free setup is the one where the agent cannot make predictions about what the next state and reward will be before it takes each action since the transition function to move from state A to state B is not learned. <br />
<br />
The disadvantage of the model-free approach comes to light when training stochastic policies. The presence of stochastic elements breaks the flow of information (gradients) from one neural network to the other, thus prohibiting the use of backpropagation. In this situation, a standard solution is to use gradient estimation (Williams, 1992)<sup>[[#References|[8]]]</sup>. This tends to suffer from high variance, resulting in a need for larger sample sizes as well as variance reduction methods. This paper proposes a model-based imitation learning algorithm (MGAIL), in which information propagates from the guiding neural network (D) to the generative model (G), which in this case represents the policy <math>\pi</math> that is to be trained. Training policy <math>\pi</math> assumes the existence of an expert policy <math>\pi_{E}</math> with given trajectories <math>\{s_{0},a_{0},s_{1},...\}^{N}_{i=0}</math> which it aims to imitate without access to the original reward signal <math>r_{e}</math>. This is achieved by two steps: (1) learning a forward model that approximates the environment’s dynamics (2) building an end-to-end differentiable computation graph that spans over multiple time-steps. The gradient in such a graph carries information from future states to earlier time-steps, helping the policy to account for compounding errors.<br />
<br />
<br />
[[File:GeneratorFollowingDiscriminator.png|center]]<br />
<br />
Figure 1: '''Illustration of GANs.''' The generative model follows the discriminating hyper-plane defined by the discriminator. Eventually, G will produce patterns similar to the expert patterns.<br />
<br />
= Background =<br />
== Markov Decision Process ==<br />
Consider an infinite-horizon discounted Markov decision process (MDP), defined by the tuple <math>(S, A, P, r, \rho_0, \gamma)</math> where <math>S</math> is the set of states, <math>A</math> is a set of actions, <math>P :<br />
S × A × S → [0, 1]</math> is the transition probability distribution, <math>r : (S × A) → R</math> is the reward function, <math>\rho_0 : S → [0, 1]</math> is the distribution over initial states, and <math>γ ∈ (0, 1)</math> is the discount factor. Let <math>π</math> denote a stochastic policy <math>π : S × A → [0, 1]</math>, <math>R(π)</math> denote its expected discounted reward: <math>E_πR = E_π [\sum_{t=0}^T \gamma^t r_t]</math> and <math>τ</math> denote a trajectory of states and actions <math>τ = {s_0, a_0, s_1, a_1, ...}</math>.<br />
<br />
== Imitation Learning ==<br />
A common technique for performing imitation learning is to train a policy <math> \pi </math> that minimizes some loss function <math> l(s, \pi(s)) </math> with respect to a discounted state distribution encountered by the expert: <math> d_\pi(s) = (1-\gamma)\sum_{t=0}^{\infty}\gamma^t p(s_t) </math>. This can be obtained using any supervised learning (SL) algorithm with <math> d_\pi(s) = argmin_{\pi \in \prod}\mathbb{E}_{s \sim d_{\pi}}[l(s,\pi (s))]</math>, but the policy's prediction affects future state distributions; this violates the independent and identically distributed (i.i.d) assumption made by most SL algorithms. This process is susceptible to compounding errors since a slight deviation in the learner's behavior can lead to different state distributions not encountered by the expert policy. <br />
<br />
This issue was overcome through the use of the Forward Training (FT) algorithm which trains a non-stationary policy iteratively over time. At each time step a new policy is trained on the state distribution induced by the previously trained policies <math>\pi_0</math>, <math>\pi_1</math>, ...<math>\pi_{t-1}</math>. This is continued till the end of the time horizon to obtain a policy that can mimic the expert policy. This requirement to train a policy at each time step till the end makes the FT algorithm impractical for cases where the time horizon is very large or undefined. This shortcoming is resolved using the Stochastic Mixing Iterative Learning (SMILe) algorithm. SMILe trains a stochastic stationary policy over several iterations under the trajectory distribution induced by the previously trained policy: <math> \pi_t = \pi_{t-1} + \alpha (1 - \alpha)^{t-1}(\hat{\pi}_t - \pi_0)</math>, with <math>\pi_0</math> following expert's policy at the start of training.<br />
<br />
== Generative Adversarial Networks ==<br />
GANs learn a generative model that can fool the discriminator by using a two-player zero-sum game:<br />
<br />
\begin{align} <br />
\underset{G}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{x\sim p_E}[log(D(x)]\ +\ \mathbb{E}_{z\sim p_z}[log(1 - D(G(z)))]<br />
\end{align}<br />
<br />
In the above equation, <math> p_E </math> represents the expert distribution and <math> p_z </math> represents the input noise distribution from which the input to the generator is sampled. The generator produces patterns and the discriminator judges if the pattern was generated or from the expert data. When the discriminator cannot distinguish between the two distributions the game ends and the generator has learned to mimic the expert. GANs rely on basic ideas such as binary classification and algorithms such as backpropagation in order to learn the expert distribution.<br />
<br />
GAIL applies GANs to the task of imitating an expert policy in a model-free approach. GAIL uses similar objective functions like GANs, but the expert distribution in GAIL represents the joint distribution over state action tuples:<br />
<br />
\begin{align} <br />
\underset{\pi}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{\pi}[log(D(s,a)]\ +\ \mathbb{E}_{\pi_E}[log(1 - D(s,a))] - \lambda H(\pi))<br />
\end{align}<br />
<br />
where <math> H(\pi) \triangleq \mathbb{E}_{\pi}[-log\: \pi(a|s)]</math> is the entropy.<br />
<br />
This problem cannot be solved using the standard methods described for GANs because the generator in GAIL represents a stochastic policy. The exact form of the first term in the above equation is given by: <math> \mathbb{E}_{s\sim \rho_\pi(s)}\mathbb{E}_{a\sim \pi(\cdot |s)} [log(D(s,a)] </math>.<br />
<br />
The two-player game now depends on the stochastic properties (<math> \theta </math>) of the policy, and it is unclear how to differentiate the above equation with respect to <math> \theta </math>. This problem can be overcome using score functions such as REINFORCE to obtain an unbiased gradient estimation:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi} [log\; D(s,a)] \cong \hat{\mathbb{E}}_{\tau_i}[\nabla_\theta\; log\; \pi_\theta(a|s)Q(s,a)]<br />
\end{align}<br />
<br />
where <math> Q(\hat{s},\hat{a}) </math> is the score function of the gradient:<br />
<br />
\begin{align}<br />
Q(\hat{s},\hat{a}) = \hat{\mathbb{E}}_{\tau_i}[log\; D(s,a) | s_0 = \hat{s}, a_0 = \hat{a}]<br />
\end{align}<br />
<br />
<br />
REINFORCE gradients suffer from high variance which makes them difficult to work with even after applying variance reduction techniques. While recent general variance reduction techniques like RELAX (Grathwohl et al., 2017)<sup>[[#References|[7]]]</sup> work well, they rely on multiple evaluations of the loss function or learning a surrogate neural network. Unfortunately, this is too computationally difficult for our task. In order to better understand the changes required to fool the discriminator we need access to the gradients of the discriminator network, which can be obtained from the Jacobian of the discriminator. This paper demonstrates the use of a forward model along with the Jacobian of the discriminator to train a policy, without using high-variance gradient estimations.<br />
<br />
= Algorithm =<br />
This section first analyzes the characteristics of the discriminator network, then describes how a forward model can enable policy imitation through GANs. Lastly, the model based adversarial imitation learning algorithm is presented.<br />
<br />
== The discriminator network ==<br />
The discriminator network is trained to predict the conditional distribution: <math> D(s,a) = p(y|s,a) </math> where <math> y \in (\pi_E, \pi) </math>. <math>D(s,a)</math> here is the likelihood ratio with the pair <math>{s,a}</math> generated by <math>\pi</math>.<br />
<br />
The discriminator is trained on an even distribution of expert and generated examples; hence <math> p(\pi) = p(\pi_E) = \frac{1}{2} </math>. Given this and applying Bayes' theorem, we can rearrange and factor <math> D(s,a) </math> to obtain:<br />
<br />
\begin{aligned}<br />
D(s,a) &= p(\pi|s,a) \\<br />
& = \frac{p(s,a|\pi)p(\pi)}{p(s,a|\pi)p(\pi) + p(s,a|\pi_E)p(\pi_E)} \\<br />
& = \frac{p(s,a|\pi)}{p(s,a|\pi) + p(s,a|\pi_E)} \\<br />
& = \frac{1}{1 + \frac{p(s,a|\pi_E)}{p(s,a|\pi)}} \\<br />
& = \frac{1}{1 + \frac{p(a|s,\pi_E)}{p(a|s,\pi)} \cdot \frac{p(s|\pi_E)}{p(s|\pi)}} \\<br />
\end{aligned}<br />
<br />
Define <math> \varphi(s,a) </math> and <math> \psi(s) </math> to be:<br />
<br />
\begin{aligned}<br />
\varphi(s,a) = \frac{p(a|s,\pi_E)}{p(a|s,\pi)}, \psi(s) = \frac{p(s|\pi_E)}{p(s|\pi)}<br />
\end{aligned}<br />
<br />
to get the final expression for <math> D(s,a) </math>:<br />
\begin{aligned}<br />
D(s,a) = \frac{1}{1 + \varphi(s,a)\cdot \psi(s)}<br />
\end{aligned}<br />
<br />
<math> \varphi(s,a) </math> represents a policy likelihood ratio, and <math> \psi(s) </math> represents a state distribution likelihood ratio. Based on these expressions, the paper states that the discriminator makes its decisions by answering two questions. The first question relates to state distribution: what is the likelihood of encountering state <math> s </math> under the distribution induces by <math> \pi_E </math> vs <math> \pi </math>? The second question is about behavior: given a state <math> s </math>, how likely is action a under <math> \pi_E </math> vs <math> \pi </math>? The desired change in state is given by <math> \psi_s \equiv \partial \psi / \partial s </math>; this information can by obtained from the partial derivatives of <math> D(s,a) </math>, which is why these derivatives are proposed to be used for training policies (see following sections):<br />
<br />
\begin{aligned}<br />
\nabla_aD &= - \frac{\varphi_a(s,a)\psi(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\nabla_sD &= - \frac{\varphi_s(s,a)\psi(s) + \varphi(s,a)\psi_s(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\end{aligned}<br />
<br />
== Backpropagating through stochastic units ==<br />
There is interest in training stochastic policies because stochasticity encourages exploration for Policy Gradient methods. This is a problem for algorithms that build differentiable computation graphs where the gradients flow from one component to another since it is unclear how to backpropagate through stochastic units. The following subsections show how to estimate the gradients of continuous and categorical stochastic elements for continuous and discrete action domains respectively.<br />
<br />
=== Continuous Action Distributions ===<br />
In the case of continuous action policies, re-parameterization was used to enable computing the derivatives of stochastic models. Assuming that the stochastic policy has a Gaussian distribution <math> \mathcal{N}(\mu_{\theta} (s), \sigma_{\theta}^2 (s))</math>, where the mean and variance are given by some deterministic functions <math>\mu_{\theta}</math> and <math>\sigma_{\theta}</math>, then the policy <math> \pi </math> can be written as <math> \pi_\theta(a|s) = \mu_\theta(s) + \xi \sigma_\theta(s) </math>, where <math> \xi \sim N(0,1) </math>. This way, the authors are able to get a Monte-Carlo estimator of the derivative of the expected value of <math> D(s, a) </math> with respect to <math> \theta </math>:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi(a|s)}D(s,a) = \mathbb{E}_{\rho (\xi )}\nabla_a D(a,s) \nabla_\theta \pi_\theta(a|s) \cong \frac{1}{M}\sum_{i=1}^{M} \nabla_a D(s,a) \nabla_\theta \pi_\theta(a|s)\Bigr|_{\substack{\xi=\xi_i}}<br />
\end{align}<br />
<br />
=== Categorical Action Distributions ===<br />
In the case of discrete action domains, the paper uses categorical re-parameterization with Gumbel-Softmax. This method relies on the Gumbel-Max trick which is a method for drawing samples from a categorical distribution with class probabilities <math> \pi(a_1|s),\pi(a_2|s),...,\pi(a_N|s) </math>:<br />
<br />
\begin{align}<br />
a_{argmax} = \underset{i}{argmax}[g_i + log\ \pi(a_i|s)]\textrm{, where } g_i \sim Gumbel(0, 1).<br />
\end{align}<br />
<br />
Gumbel-Softmax provides a differentiable approximation of the samples obtained using the Gumbel-Max trick (Gumbel-softmax allows us to generate a differentiable sample from a discrete distribution, which is needed in this trajectory imitation setting.):<br />
<br />
\begin{align}<br />
a_{softmax} = \frac{exp[\frac{1}{\tau}(g_i + log\ \pi(a_i|s))]}{\sum_{j=1}^{k}exp[\frac{1}{\tau}(g_j + log\ \pi(a_i|s))]}<br />
\end{align}<br />
<br />
<br />
In the above equation, the hyper-parameter <math> \tau </math> (temperature) trades bias for variance. When <math> \tau </math> gets closer to zero, the softmax operator acts like argmax resulting in a low bias, but high variance; vice versa when the <math> \tau </math> is large.<br />
<br />
The authors use <math> a_{softmax} </math> to interact with the environment; argmax is applied over <math> a_{softmax} </math> to obtain a single “pure” action, but the continuous approximation is used in the backward pass using the estimation: <math> \nabla_\theta\; a_{argmax} \approx \nabla_\theta\; a_{softmax} </math>.<br />
<br />
== Backpropagating through a Forward model ==<br />
The above subsections presented the means for extracting the partial derivative <math> \nabla_aD </math>. The main contribution of this paper is incorporating the use of <math> \nabla_sD </math>. In a model-free approach the state <math> s </math> is treated as a fixed input, therefore <math> \nabla_sD </math> is discarded. This is illustrated in Figure 2. This work uses a model-based approach which makes incorporating <math> \nabla_sD </math> more involved. In the model-based approach, a state <math> s_t </math> can be written as a function of the previous state action pair: <math> s_t = f(s_{t-1}, a_{t-1}) </math>, where <math> f </math> represents the forward model. Using the forward model and the law of total derivatives we get:<br />
<br />
\begin{align}<br />
\nabla_\theta D(s_t,a_t)\Bigr|_{\substack{s=s_t, a=a_t}} &= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_t}} \\<br />
&= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\left (\frac{\partial f}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_{t-1}}} + \frac{\partial f}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_{t-1}}} \right )<br />
\end{align}<br />
<br />
<br />
Using this formula, the error regarding deviations of future states <math> (\psi_s) </math> propagate back in time and influence the actions of policies in earlier times. This is summarized in Figure 3.<br />
<br />
[[File:modelFree_blockDiagram.PNG|400px|center]]<br />
<br />
Figure 2: Block-diagram of the model-free approach: given a state <math> s </math>, the policy outputs <math> \mu </math> which is fed to a stochastic sampling unit. An action <math> a </math> is sampled, and together with <math> s </math> are presented to the discriminator network. In the backward phase, the error message <math> \delta_a </math> is blocked at the stochastic sampling unit. From there, a high-variance gradient estimation is used (<math> \delta_{HV} </math>). Meanwhile, the error message <math> \delta_s </math> is flushed.<br />
<br />
[[File:modelBased_blockDiagram.PNG|700px|center]]<br />
<br />
Figure 3: Block diagram of model-based adversarial imitation learning. <br />
<br />
Figure 3 describes the computation graph for training the policy (i.e. G). The discriminator network D is fixed at this stage and is trained separately. At time <math> t </math> of the forward pass, <math> \pi </math> outputs a distribution over actions: <math> \mu_t = \pi(s_t) </math>, from which an action at is sampled. For example, in the continuous case, this is done using the re-parametrization trick: <math> a_t = \mu_t + \xi \cdot \sigma </math>, where <math> \xi \sim N(0,1) </math>. The next state <math> s_{t+1} = f(s_t, a_t) </math> is computed using the forward model (which is also trained separately), and the entire process repeats for time <math> t+1 </math>. In the backward pass, the gradient of <math> \pi </math> is comprised of a.) the error message <math> \delta_a </math> (Green) that propagates fluently through the differentiable approximation of the sampling process. And b.) the error message <math> \delta_s </math> (Blue) of future time-steps, that propagate back through the differentiable forward model.<br />
<br />
== MGAIL Algorithm ==<br />
Shalev-Shwartz et al. (2016)<sup>[[#References|[3]]]</sup> and Heess et al. (2015)<sup>[[#References|[4]]]</sup> built a multi-step computation graph for describing the familiar policy gradient objective; in this case it is given by:<br />
<br />
\begin{align}<br />
J(\theta) = \mathbb{E}\left [ \sum_{t=0}^{T} \gamma ^t D(s_t,a_t)|\theta\right ]<br />
\end{align}<br />
<br />
<br />
Using the results from Heess et al. (2015)<sup>[[#References|[4]]]</sup> this paper demonstrates how to differentiate <math> J(\theta) </math> over a trajectory of <math>(s,a,s’) </math> transitions:<br />
<br />
\begin{align}<br />
J_s &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_s + D_a \pi_s + \gamma J'_{s'}(f_s + f_a \pi_s) \right] \\<br />
J_\theta &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_a \pi_\theta + \gamma (J'_{s'} f_a \pi_\theta + J'_\theta) \right]<br />
\end{align}<br />
<br />
The policy gradient <math> \nabla_\theta J </math> is calculated by applying equations 12 and 13 recursively for <math> T </math> iterations. The MGAIL algorithm is presented below.<br />
<br />
[[File:MGAIL_alg.PNG]]<br />
<br />
== Forward Model Structure ==<br />
The stability of the learning process depends on the prediction accuracy of the forward model, but learning an accurate forward model is challenging by itself. The authors propose methods for improving the performance of the forward model based on two aspects of its functionality. First, the forward model should learn to use the action as an operator over the state space. To accomplish this, the actions and states, which are sampled form different distributions need to be first represented in a shared space. This is done by encoding the state and action using two separate neural networks and combining their outputs to form a single vector. Additionally, multiple previous states are used to predict the next state by representing the environment as an <math> n^{th} </math> order MDP. A gated recurrent units (GRU, a simpler variant on the LSTM model) layer is incorporated into the state encoder to enable recurrent connections from previous states. Using these modifications, the model is able to achieve better, and more stable results compared to the standard forward model based on a feed forward neural network. The comparison is presented in Figure 4.<br />
<br />
[[File:performance_comparison.PNG]]<br />
<br />
Figure 4: Performance comparison between a basic forward model (Blue), and the advanced forward model (Green).<br />
<br />
= Experiments =<br />
The proposed algorithm is evaluated on three discrete control tasks (Cartpole, Mountain-Car, Acrobot) and five continuous control tasks (Hopper, Walker, Half-Cheetah, Ant, and Humanoid). These tasks are modelled by the MuJoCo physics simulator (Todorov et al., 2012)<sup>[[#References|[9]]]</sup>, contain second order dynamics and utilize direct torque control. Expert policies are trained using the Trust Region Policy Optimization (TRPO) algorithm (Schulman et al., 2015)<sup>[[#References|[5]]]</sup>. Different number of trajectories are used to train the expert for each task, but all trajectories are of length 1000.<br />
The discriminator and generator (policy) networks contains two hidden layers with ReLU non-linearities and are trained using the ADAM optimizer. The total reward received over a period of <math> N </math> steps using BC, GAIL and MGAIL is presented in Table 1. The proposed algorithm achieved the highest reward for most environments while exhibiting performance comparable to the expert over all of them. A comparison between the basic forward model and the more advanced forward model is also made and described in the previous section of this summary. The two models compared are shown below.<br />
<br />
[[File:baram17_forward.PNG]]<br />
<br />
[[File:mgail_test_results_1.PNG]]<br />
<br />
[[File:mgail_test_results.PNG]]<br />
<br />
Table 1. Policy performance, <math> \pm </math> represents one standard deviation, a higher (reward) value is better. MGAIL consistently outperforms both GAIL and Behavioural cloning approaches, except on the Cartpole, where MGAIL and GAIL perform equally.<br />
<br />
= Discussion =<br />
This paper presented a model-free algorithm for imitation learning. It demonstrated how a forward model can be used to train policies using the exact gradient of the discriminator network. A downside of this approach is the need to learn a forward model; this could be difficult in certain domains. Learning the system dynamics directly from raw images is considered as one line of future work. Another future work is to address the violation of the fundamental assumption made by all supervised learning algorithms, which requires the data to be i.i.d. This problem arises because the discriminator and forward models are trained in a supervised learning fashion using data sampled from a dynamic distribution. The authors tried a solution proposed by another paper (Loshchilov & Hutter, 2016)<sup>[[#References|[10]]]</sup>, which is to reset the learning rate several times during training period, but it did not result in significant improvements.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper: https://github.com/itaicaspi/mgail. The repository provides the source code written by the authors, in Tensorflow.<br />
<br />
= Source =<br />
# Baram, Nir, et al. "End-to-end differentiable adversarial imitation learning." International Conference on Machine Learning. 2017.<br />
# Ho, Jonathan, and Stefano Ermon. "Generative adversarial imitation learning." Advances in Neural Information Processing Systems. 2016.<br />
# Shalev-Shwartz, Shai, et al. "Long-term planning by short-term prediction." arXiv preprint arXiv:1602.01580 (2016).<br />
# Heess, Nicolas, et al. "Learning continuous control policies by stochastic value gradients." Advances in Neural Information Processing Systems. 2015.<br />
# Schulman, John, et al. "Trust region policy optimization." International Conference on Machine Learning. 2015.<br />
# Caspi, I. (n.d.). Itaicaspi/mgail. Retrieved March 25, 2018, from https://github.com/itaicaspi/mgail.<br />
# Grathwohl, W., Choi, D., Wu, Y., Roeder, G., & Duvenaud, D. (2017). Backpropagation through the Void: Optimizing control variates for black-box gradient estimation. arXiv preprint arXiv:1711.00123.<br />
# Williams, Ronald J. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.<br />
# Todorov, Emanuel, Erez, Tom, and Tassa, Yuval. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026–5033. IEEE, 2012.<br />
# Loshchilov, Ilya and Hutter, Frank. Sgdr: Stochastic gradient descent with restarts. arXiv preprint arXiv:1608.03983, 2016.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=End-to-End_Differentiable_Adversarial_Imitation_Learning&diff=36433End-to-End Differentiable Adversarial Imitation Learning2018-04-21T03:00:18Z<p>W285liu: /* Continuous Action Distributions */</p>
<hr />
<div>= Introduction =<br />
The ability to imitate an expert policy is very beneficial in the case of automating human demonstrated tasks. Assuming that a sequence of state action pairs (trajectories) of an expert policy are available, a new policy can be trained that imitates the expert without having access to the original reward signal used by the expert. There are two main approaches to solve the problem of imitating a policy; they are Behavioural Cloning (BC) and Inverse Reinforcement Learning (IRL). BC directly learns the conditional distribution of actions over states in a supervised fashion by training on single time-step state-action pairs. The disadvantage of BC is that the training requires large amounts of expert data, which is hard to obtain. In addition, an agent trained using BC is unaware of how its action can affect future state distribution. The second method using IRL involves recovering a reward signal under which the expert is uniquely optimal; the main disadvantage is that it’s an ill-posed problem.<br />
<br />
To address the problem of imitating an expert policy, techniques based on Generative Adversarial Networks (GANs) have been proposed in recent years. GANs use a discriminator to guide the generative model towards producing patterns like those of the expert. The generator is guided as it tries to produce samples on the correct side of the discriminators decision boundary hyper-plane, as seen in Figure 1. This idea was used by (Ho & Ermon, 2016)<sup>[[#References|[2]]]</sup> in their work titled Generative Adversarial Imitation Learning (GAIL) to imitate an expert policy in a model-free setup. A model free setup is the one where the agent cannot make predictions about what the next state and reward will be before it takes each action since the transition function to move from state A to state B is not learned. <br />
<br />
The disadvantage of the model-free approach comes to light when training stochastic policies. The presence of stochastic elements breaks the flow of information (gradients) from one neural network to the other, thus prohibiting the use of backpropagation. In this situation, a standard solution is to use gradient estimation (Williams, 1992)<sup>[[#References|[8]]]</sup>. This tends to suffer from high variance, resulting in a need for larger sample sizes as well as variance reduction methods. This paper proposes a model-based imitation learning algorithm (MGAIL), in which information propagates from the guiding neural network (D) to the generative model (G), which in this case represents the policy <math>\pi</math> that is to be trained. Training policy <math>\pi</math> assumes the existence of an expert policy <math>\pi_{E}</math> with given trajectories <math>\{s_{0},a_{0},s_{1},...\}^{N}_{i=0}</math> which it aims to imitate without access to the original reward signal <math>r_{e}</math>. This is achieved by two steps: (1) learning a forward model that approximates the environment’s dynamics (2) building an end-to-end differentiable computation graph that spans over multiple time-steps. The gradient in such a graph carries information from future states to earlier time-steps, helping the policy to account for compounding errors.<br />
<br />
<br />
[[File:GeneratorFollowingDiscriminator.png|center]]<br />
<br />
Figure 1: '''Illustration of GANs.''' The generative model follows the discriminating hyper-plane defined by the discriminator. Eventually, G will produce patterns similar to the expert patterns.<br />
<br />
= Background =<br />
== Markov Decision Process ==<br />
Consider an infinite-horizon discounted Markov decision process (MDP), defined by the tuple <math>(S, A, P, r, \rho_0, \gamma)</math> where <math>S</math> is the set of states, <math>A</math> is a set of actions, <math>P :<br />
S × A × S → [0, 1]</math> is the transition probability distribution, <math>r : (S × A) → R</math> is the reward function, <math>\rho_0 : S → [0, 1]</math> is the distribution over initial states, and <math>γ ∈ (0, 1)</math> is the discount factor. Let <math>π</math> denote a stochastic policy <math>π : S × A → [0, 1]</math>, <math>R(π)</math> denote its expected discounted reward: <math>E_πR = E_π [\sum_{t=0}^T \gamma^t r_t]</math> and <math>τ</math> denote a trajectory of states and actions <math>τ = {s_0, a_0, s_1, a_1, ...}</math>.<br />
<br />
== Imitation Learning ==<br />
A common technique for performing imitation learning is to train a policy <math> \pi </math> that minimizes some loss function <math> l(s, \pi(s)) </math> with respect to a discounted state distribution encountered by the expert: <math> d_\pi(s) = (1-\gamma)\sum_{t=0}^{\infty}\gamma^t p(s_t) </math>. This can be obtained using any supervised learning (SL) algorithm with <math> d_\pi(s) = argmin_{\pi \in \prod}\mathbb{E}_{s \sim d_{\pi}}[l(s,\pi (s))]</math>, but the policy's prediction affects future state distributions; this violates the independent and identically distributed (i.i.d) assumption made by most SL algorithms. This process is susceptible to compounding errors since a slight deviation in the learner's behavior can lead to different state distributions not encountered by the expert policy. <br />
<br />
This issue was overcome through the use of the Forward Training (FT) algorithm which trains a non-stationary policy iteratively over time. At each time step a new policy is trained on the state distribution induced by the previously trained policies <math>\pi_0</math>, <math>\pi_1</math>, ...<math>\pi_{t-1}</math>. This is continued till the end of the time horizon to obtain a policy that can mimic the expert policy. This requirement to train a policy at each time step till the end makes the FT algorithm impractical for cases where the time horizon is very large or undefined. This shortcoming is resolved using the Stochastic Mixing Iterative Learning (SMILe) algorithm. SMILe trains a stochastic stationary policy over several iterations under the trajectory distribution induced by the previously trained policy: <math> \pi_t = \pi_{t-1} + \alpha (1 - \alpha)^{t-1}(\hat{\pi}_t - \pi_0)</math>, with <math>\pi_0</math> following expert's policy at the start of training.<br />
<br />
== Generative Adversarial Networks ==<br />
GANs learn a generative model that can fool the discriminator by using a two-player zero-sum game:<br />
<br />
\begin{align} <br />
\underset{G}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{x\sim p_E}[log(D(x)]\ +\ \mathbb{E}_{z\sim p_z}[log(1 - D(G(z)))]<br />
\end{align}<br />
<br />
In the above equation, <math> p_E </math> represents the expert distribution and <math> p_z </math> represents the input noise distribution from which the input to the generator is sampled. The generator produces patterns and the discriminator judges if the pattern was generated or from the expert data. When the discriminator cannot distinguish between the two distributions the game ends and the generator has learned to mimic the expert. GANs rely on basic ideas such as binary classification and algorithms such as backpropagation in order to learn the expert distribution.<br />
<br />
GAIL applies GANs to the task of imitating an expert policy in a model-free approach. GAIL uses similar objective functions like GANs, but the expert distribution in GAIL represents the joint distribution over state action tuples:<br />
<br />
\begin{align} <br />
\underset{\pi}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{\pi}[log(D(s,a)]\ +\ \mathbb{E}_{\pi_E}[log(1 - D(s,a))] - \lambda H(\pi))<br />
\end{align}<br />
<br />
where <math> H(\pi) \triangleq \mathbb{E}_{\pi}[-log\: \pi(a|s)]</math> is the entropy.<br />
<br />
This problem cannot be solved using the standard methods described for GANs because the generator in GAIL represents a stochastic policy. The exact form of the first term in the above equation is given by: <math> \mathbb{E}_{s\sim \rho_\pi(s)}\mathbb{E}_{a\sim \pi(\cdot |s)} [log(D(s,a)] </math>.<br />
<br />
The two-player game now depends on the stochastic properties (<math> \theta </math>) of the policy, and it is unclear how to differentiate the above equation with respect to <math> \theta </math>. This problem can be overcome using score functions such as REINFORCE to obtain an unbiased gradient estimation:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi} [log\; D(s,a)] \cong \hat{\mathbb{E}}_{\tau_i}[\nabla_\theta\; log\; \pi_\theta(a|s)Q(s,a)]<br />
\end{align}<br />
<br />
where <math> Q(\hat{s},\hat{a}) </math> is the score function of the gradient:<br />
<br />
\begin{align}<br />
Q(\hat{s},\hat{a}) = \hat{\mathbb{E}}_{\tau_i}[log\; D(s,a) | s_0 = \hat{s}, a_0 = \hat{a}]<br />
\end{align}<br />
<br />
<br />
REINFORCE gradients suffer from high variance which makes them difficult to work with even after applying variance reduction techniques. While recent general variance reduction techniques like RELAX (Grathwohl et al., 2017)<sup>[[#References|[7]]]</sup> work well, they rely on multiple evaluations of the loss function or learning a surrogate neural network. Unfortunately, this is too computationally difficult for our task. In order to better understand the changes required to fool the discriminator we need access to the gradients of the discriminator network, which can be obtained from the Jacobian of the discriminator. This paper demonstrates the use of a forward model along with the Jacobian of the discriminator to train a policy, without using high-variance gradient estimations.<br />
<br />
= Algorithm =<br />
This section first analyzes the characteristics of the discriminator network, then describes how a forward model can enable policy imitation through GANs. Lastly, the model based adversarial imitation learning algorithm is presented.<br />
<br />
== The discriminator network ==<br />
The discriminator network is trained to predict the conditional distribution: <math> D(s,a) = p(y|s,a) </math> where <math> y \in (\pi_E, \pi) </math>. <math>D(s,a)</math> here is the likelihood ratio with the pair <math>{s,a}</math> generated by <math>\pi</math>.<br />
<br />
The discriminator is trained on an even distribution of expert and generated examples; hence <math> p(\pi) = p(\pi_E) = \frac{1}{2} </math>. Given this and applying Bayes' theorem, we can rearrange and factor <math> D(s,a) </math> to obtain:<br />
<br />
\begin{aligned}<br />
D(s,a) &= p(\pi|s,a) \\<br />
& = \frac{p(s,a|\pi)p(\pi)}{p(s,a|\pi)p(\pi) + p(s,a|\pi_E)p(\pi_E)} \\<br />
& = \frac{p(s,a|\pi)}{p(s,a|\pi) + p(s,a|\pi_E)} \\<br />
& = \frac{1}{1 + \frac{p(s,a|\pi_E)}{p(s,a|\pi)}} \\<br />
& = \frac{1}{1 + \frac{p(a|s,\pi_E)}{p(a|s,\pi)} \cdot \frac{p(s|\pi_E)}{p(s|\pi)}} \\<br />
\end{aligned}<br />
<br />
Define <math> \varphi(s,a) </math> and <math> \psi(s) </math> to be:<br />
<br />
\begin{aligned}<br />
\varphi(s,a) = \frac{p(a|s,\pi_E)}{p(a|s,\pi)}, \psi(s) = \frac{p(s|\pi_E)}{p(s|\pi)}<br />
\end{aligned}<br />
<br />
to get the final expression for <math> D(s,a) </math>:<br />
\begin{aligned}<br />
D(s,a) = \frac{1}{1 + \varphi(s,a)\cdot \psi(s)}<br />
\end{aligned}<br />
<br />
<math> \varphi(s,a) </math> represents a policy likelihood ratio, and <math> \psi(s) </math> represents a state distribution likelihood ratio. Based on these expressions, the paper states that the discriminator makes its decisions by answering two questions. The first question relates to state distribution: what is the likelihood of encountering state <math> s </math> under the distribution induces by <math> \pi_E </math> vs <math> \pi </math>? The second question is about behavior: given a state <math> s </math>, how likely is action a under <math> \pi_E </math> vs <math> \pi </math>? The desired change in state is given by <math> \psi_s \equiv \partial \psi / \partial s </math>; this information can by obtained from the partial derivatives of <math> D(s,a) </math>, which is why these derivatives are proposed to be used for training policies (see following sections):<br />
<br />
\begin{aligned}<br />
\nabla_aD &= - \frac{\varphi_a(s,a)\psi(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\nabla_sD &= - \frac{\varphi_s(s,a)\psi(s) + \varphi(s,a)\psi_s(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\end{aligned}<br />
<br />
== Backpropagating through stochastic units ==<br />
There is interest in training stochastic policies because stochasticity encourages exploration for Policy Gradient methods. This is a problem for algorithms that build differentiable computation graphs where the gradients flow from one component to another since it is unclear how to backpropagate through stochastic units. The following subsections show how to estimate the gradients of continuous and categorical stochastic elements for continuous and discrete action domains respectively.<br />
<br />
=== Continuous Action Distributions ===<br />
re-parameterization was used to enable computing the derivatives of stochastic models. Assuming that the stochastic policy has a Gaussian distribution <math> \mathcal{N}(\mu_{\theta} (s), \sigma_{\theta}^2 (s))</math>, where the mean and variance are given by some deterministic functions <math>\mu_{\theta}</math> and <math>\sigma_{\theta}</math>, then the policy <math> \pi </math> can be written as <math> \pi_\theta(a|s) = \mu_\theta(s) + \xi \sigma_\theta(s) </math>, where <math> \xi \sim N(0,1) </math>. This way, the authors are able to get a Monte-Carlo estimator of the derivative of the expected value of <math> D(s, a) </math> with respect to <math> \theta </math>:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi(a|s)}D(s,a) = \mathbb{E}_{\rho (\xi )}\nabla_a D(a,s) \nabla_\theta \pi_\theta(a|s) \cong \frac{1}{M}\sum_{i=1}^{M} \nabla_a D(s,a) \nabla_\theta \pi_\theta(a|s)\Bigr|_{\substack{\xi=\xi_i}}<br />
\end{align}<br />
<br />
=== Categorical Action Distributions ===<br />
In the case of discrete action domains, the paper uses categorical re-parameterization with Gumbel-Softmax. This method relies on the Gumbel-Max trick which is a method for drawing samples from a categorical distribution with class probabilities <math> \pi(a_1|s),\pi(a_2|s),...,\pi(a_N|s) </math>:<br />
<br />
\begin{align}<br />
a_{argmax} = \underset{i}{argmax}[g_i + log\ \pi(a_i|s)]\textrm{, where } g_i \sim Gumbel(0, 1).<br />
\end{align}<br />
<br />
Gumbel-Softmax provides a differentiable approximation of the samples obtained using the Gumbel-Max trick (Gumbel-softmax allows us to generate a differentiable sample from a discrete distribution, which is needed in this trajectory imitation setting.):<br />
<br />
\begin{align}<br />
a_{softmax} = \frac{exp[\frac{1}{\tau}(g_i + log\ \pi(a_i|s))]}{\sum_{j=1}^{k}exp[\frac{1}{\tau}(g_j + log\ \pi(a_i|s))]}<br />
\end{align}<br />
<br />
<br />
In the above equation, the hyper-parameter <math> \tau </math> (temperature) trades bias for variance. When <math> \tau </math> gets closer to zero, the softmax operator acts like argmax resulting in a low bias, but high variance; vice versa when the <math> \tau </math> is large.<br />
<br />
The authors use <math> a_{softmax} </math> to interact with the environment; argmax is applied over <math> a_{softmax} </math> to obtain a single “pure” action, but the continuous approximation is used in the backward pass using the estimation: <math> \nabla_\theta\; a_{argmax} \approx \nabla_\theta\; a_{softmax} </math>.<br />
<br />
== Backpropagating through a Forward model ==<br />
The above subsections presented the means for extracting the partial derivative <math> \nabla_aD </math>. The main contribution of this paper is incorporating the use of <math> \nabla_sD </math>. In a model-free approach the state <math> s </math> is treated as a fixed input, therefore <math> \nabla_sD </math> is discarded. This is illustrated in Figure 2. This work uses a model-based approach which makes incorporating <math> \nabla_sD </math> more involved. In the model-based approach, a state <math> s_t </math> can be written as a function of the previous state action pair: <math> s_t = f(s_{t-1}, a_{t-1}) </math>, where <math> f </math> represents the forward model. Using the forward model and the law of total derivatives we get:<br />
<br />
\begin{align}<br />
\nabla_\theta D(s_t,a_t)\Bigr|_{\substack{s=s_t, a=a_t}} &= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_t}} \\<br />
&= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\left (\frac{\partial f}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_{t-1}}} + \frac{\partial f}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_{t-1}}} \right )<br />
\end{align}<br />
<br />
<br />
Using this formula, the error regarding deviations of future states <math> (\psi_s) </math> propagate back in time and influence the actions of policies in earlier times. This is summarized in Figure 3.<br />
<br />
[[File:modelFree_blockDiagram.PNG|400px|center]]<br />
<br />
Figure 2: Block-diagram of the model-free approach: given a state <math> s </math>, the policy outputs <math> \mu </math> which is fed to a stochastic sampling unit. An action <math> a </math> is sampled, and together with <math> s </math> are presented to the discriminator network. In the backward phase, the error message <math> \delta_a </math> is blocked at the stochastic sampling unit. From there, a high-variance gradient estimation is used (<math> \delta_{HV} </math>). Meanwhile, the error message <math> \delta_s </math> is flushed.<br />
<br />
[[File:modelBased_blockDiagram.PNG|700px|center]]<br />
<br />
Figure 3: Block diagram of model-based adversarial imitation learning. <br />
<br />
Figure 3 describes the computation graph for training the policy (i.e. G). The discriminator network D is fixed at this stage and is trained separately. At time <math> t </math> of the forward pass, <math> \pi </math> outputs a distribution over actions: <math> \mu_t = \pi(s_t) </math>, from which an action at is sampled. For example, in the continuous case, this is done using the re-parametrization trick: <math> a_t = \mu_t + \xi \cdot \sigma </math>, where <math> \xi \sim N(0,1) </math>. The next state <math> s_{t+1} = f(s_t, a_t) </math> is computed using the forward model (which is also trained separately), and the entire process repeats for time <math> t+1 </math>. In the backward pass, the gradient of <math> \pi </math> is comprised of a.) the error message <math> \delta_a </math> (Green) that propagates fluently through the differentiable approximation of the sampling process. And b.) the error message <math> \delta_s </math> (Blue) of future time-steps, that propagate back through the differentiable forward model.<br />
<br />
== MGAIL Algorithm ==<br />
Shalev-Shwartz et al. (2016)<sup>[[#References|[3]]]</sup> and Heess et al. (2015)<sup>[[#References|[4]]]</sup> built a multi-step computation graph for describing the familiar policy gradient objective; in this case it is given by:<br />
<br />
\begin{align}<br />
J(\theta) = \mathbb{E}\left [ \sum_{t=0}^{T} \gamma ^t D(s_t,a_t)|\theta\right ]<br />
\end{align}<br />
<br />
<br />
Using the results from Heess et al. (2015)<sup>[[#References|[4]]]</sup> this paper demonstrates how to differentiate <math> J(\theta) </math> over a trajectory of <math>(s,a,s’) </math> transitions:<br />
<br />
\begin{align}<br />
J_s &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_s + D_a \pi_s + \gamma J'_{s'}(f_s + f_a \pi_s) \right] \\<br />
J_\theta &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_a \pi_\theta + \gamma (J'_{s'} f_a \pi_\theta + J'_\theta) \right]<br />
\end{align}<br />
<br />
The policy gradient <math> \nabla_\theta J </math> is calculated by applying equations 12 and 13 recursively for <math> T </math> iterations. The MGAIL algorithm is presented below.<br />
<br />
[[File:MGAIL_alg.PNG]]<br />
<br />
== Forward Model Structure ==<br />
The stability of the learning process depends on the prediction accuracy of the forward model, but learning an accurate forward model is challenging by itself. The authors propose methods for improving the performance of the forward model based on two aspects of its functionality. First, the forward model should learn to use the action as an operator over the state space. To accomplish this, the actions and states, which are sampled form different distributions need to be first represented in a shared space. This is done by encoding the state and action using two separate neural networks and combining their outputs to form a single vector. Additionally, multiple previous states are used to predict the next state by representing the environment as an <math> n^{th} </math> order MDP. A gated recurrent units (GRU, a simpler variant on the LSTM model) layer is incorporated into the state encoder to enable recurrent connections from previous states. Using these modifications, the model is able to achieve better, and more stable results compared to the standard forward model based on a feed forward neural network. The comparison is presented in Figure 4.<br />
<br />
[[File:performance_comparison.PNG]]<br />
<br />
Figure 4: Performance comparison between a basic forward model (Blue), and the advanced forward model (Green).<br />
<br />
= Experiments =<br />
The proposed algorithm is evaluated on three discrete control tasks (Cartpole, Mountain-Car, Acrobot) and five continuous control tasks (Hopper, Walker, Half-Cheetah, Ant, and Humanoid). These tasks are modelled by the MuJoCo physics simulator (Todorov et al., 2012)<sup>[[#References|[9]]]</sup>, contain second order dynamics and utilize direct torque control. Expert policies are trained using the Trust Region Policy Optimization (TRPO) algorithm (Schulman et al., 2015)<sup>[[#References|[5]]]</sup>. Different number of trajectories are used to train the expert for each task, but all trajectories are of length 1000.<br />
The discriminator and generator (policy) networks contains two hidden layers with ReLU non-linearities and are trained using the ADAM optimizer. The total reward received over a period of <math> N </math> steps using BC, GAIL and MGAIL is presented in Table 1. The proposed algorithm achieved the highest reward for most environments while exhibiting performance comparable to the expert over all of them. A comparison between the basic forward model and the more advanced forward model is also made and described in the previous section of this summary. The two models compared are shown below.<br />
<br />
[[File:baram17_forward.PNG]]<br />
<br />
[[File:mgail_test_results_1.PNG]]<br />
<br />
[[File:mgail_test_results.PNG]]<br />
<br />
Table 1. Policy performance, <math> \pm </math> represents one standard deviation, a higher (reward) value is better. MGAIL consistently outperforms both GAIL and Behavioural cloning approaches, except on the Cartpole, where MGAIL and GAIL perform equally.<br />
<br />
= Discussion =<br />
This paper presented a model-free algorithm for imitation learning. It demonstrated how a forward model can be used to train policies using the exact gradient of the discriminator network. A downside of this approach is the need to learn a forward model; this could be difficult in certain domains. Learning the system dynamics directly from raw images is considered as one line of future work. Another future work is to address the violation of the fundamental assumption made by all supervised learning algorithms, which requires the data to be i.i.d. This problem arises because the discriminator and forward models are trained in a supervised learning fashion using data sampled from a dynamic distribution. The authors tried a solution proposed by another paper (Loshchilov & Hutter, 2016)<sup>[[#References|[10]]]</sup>, which is to reset the learning rate several times during training period, but it did not result in significant improvements.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper: https://github.com/itaicaspi/mgail. The repository provides the source code written by the authors, in Tensorflow.<br />
<br />
= Source =<br />
# Baram, Nir, et al. "End-to-end differentiable adversarial imitation learning." International Conference on Machine Learning. 2017.<br />
# Ho, Jonathan, and Stefano Ermon. "Generative adversarial imitation learning." Advances in Neural Information Processing Systems. 2016.<br />
# Shalev-Shwartz, Shai, et al. "Long-term planning by short-term prediction." arXiv preprint arXiv:1602.01580 (2016).<br />
# Heess, Nicolas, et al. "Learning continuous control policies by stochastic value gradients." Advances in Neural Information Processing Systems. 2015.<br />
# Schulman, John, et al. "Trust region policy optimization." International Conference on Machine Learning. 2015.<br />
# Caspi, I. (n.d.). Itaicaspi/mgail. Retrieved March 25, 2018, from https://github.com/itaicaspi/mgail.<br />
# Grathwohl, W., Choi, D., Wu, Y., Roeder, G., & Duvenaud, D. (2017). Backpropagation through the Void: Optimizing control variates for black-box gradient estimation. arXiv preprint arXiv:1711.00123.<br />
# Williams, Ronald J. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.<br />
# Todorov, Emanuel, Erez, Tom, and Tassa, Yuval. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026–5033. IEEE, 2012.<br />
# Loshchilov, Ilya and Hutter, Frank. Sgdr: Stochastic gradient descent with restarts. arXiv preprint arXiv:1608.03983, 2016.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wavelet_Pooling_CNN&diff=36431Wavelet Pooling CNN2018-04-21T02:57:08Z<p>W285liu: /* Back Propagation */</p>
<hr />
<div>== Introduction ==<br />
Convolutional neural networks (CNN) have been proven to be powerful in image classification. Over the past few years, researchers have put efforts in improving fundamental components of CNNs such as the pooling operation. Various pooling methods exist; deterministic methods include max pooling and average pooling and probabilistic methods include mixed pooling and stochastic pooling. All these methods employ a neighborhood approach to the sub-sampling which, albeit fast and simple, can produce artifacts such as blurring, aliasing, and edge halos (Parker et al., 1983).<br />
<br />
This paper introduces a novel pooling method based on the discrete wavelet transform. Specifically, it uses a second-level wavelet decomposition for the sub-sampling. This method, instead of nearest neighbor interpolation uses a sub-band method that the authors' claim produces fewer artifacts and represents the underlying features more accurately. Therefore, if pooling is viewed as a lossy process, the reason for employing a wavelet approach is to try to minimize this loss.<br />
<br />
== Pooling Background ==<br />
Pooling essentially means sub-sampling. After the pooling layer, the spatial dimensions of the data are reduced to some degree, with the goal being to compress the data rather than discard some of it. Typical approaches to pooling reduce the dimensionality by using some method to combine a region of values into one value. Max pooling and Mean/Average pooling are the 2 most commonly used pooling methods. For max pooling, this can be represented by the equation <math>a_{kij} = max_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> where <math>a_{kij}</math> is the output activation of the <math>k^th</math> feature map at <math>(i,j)</math>, <math>a_{kpq}</math> is input activation at <math>(p,q)</math> within <math>R_{ij}</math>, and <math>|R_{ij}|</math> is the size of the pooling region. Mean pooling can be represented by the equation <math>a_{kij} = \frac{1}{|R_{ij}|} \sum_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> with everything defined as before. Figure 1 provides a numerical example that can be followed.<br />
<br />
[[File:WT_Fig1.PNG|650px|center|]]<br />
<br />
The paper mentions that these pooling methods, although simple and effective, have shortcomings. Max pooling can omit details from an image if the important features have less intensity than the insignificant ones (unless the neural network has learnt to inverse the intensity level), and also commonly overfits. On the other hand, average pooling can dilute important features if the data is averaged with values of significantly lower intensities. Figure 2 displays an image of this.<br />
<br />
[[File:WT_Fig2.PNG|650px|center|]]<br />
<br />
To account for the above-mentioned issues, probabilistic pooling methods were introduced, namely mixed pooling and; stochastic pooling. Mixed pooling is a simple method which just combines the max and the average pooling by randomly selecting one method over the other during training. Mixed pooling can be applied for all features, mixed between features, or mixed between regions for different features. Stochastic pooling on the other hand randomly samples within a receptive field with the activation values as the probabilities. These are calculated by taking each activation value and dividing it by the sum of all activation values in the grid so that the probabilities sum to 1.<br />
<br />
Figure 3 shows an example of how stochastic pooling works. On the left is a 3x3 grid filled with activations. The middle grid is the corresponding probability for each activation. The activation in the middle was randomly selected (it had a 13% chance of getting selected). Because the stochastic pooling is based on the probability of the pixels, it is able to avoid the shortcomings of max and mean pooling mentioned above.<br />
<br />
[[File:paper21-stochasticpooling.png|650px|center|]]<br />
<br />
== Wavelet Background ==<br />
Data or signals tend to be composed of slowly changing trends (low frequency) as well as fast-changing transients (high frequency). Similarly, images have smooth regions of intensity which are perturbed by edges or abrupt changes. We know that these abrupt changes can represent features that are of great importance to us when we perform deep learning. Wavelets are a class of functions that are well localized in time and frequency. The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet <math>\psi(t)=e^{-2 \pi it}</math>. The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The ability of wavelets to be localized in time and space is what makes it suitable for detecting the abrupt changes in an image well. <br />
<br />
Essentially, a wavelet is a fast decaying oscillating signal with zero mean that only exists for a fixed duration and can be scaled and shifted in time. There are some well-defined types of wavelets as shown in Figure 3. The key characteristic of wavelets for us is that they have a band-pass characteristic, and the band can be adjusted based on the scaling and shifting. <br />
<br />
[[File:WT_Fig3.jpg|650px|center|]]<br />
<br />
The paper uses discrete wavelet transform and more specifically a faster variation called Fast Wavelet Transform (FWT) using the Haar wavelet. There also exists a continuous wavelet transform. The main difference in these is how the scale and shift parameters are selected.<br />
<br />
== Discrete Wavelet Transform General==<br />
The discrete wavelet transforms for images is essentially applying a low pass and high pass filter to your image where the transfer functions of the filters are related and defined by the type of wavelet used (Haar in this paper). This is shown in the figures below, which also show the recursive nature of the transform. For an image, the per-row transform is taken first. This results in a new image where the first half is a low-frequency sub-band and the second half is the high-frequency sub-band. Then this new image is transformed again per column, resulting in four sub-bands. Generally, the low-frequency content approximates the image and the high-frequency content represents abrupt changes. Therefore, one can simply take the LL band and perform the transformation again to sub-sample even more.<br />
<br />
[[File:WT_Fig8.png|650px|center|]]<br />
<br />
[[File:WT_Fig9.png|650px|center|]]<br />
<br />
In left half of the above image we see a grid containing four different transformations of the same initial image. Each transform has been done by applying a row wise convolution with a wavelet of either high or low frequency, then a column wise convolution with another wavelet of either high or low frequency. The four choices of frequency (LL, LH, HL, HH) result in four different images. The top left image is the result of applying a low frequency wavelet convolution to the original image both row wise and column wise. The bottom left image is the result of first applying a high frequency wavelet convolution row wise and then applying a low frequency wavelet convolution column wise. Since the LL (top right) transformation preserves the original image best, it is then used in this process again to generate the grid of smaller images that can be seen in the top center-right of the above image. The images in this smaller grid are called second order coefficients.<br />
<br />
== DWT example using Haar Wavelet ==<br />
Suppose we have an image represented by the following pixels:<br />
<br />
\begin{align}<br />
\begin{bmatrix} <br />
100 & 50 & 60 & 150 \\<br />
20 & 60 & 40 & 30 \\<br />
50 & 90 & 70 & 82 \\<br />
74 & 66 & 90 & 58 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
For each level of the DWT using the Haar wavelet, we will perform the transform on the rows first and then the columns. For the row pass, we transform each row as follows:<br />
* For each row i = <math>[i_{1}, i_{2}, i_{3}, i_{4}]</math> of the input image, transform the row to <math>i_{t}</math> via<br />
<br />
\begin{align}<br />
i_{t} = [(i_{1} + i_{2}) / 2, (i_{3} + i_{4}) / 2, (i_{1}, - i_{2}) / 2, (i_{3} - i_{4}) / 2]<br />
\end{align}<br />
<br />
After the row transforms, the images looks as follows:<br />
\begin{align}<br />
\begin{bmatrix} <br />
75 & 105 & 25 & -45 \\<br />
40 & 35 & -20 & 5 \\<br />
70 & 76 & -20 & -6 \\<br />
70 & 74 & 4 & 16 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
Now we apply the same method to the columns in the exact same way.<br />
<br />
== Proposed Method ==<br />
The proposed method uses subbands from the second level FWT and discards the first level subbands. The authors postulate that this method is more 'organic' in capturing the data compression and will create less artifacts that may affect the image classification.<br />
=== Forward Propagation ===<br />
FWT can be expressed by <math>W_\varphi[j + 1, k] = h_\varphi[-n]*W_\varphi[j,n]|_{n = 2k, k <= 0}</math> and <math>W_\psi[j + 1, k] = h_\psi[-n]*W_\psi[j,n]|_{n = 2k, k <= 0}</math> where <math>\varphi</math> is the approximation function, <math>\psi</math> is the detail function, <math>W_\varphi</math>, <math>W_\psi</math>, are approximation and detail coefficients, <math>h_\varphi[-n]</math> and <math>h_\psi[-n]</math> are time reversed scaling and wavelet vectors, <math>(n)</math> represents the sample in the vector, and <math>j</math> denotes the resolution level. To apply to images, FWT is first applied on the rows and then the columns. If a low (L) and high(H) sub-band is extracted from the rows and similarly for the columns than at each level there is 4 sub-bands (LH, HL, HH, and LL) where LL will further be decomposed into the level 2 decomposition. <br />
<br />
Using the level 2 decomposition sub-bands, the Inverse Fast Wavelet Transform (IFWT) is used to obtain the resulting sub-sampled image, which is sub-sampled by a factor of two. The Equation for IFWT is <math>W_\varphi[j, k] = h_\varphi[-n]*W_\varphi[j + 1,n] + h_\psi[-n]*W_\psi[j + 1,n]|_{n = \frac{k}{2}, k <= 0}</math> where the parameters are the same as previously explained. Figure 4 displays the algorithm for the forward propagation.<br />
<br />
[[File:WT_Fig6.PNG|650px|center|]]<br />
<br />
=== Back Propagation ===<br />
This is simply the reverse of the forward propagation. The image features first have to be converted into the sub-bands using the 1st order wavelet decomposition. The sub-bands are upsampled by a factor of 2 and then backpropagated through the IDWT to achieve the final image. Figure 5 displays the algorithm.<br />
<br />
[[File:WT_Fig7.PNG|650px|center|]]<br />
<br />
== Results ==<br />
The authors tested on MNIST, CIFAR-10, SHVN, and KDEF and the paper provides comprehensive results for each. Stochastic gradient descent was used and the Haar wavelet is used due to its even, square subbands. The network for all datasets except MNIST is loosely based on (Zeiler & Fergus, 2013). The authors keep the network consistent but change the pooling method for each dataset. They also experiment with dropout and Batch Normalization to examine the effects of regularization on their method. All pooling methods compared use a 2x2 window, and a consistent pooling method was used for all pooling layers of a network. The overall results teach us that the pooling method should be chosen specifically for the type of data we have. In some cases, wavelet pooling may perform the best, and in other cases, other methods may perform better, if the data is more suited for those types of pooling.<br />
<br />
=== MNIST ===<br />
Figure 7 shows the network how's architecture was based on an example of MNIST structure from MatConvNet, with batch normalization. Table 1 shows the algorithms accuracy. It can be seen that wavelet pooling achieves the best accuracy from all pooling methods compared. Figure 8 shows the energy of each method per epoch. As can be noted by Figure 8 average and wavelet pooling show a smoother descent in learning and error reduction.<br />
<br />
[[File:WT_Fig4.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig8.png|800px|center]]<br />
<br />
[[File:WT_Tab1.PNG|650px|center|]]<br />
<br />
=== CIFAR-10 ===<br />
In order to investigate the performance of different pooling methods, two types of networks are trained based on CIFAR-10. The first one is the regular CNN and the second one is the network with dropout and batch normalization. Figure 9 shows the network and Tables 2 and 3 shows the accuracy without and with dropout. Average pooling achieves the best accuracy but wavelet pooling is still competitive, while max pooling overfitted on the validation data fairly quickly as shown by the right energy curve in Figure 10 (although the accuracy performance is not significantly worse when dropout and batch normalization are applied).<br />
<br />
[[File:WT_Fig5.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig10.png|800px|center]]<br />
<br />
[[File:WT_Tab2.PNG|650px|center|]]<br />
<br />
[[File:WT_Tab3.PNG|650px|center|]]<br />
<br />
===SHVN===<br />
Figure 11 shows the network and Table 4 and 5 shows the accuracy without and with dropout. The proposed method does not perform well in this experiment. <br />
<br />
[[File: a.png|650px|center|]]<br />
<br />
[[File:paper21_fig12.png|800px|center]]<br />
<br />
[[File: b.png|650px|center|]]<br />
<br />
===KDEF===<br />
The authors experimented with pooling methods + dropout on the KDEF dataset (which consists of 4,900 images of 35 people portraying varying emotions through facial expressions under different poses, 3,900 of which were randomly assigned to be used for training). The data was treated for errors (e.g. corrupt images) and resized to 128x128 for memory and time constraints. <br />
<br />
Figure 13 below shows the network structure. Figure 14 shows the energy curve of the competing models on training and validation sets as the number of epochs increases, and Table 6 shows the accuracy performance. Average pooling demonstrated the highest accuracy, with wavelet pooling coming in second and max-pooling a close third. However, stochastic and wavelet pooling exhibited more stable learning progression compared to the other methods, and max-pooling eventually overfitted. <br />
<br />
[[File:kdef_struc.PNG|700px|center|]]<br />
[[File:kdef_curve.PNG|750px|center|]]<br />
[[File:kdef_accu.PNG|550px|center|]]<br />
<br />
== Computational Complexity ==<br />
The authors explain that their paper is a proof of concept and is not meant to implement wavelet pooling in the most efficient way. The table below displays a comparison of the number of mathematical operations for each method according to the dataset. It can be seen that wavelet pooling is significantly worse. The authors explain that through good implementation and coding practices, the method can prove to be viable.<br />
<br />
[[File:WT_Tab4.PNG|650px|center|]]<br />
<br />
== Criticism ==<br />
=== Positive ===<br />
* Wavelet Pooling achieves competitive performance with standard go-to pooling methods<br />
* Leads to a comparison of discrete transformation techniques for pooling (DCT, DFT)<br />
=== Negative ===<br />
* Only 2x2 pooling window used for comparison<br />
* Highly computationally extensive<br />
* Not as simple as other pooling methods<br />
* Only one wavelet used (HAAR wavelet)<br />
<br />
== References ==<br />
* Travis Williams and Robert Li. Wavelet Pooling for Convolutional Neural Networks. ICLR 2018.<br />
* J. Anthony Parker, Robert V. Kenyon, and Donald E. Troxel. Comparison of interpolating methods for image resampling. IEEE Transactions on Medical Imaging, 2(1):31–39, 1983.<br />
* Matthew Zeiler and Robert Fergus. Stochastic pooling for regularization of deep convolutional neural networks. In Proceedings of the International Conference on Learning Representation (ICLR), 2013.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wavelet_Pooling_CNN&diff=36430Wavelet Pooling CNN2018-04-21T02:56:14Z<p>W285liu: /* SHVN */</p>
<hr />
<div>== Introduction ==<br />
Convolutional neural networks (CNN) have been proven to be powerful in image classification. Over the past few years, researchers have put efforts in improving fundamental components of CNNs such as the pooling operation. Various pooling methods exist; deterministic methods include max pooling and average pooling and probabilistic methods include mixed pooling and stochastic pooling. All these methods employ a neighborhood approach to the sub-sampling which, albeit fast and simple, can produce artifacts such as blurring, aliasing, and edge halos (Parker et al., 1983).<br />
<br />
This paper introduces a novel pooling method based on the discrete wavelet transform. Specifically, it uses a second-level wavelet decomposition for the sub-sampling. This method, instead of nearest neighbor interpolation uses a sub-band method that the authors' claim produces fewer artifacts and represents the underlying features more accurately. Therefore, if pooling is viewed as a lossy process, the reason for employing a wavelet approach is to try to minimize this loss.<br />
<br />
== Pooling Background ==<br />
Pooling essentially means sub-sampling. After the pooling layer, the spatial dimensions of the data are reduced to some degree, with the goal being to compress the data rather than discard some of it. Typical approaches to pooling reduce the dimensionality by using some method to combine a region of values into one value. Max pooling and Mean/Average pooling are the 2 most commonly used pooling methods. For max pooling, this can be represented by the equation <math>a_{kij} = max_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> where <math>a_{kij}</math> is the output activation of the <math>k^th</math> feature map at <math>(i,j)</math>, <math>a_{kpq}</math> is input activation at <math>(p,q)</math> within <math>R_{ij}</math>, and <math>|R_{ij}|</math> is the size of the pooling region. Mean pooling can be represented by the equation <math>a_{kij} = \frac{1}{|R_{ij}|} \sum_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> with everything defined as before. Figure 1 provides a numerical example that can be followed.<br />
<br />
[[File:WT_Fig1.PNG|650px|center|]]<br />
<br />
The paper mentions that these pooling methods, although simple and effective, have shortcomings. Max pooling can omit details from an image if the important features have less intensity than the insignificant ones (unless the neural network has learnt to inverse the intensity level), and also commonly overfits. On the other hand, average pooling can dilute important features if the data is averaged with values of significantly lower intensities. Figure 2 displays an image of this.<br />
<br />
[[File:WT_Fig2.PNG|650px|center|]]<br />
<br />
To account for the above-mentioned issues, probabilistic pooling methods were introduced, namely mixed pooling and; stochastic pooling. Mixed pooling is a simple method which just combines the max and the average pooling by randomly selecting one method over the other during training. Mixed pooling can be applied for all features, mixed between features, or mixed between regions for different features. Stochastic pooling on the other hand randomly samples within a receptive field with the activation values as the probabilities. These are calculated by taking each activation value and dividing it by the sum of all activation values in the grid so that the probabilities sum to 1.<br />
<br />
Figure 3 shows an example of how stochastic pooling works. On the left is a 3x3 grid filled with activations. The middle grid is the corresponding probability for each activation. The activation in the middle was randomly selected (it had a 13% chance of getting selected). Because the stochastic pooling is based on the probability of the pixels, it is able to avoid the shortcomings of max and mean pooling mentioned above.<br />
<br />
[[File:paper21-stochasticpooling.png|650px|center|]]<br />
<br />
== Wavelet Background ==<br />
Data or signals tend to be composed of slowly changing trends (low frequency) as well as fast-changing transients (high frequency). Similarly, images have smooth regions of intensity which are perturbed by edges or abrupt changes. We know that these abrupt changes can represent features that are of great importance to us when we perform deep learning. Wavelets are a class of functions that are well localized in time and frequency. The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet <math>\psi(t)=e^{-2 \pi it}</math>. The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The ability of wavelets to be localized in time and space is what makes it suitable for detecting the abrupt changes in an image well. <br />
<br />
Essentially, a wavelet is a fast decaying oscillating signal with zero mean that only exists for a fixed duration and can be scaled and shifted in time. There are some well-defined types of wavelets as shown in Figure 3. The key characteristic of wavelets for us is that they have a band-pass characteristic, and the band can be adjusted based on the scaling and shifting. <br />
<br />
[[File:WT_Fig3.jpg|650px|center|]]<br />
<br />
The paper uses discrete wavelet transform and more specifically a faster variation called Fast Wavelet Transform (FWT) using the Haar wavelet. There also exists a continuous wavelet transform. The main difference in these is how the scale and shift parameters are selected.<br />
<br />
== Discrete Wavelet Transform General==<br />
The discrete wavelet transforms for images is essentially applying a low pass and high pass filter to your image where the transfer functions of the filters are related and defined by the type of wavelet used (Haar in this paper). This is shown in the figures below, which also show the recursive nature of the transform. For an image, the per-row transform is taken first. This results in a new image where the first half is a low-frequency sub-band and the second half is the high-frequency sub-band. Then this new image is transformed again per column, resulting in four sub-bands. Generally, the low-frequency content approximates the image and the high-frequency content represents abrupt changes. Therefore, one can simply take the LL band and perform the transformation again to sub-sample even more.<br />
<br />
[[File:WT_Fig8.png|650px|center|]]<br />
<br />
[[File:WT_Fig9.png|650px|center|]]<br />
<br />
In left half of the above image we see a grid containing four different transformations of the same initial image. Each transform has been done by applying a row wise convolution with a wavelet of either high or low frequency, then a column wise convolution with another wavelet of either high or low frequency. The four choices of frequency (LL, LH, HL, HH) result in four different images. The top left image is the result of applying a low frequency wavelet convolution to the original image both row wise and column wise. The bottom left image is the result of first applying a high frequency wavelet convolution row wise and then applying a low frequency wavelet convolution column wise. Since the LL (top right) transformation preserves the original image best, it is then used in this process again to generate the grid of smaller images that can be seen in the top center-right of the above image. The images in this smaller grid are called second order coefficients.<br />
<br />
== DWT example using Haar Wavelet ==<br />
Suppose we have an image represented by the following pixels:<br />
<br />
\begin{align}<br />
\begin{bmatrix} <br />
100 & 50 & 60 & 150 \\<br />
20 & 60 & 40 & 30 \\<br />
50 & 90 & 70 & 82 \\<br />
74 & 66 & 90 & 58 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
For each level of the DWT using the Haar wavelet, we will perform the transform on the rows first and then the columns. For the row pass, we transform each row as follows:<br />
* For each row i = <math>[i_{1}, i_{2}, i_{3}, i_{4}]</math> of the input image, transform the row to <math>i_{t}</math> via<br />
<br />
\begin{align}<br />
i_{t} = [(i_{1} + i_{2}) / 2, (i_{3} + i_{4}) / 2, (i_{1}, - i_{2}) / 2, (i_{3} - i_{4}) / 2]<br />
\end{align}<br />
<br />
After the row transforms, the images looks as follows:<br />
\begin{align}<br />
\begin{bmatrix} <br />
75 & 105 & 25 & -45 \\<br />
40 & 35 & -20 & 5 \\<br />
70 & 76 & -20 & -6 \\<br />
70 & 74 & 4 & 16 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
Now we apply the same method to the columns in the exact same way.<br />
<br />
== Proposed Method ==<br />
The proposed method uses subbands from the second level FWT and discards the first level subbands. The authors postulate that this method is more 'organic' in capturing the data compression and will create less artifacts that may affect the image classification.<br />
=== Forward Propagation ===<br />
FWT can be expressed by <math>W_\varphi[j + 1, k] = h_\varphi[-n]*W_\varphi[j,n]|_{n = 2k, k <= 0}</math> and <math>W_\psi[j + 1, k] = h_\psi[-n]*W_\psi[j,n]|_{n = 2k, k <= 0}</math> where <math>\varphi</math> is the approximation function, <math>\psi</math> is the detail function, <math>W_\varphi</math>, <math>W_\psi</math>, are approximation and detail coefficients, <math>h_\varphi[-n]</math> and <math>h_\psi[-n]</math> are time reversed scaling and wavelet vectors, <math>(n)</math> represents the sample in the vector, and <math>j</math> denotes the resolution level. To apply to images, FWT is first applied on the rows and then the columns. If a low (L) and high(H) sub-band is extracted from the rows and similarly for the columns than at each level there is 4 sub-bands (LH, HL, HH, and LL) where LL will further be decomposed into the level 2 decomposition. <br />
<br />
Using the level 2 decomposition sub-bands, the Inverse Fast Wavelet Transform (IFWT) is used to obtain the resulting sub-sampled image, which is sub-sampled by a factor of two. The Equation for IFWT is <math>W_\varphi[j, k] = h_\varphi[-n]*W_\varphi[j + 1,n] + h_\psi[-n]*W_\psi[j + 1,n]|_{n = \frac{k}{2}, k <= 0}</math> where the parameters are the same as previously explained. Figure 4 displays the algorithm for the forward propagation.<br />
<br />
[[File:WT_Fig6.PNG|650px|center|]]<br />
<br />
=== Back Propagation ===<br />
This is simply the reverse of the forward propagation. The image feature first has to be converted into the sub-bands using the 1st order wavelet decomposition. The sub-bands are upsampled by a factor of 2 and then backpropagated through the IDWT to achieve the final image. Figure 5 displays the algorithm.<br />
<br />
[[File:WT_Fig7.PNG|650px|center|]]<br />
<br />
== Results ==<br />
The authors tested on MNIST, CIFAR-10, SHVN, and KDEF and the paper provides comprehensive results for each. Stochastic gradient descent was used and the Haar wavelet is used due to its even, square subbands. The network for all datasets except MNIST is loosely based on (Zeiler & Fergus, 2013). The authors keep the network consistent but change the pooling method for each dataset. They also experiment with dropout and Batch Normalization to examine the effects of regularization on their method. All pooling methods compared use a 2x2 window, and a consistent pooling method was used for all pooling layers of a network. The overall results teach us that the pooling method should be chosen specifically for the type of data we have. In some cases, wavelet pooling may perform the best, and in other cases, other methods may perform better, if the data is more suited for those types of pooling.<br />
<br />
=== MNIST ===<br />
Figure 7 shows the network how's architecture was based on an example of MNIST structure from MatConvNet, with batch normalization. Table 1 shows the algorithms accuracy. It can be seen that wavelet pooling achieves the best accuracy from all pooling methods compared. Figure 8 shows the energy of each method per epoch. As can be noted by Figure 8 average and wavelet pooling show a smoother descent in learning and error reduction.<br />
<br />
[[File:WT_Fig4.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig8.png|800px|center]]<br />
<br />
[[File:WT_Tab1.PNG|650px|center|]]<br />
<br />
=== CIFAR-10 ===<br />
In order to investigate the performance of different pooling methods, two types of networks are trained based on CIFAR-10. The first one is the regular CNN and the second one is the network with dropout and batch normalization. Figure 9 shows the network and Tables 2 and 3 shows the accuracy without and with dropout. Average pooling achieves the best accuracy but wavelet pooling is still competitive, while max pooling overfitted on the validation data fairly quickly as shown by the right energy curve in Figure 10 (although the accuracy performance is not significantly worse when dropout and batch normalization are applied).<br />
<br />
[[File:WT_Fig5.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig10.png|800px|center]]<br />
<br />
[[File:WT_Tab2.PNG|650px|center|]]<br />
<br />
[[File:WT_Tab3.PNG|650px|center|]]<br />
<br />
===SHVN===<br />
Figure 11 shows the network and Table 4 and 5 shows the accuracy without and with dropout. The proposed method does not perform well in this experiment. <br />
<br />
[[File: a.png|650px|center|]]<br />
<br />
[[File:paper21_fig12.png|800px|center]]<br />
<br />
[[File: b.png|650px|center|]]<br />
<br />
===KDEF===<br />
The authors experimented with pooling methods + dropout on the KDEF dataset (which consists of 4,900 images of 35 people portraying varying emotions through facial expressions under different poses, 3,900 of which were randomly assigned to be used for training). The data was treated for errors (e.g. corrupt images) and resized to 128x128 for memory and time constraints. <br />
<br />
Figure 13 below shows the network structure. Figure 14 shows the energy curve of the competing models on training and validation sets as the number of epochs increases, and Table 6 shows the accuracy performance. Average pooling demonstrated the highest accuracy, with wavelet pooling coming in second and max-pooling a close third. However, stochastic and wavelet pooling exhibited more stable learning progression compared to the other methods, and max-pooling eventually overfitted. <br />
<br />
[[File:kdef_struc.PNG|700px|center|]]<br />
[[File:kdef_curve.PNG|750px|center|]]<br />
[[File:kdef_accu.PNG|550px|center|]]<br />
<br />
== Computational Complexity ==<br />
The authors explain that their paper is a proof of concept and is not meant to implement wavelet pooling in the most efficient way. The table below displays a comparison of the number of mathematical operations for each method according to the dataset. It can be seen that wavelet pooling is significantly worse. The authors explain that through good implementation and coding practices, the method can prove to be viable.<br />
<br />
[[File:WT_Tab4.PNG|650px|center|]]<br />
<br />
== Criticism ==<br />
=== Positive ===<br />
* Wavelet Pooling achieves competitive performance with standard go-to pooling methods<br />
* Leads to a comparison of discrete transformation techniques for pooling (DCT, DFT)<br />
=== Negative ===<br />
* Only 2x2 pooling window used for comparison<br />
* Highly computationally extensive<br />
* Not as simple as other pooling methods<br />
* Only one wavelet used (HAAR wavelet)<br />
<br />
== References ==<br />
* Travis Williams and Robert Li. Wavelet Pooling for Convolutional Neural Networks. ICLR 2018.<br />
* J. Anthony Parker, Robert V. Kenyon, and Donald E. Troxel. Comparison of interpolating methods for image resampling. IEEE Transactions on Medical Imaging, 2(1):31–39, 1983.<br />
* Matthew Zeiler and Robert Fergus. Stochastic pooling for regularization of deep convolutional neural networks. In Proceedings of the International Conference on Learning Representation (ICLR), 2013.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wavelet_Pooling_CNN&diff=36429Wavelet Pooling CNN2018-04-21T02:55:06Z<p>W285liu: /* Pooling Background */</p>
<hr />
<div>== Introduction ==<br />
Convolutional neural networks (CNN) have been proven to be powerful in image classification. Over the past few years, researchers have put efforts in improving fundamental components of CNNs such as the pooling operation. Various pooling methods exist; deterministic methods include max pooling and average pooling and probabilistic methods include mixed pooling and stochastic pooling. All these methods employ a neighborhood approach to the sub-sampling which, albeit fast and simple, can produce artifacts such as blurring, aliasing, and edge halos (Parker et al., 1983).<br />
<br />
This paper introduces a novel pooling method based on the discrete wavelet transform. Specifically, it uses a second-level wavelet decomposition for the sub-sampling. This method, instead of nearest neighbor interpolation uses a sub-band method that the authors' claim produces fewer artifacts and represents the underlying features more accurately. Therefore, if pooling is viewed as a lossy process, the reason for employing a wavelet approach is to try to minimize this loss.<br />
<br />
== Pooling Background ==<br />
Pooling essentially means sub-sampling. After the pooling layer, the spatial dimensions of the data are reduced to some degree, with the goal being to compress the data rather than discard some of it. Typical approaches to pooling reduce the dimensionality by using some method to combine a region of values into one value. Max pooling and Mean/Average pooling are the 2 most commonly used pooling methods. For max pooling, this can be represented by the equation <math>a_{kij} = max_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> where <math>a_{kij}</math> is the output activation of the <math>k^th</math> feature map at <math>(i,j)</math>, <math>a_{kpq}</math> is input activation at <math>(p,q)</math> within <math>R_{ij}</math>, and <math>|R_{ij}|</math> is the size of the pooling region. Mean pooling can be represented by the equation <math>a_{kij} = \frac{1}{|R_{ij}|} \sum_{(p,q) \epsilon R_{ij}} (a_{kpq})</math> with everything defined as before. Figure 1 provides a numerical example that can be followed.<br />
<br />
[[File:WT_Fig1.PNG|650px|center|]]<br />
<br />
The paper mentions that these pooling methods, although simple and effective, have shortcomings. Max pooling can omit details from an image if the important features have less intensity than the insignificant ones (unless the neural network has learnt to inverse the intensity level), and also commonly overfits. On the other hand, average pooling can dilute important features if the data is averaged with values of significantly lower intensities. Figure 2 displays an image of this.<br />
<br />
[[File:WT_Fig2.PNG|650px|center|]]<br />
<br />
To account for the above-mentioned issues, probabilistic pooling methods were introduced, namely mixed pooling and; stochastic pooling. Mixed pooling is a simple method which just combines the max and the average pooling by randomly selecting one method over the other during training. Mixed pooling can be applied for all features, mixed between features, or mixed between regions for different features. Stochastic pooling on the other hand randomly samples within a receptive field with the activation values as the probabilities. These are calculated by taking each activation value and dividing it by the sum of all activation values in the grid so that the probabilities sum to 1.<br />
<br />
Figure 3 shows an example of how stochastic pooling works. On the left is a 3x3 grid filled with activations. The middle grid is the corresponding probability for each activation. The activation in the middle was randomly selected (it had a 13% chance of getting selected). Because the stochastic pooling is based on the probability of the pixels, it is able to avoid the shortcomings of max and mean pooling mentioned above.<br />
<br />
[[File:paper21-stochasticpooling.png|650px|center|]]<br />
<br />
== Wavelet Background ==<br />
Data or signals tend to be composed of slowly changing trends (low frequency) as well as fast-changing transients (high frequency). Similarly, images have smooth regions of intensity which are perturbed by edges or abrupt changes. We know that these abrupt changes can represent features that are of great importance to us when we perform deep learning. Wavelets are a class of functions that are well localized in time and frequency. The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet <math>\psi(t)=e^{-2 \pi it}</math>. The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The ability of wavelets to be localized in time and space is what makes it suitable for detecting the abrupt changes in an image well. <br />
<br />
Essentially, a wavelet is a fast decaying oscillating signal with zero mean that only exists for a fixed duration and can be scaled and shifted in time. There are some well-defined types of wavelets as shown in Figure 3. The key characteristic of wavelets for us is that they have a band-pass characteristic, and the band can be adjusted based on the scaling and shifting. <br />
<br />
[[File:WT_Fig3.jpg|650px|center|]]<br />
<br />
The paper uses discrete wavelet transform and more specifically a faster variation called Fast Wavelet Transform (FWT) using the Haar wavelet. There also exists a continuous wavelet transform. The main difference in these is how the scale and shift parameters are selected.<br />
<br />
== Discrete Wavelet Transform General==<br />
The discrete wavelet transforms for images is essentially applying a low pass and high pass filter to your image where the transfer functions of the filters are related and defined by the type of wavelet used (Haar in this paper). This is shown in the figures below, which also show the recursive nature of the transform. For an image, the per-row transform is taken first. This results in a new image where the first half is a low-frequency sub-band and the second half is the high-frequency sub-band. Then this new image is transformed again per column, resulting in four sub-bands. Generally, the low-frequency content approximates the image and the high-frequency content represents abrupt changes. Therefore, one can simply take the LL band and perform the transformation again to sub-sample even more.<br />
<br />
[[File:WT_Fig8.png|650px|center|]]<br />
<br />
[[File:WT_Fig9.png|650px|center|]]<br />
<br />
In left half of the above image we see a grid containing four different transformations of the same initial image. Each transform has been done by applying a row wise convolution with a wavelet of either high or low frequency, then a column wise convolution with another wavelet of either high or low frequency. The four choices of frequency (LL, LH, HL, HH) result in four different images. The top left image is the result of applying a low frequency wavelet convolution to the original image both row wise and column wise. The bottom left image is the result of first applying a high frequency wavelet convolution row wise and then applying a low frequency wavelet convolution column wise. Since the LL (top right) transformation preserves the original image best, it is then used in this process again to generate the grid of smaller images that can be seen in the top center-right of the above image. The images in this smaller grid are called second order coefficients.<br />
<br />
== DWT example using Haar Wavelet ==<br />
Suppose we have an image represented by the following pixels:<br />
<br />
\begin{align}<br />
\begin{bmatrix} <br />
100 & 50 & 60 & 150 \\<br />
20 & 60 & 40 & 30 \\<br />
50 & 90 & 70 & 82 \\<br />
74 & 66 & 90 & 58 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
For each level of the DWT using the Haar wavelet, we will perform the transform on the rows first and then the columns. For the row pass, we transform each row as follows:<br />
* For each row i = <math>[i_{1}, i_{2}, i_{3}, i_{4}]</math> of the input image, transform the row to <math>i_{t}</math> via<br />
<br />
\begin{align}<br />
i_{t} = [(i_{1} + i_{2}) / 2, (i_{3} + i_{4}) / 2, (i_{1}, - i_{2}) / 2, (i_{3} - i_{4}) / 2]<br />
\end{align}<br />
<br />
After the row transforms, the images looks as follows:<br />
\begin{align}<br />
\begin{bmatrix} <br />
75 & 105 & 25 & -45 \\<br />
40 & 35 & -20 & 5 \\<br />
70 & 76 & -20 & -6 \\<br />
70 & 74 & 4 & 16 \\<br />
\end{bmatrix}<br />
\end{align}<br />
<br />
Now we apply the same method to the columns in the exact same way.<br />
<br />
== Proposed Method ==<br />
The proposed method uses subbands from the second level FWT and discards the first level subbands. The authors postulate that this method is more 'organic' in capturing the data compression and will create less artifacts that may affect the image classification.<br />
=== Forward Propagation ===<br />
FWT can be expressed by <math>W_\varphi[j + 1, k] = h_\varphi[-n]*W_\varphi[j,n]|_{n = 2k, k <= 0}</math> and <math>W_\psi[j + 1, k] = h_\psi[-n]*W_\psi[j,n]|_{n = 2k, k <= 0}</math> where <math>\varphi</math> is the approximation function, <math>\psi</math> is the detail function, <math>W_\varphi</math>, <math>W_\psi</math>, are approximation and detail coefficients, <math>h_\varphi[-n]</math> and <math>h_\psi[-n]</math> are time reversed scaling and wavelet vectors, <math>(n)</math> represents the sample in the vector, and <math>j</math> denotes the resolution level. To apply to images, FWT is first applied on the rows and then the columns. If a low (L) and high(H) sub-band is extracted from the rows and similarly for the columns than at each level there is 4 sub-bands (LH, HL, HH, and LL) where LL will further be decomposed into the level 2 decomposition. <br />
<br />
Using the level 2 decomposition sub-bands, the Inverse Fast Wavelet Transform (IFWT) is used to obtain the resulting sub-sampled image, which is sub-sampled by a factor of two. The Equation for IFWT is <math>W_\varphi[j, k] = h_\varphi[-n]*W_\varphi[j + 1,n] + h_\psi[-n]*W_\psi[j + 1,n]|_{n = \frac{k}{2}, k <= 0}</math> where the parameters are the same as previously explained. Figure 4 displays the algorithm for the forward propagation.<br />
<br />
[[File:WT_Fig6.PNG|650px|center|]]<br />
<br />
=== Back Propagation ===<br />
This is simply the reverse of the forward propagation. The image feature first has to be converted into the sub-bands using the 1st order wavelet decomposition. The sub-bands are upsampled by a factor of 2 and then backpropagated through the IDWT to achieve the final image. Figure 5 displays the algorithm.<br />
<br />
[[File:WT_Fig7.PNG|650px|center|]]<br />
<br />
== Results ==<br />
The authors tested on MNIST, CIFAR-10, SHVN, and KDEF and the paper provides comprehensive results for each. Stochastic gradient descent was used and the Haar wavelet is used due to its even, square subbands. The network for all datasets except MNIST is loosely based on (Zeiler & Fergus, 2013). The authors keep the network consistent but change the pooling method for each dataset. They also experiment with dropout and Batch Normalization to examine the effects of regularization on their method. All pooling methods compared use a 2x2 window, and a consistent pooling method was used for all pooling layers of a network. The overall results teach us that the pooling method should be chosen specifically for the type of data we have. In some cases, wavelet pooling may perform the best, and in other cases, other methods may perform better, if the data is more suited for those types of pooling.<br />
<br />
=== MNIST ===<br />
Figure 7 shows the network how's architecture was based on an example of MNIST structure from MatConvNet, with batch normalization. Table 1 shows the algorithms accuracy. It can be seen that wavelet pooling achieves the best accuracy from all pooling methods compared. Figure 8 shows the energy of each method per epoch. As can be noted by Figure 8 average and wavelet pooling show a smoother descent in learning and error reduction.<br />
<br />
[[File:WT_Fig4.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig8.png|800px|center]]<br />
<br />
[[File:WT_Tab1.PNG|650px|center|]]<br />
<br />
=== CIFAR-10 ===<br />
In order to investigate the performance of different pooling methods, two types of networks are trained based on CIFAR-10. The first one is the regular CNN and the second one is the network with dropout and batch normalization. Figure 9 shows the network and Tables 2 and 3 shows the accuracy without and with dropout. Average pooling achieves the best accuracy but wavelet pooling is still competitive, while max pooling overfitted on the validation data fairly quickly as shown by the right energy curve in Figure 10 (although the accuracy performance is not significantly worse when dropout and batch normalization are applied).<br />
<br />
[[File:WT_Fig5.PNG|650px|center|]]<br />
<br />
[[File:paper21_fig10.png|800px|center]]<br />
<br />
[[File:WT_Tab2.PNG|650px|center|]]<br />
<br />
[[File:WT_Tab3.PNG|650px|center|]]<br />
<br />
===SHVN===<br />
Figure 11 shows the network and Tables 4 and 5 shows the accuracy without and with dropout. The proposed method does not perform well in this experiment. <br />
<br />
[[File: a.png|650px|center|]]<br />
<br />
[[File:paper21_fig12.png|800px|center]]<br />
<br />
[[File: b.png|650px|center|]]<br />
<br />
===KDEF===<br />
The authors experimented with pooling methods + dropout on the KDEF dataset (which consists of 4,900 images of 35 people portraying varying emotions through facial expressions under different poses, 3,900 of which were randomly assigned to be used for training). The data was treated for errors (e.g. corrupt images) and resized to 128x128 for memory and time constraints. <br />
<br />
Figure 13 below shows the network structure. Figure 14 shows the energy curve of the competing models on training and validation sets as the number of epochs increases, and Table 6 shows the accuracy performance. Average pooling demonstrated the highest accuracy, with wavelet pooling coming in second and max-pooling a close third. However, stochastic and wavelet pooling exhibited more stable learning progression compared to the other methods, and max-pooling eventually overfitted. <br />
<br />
[[File:kdef_struc.PNG|700px|center|]]<br />
[[File:kdef_curve.PNG|750px|center|]]<br />
[[File:kdef_accu.PNG|550px|center|]]<br />
<br />
== Computational Complexity ==<br />
The authors explain that their paper is a proof of concept and is not meant to implement wavelet pooling in the most efficient way. The table below displays a comparison of the number of mathematical operations for each method according to the dataset. It can be seen that wavelet pooling is significantly worse. The authors explain that through good implementation and coding practices, the method can prove to be viable.<br />
<br />
[[File:WT_Tab4.PNG|650px|center|]]<br />
<br />
== Criticism ==<br />
=== Positive ===<br />
* Wavelet Pooling achieves competitive performance with standard go-to pooling methods<br />
* Leads to a comparison of discrete transformation techniques for pooling (DCT, DFT)<br />
=== Negative ===<br />
* Only 2x2 pooling window used for comparison<br />
* Highly computationally extensive<br />
* Not as simple as other pooling methods<br />
* Only one wavelet used (HAAR wavelet)<br />
<br />
== References ==<br />
* Travis Williams and Robert Li. Wavelet Pooling for Convolutional Neural Networks. ICLR 2018.<br />
* J. Anthony Parker, Robert V. Kenyon, and Donald E. Troxel. Comparison of interpolating methods for image resampling. IEEE Transactions on Medical Imaging, 2(1):31–39, 1983.<br />
* Matthew Zeiler and Robert Fergus. Stochastic pooling for regularization of deep convolutional neural networks. In Proceedings of the International Conference on Learning Representation (ICLR), 2013.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Spherical_CNNs&diff=36428Spherical CNNs2018-04-21T02:53:40Z<p>W285liu: /* Related Work */</p>
<hr />
<div>= Introduction =<br />
Convolutional Neural Networks (CNNs), or network architectures involving CNNs, are the current state of the art for learning 2D image processing tasks such as semantic segmentation and object detection. CNNs work well in large part due to the property of being translationally equivariant. This property allows a network trained to detect a certain type of object to still detect the object even if it is translated to another position in the image. However, this does not correspond well to spherical signals since projecting a spherical signal onto a plane will result in distortions, as demonstrated in Figure 1. There are many different types of spherical projections onto a 2D plane, as most people know from the various types of world maps, none of which provide all the necessary properties for rotation-invariant learning. Applications where spherical CNNs can be applied include omnidirectional vision for robots, molecular regression problems, and weather/climate modelling.<br />
<br />
[[File:paper26-fig1.png|center]]<br />
<br />
The implementation of a spherical CNN is challenging mainly because no perfectly symmetrical grids for the sphere exists which makes it difficult to define the rotation of a spherical filter by one pixel and the computational efficiency of the system.<br />
<br />
The main contributions of this paper are the following:<br />
# The theory of spherical CNNs. The authors provide mathematical foundations for translation equivariance under a spherical framework.<br />
# The first automatically differentiable implementation of the generalized Fourier transform for <math>S^2</math> and SO(3). The provided PyTorch code by the authors is easy to use, fast, and memory efficient.<br />
# The first empirical support for the utility of spherical CNNs for rotation-invariant learning problems. They apply it to spherical MNIST, 3D shape classification, and molecular energy regression.<br />
<br />
=== Note: Translationally equivariant === <br />
<br />
Equivariant to translation means that a translation of input features results in an equivalent translation of outputs. So if your pattern 0,3,2,0,0 on the input results in 0,1,0,0 in the output, then the pattern 0,0,3,2,0 might lead to 0,0,1,0.<br />
<br />
= Notation =<br />
Below are listed several important terms:<br />
* '''Unit Sphere''' <math>S^2</math> is defined as a sphere where all of its points are distance of 1 from the origin. The unit sphere can be parameterized by the spherical coordinates <math>\alpha ∈ [0, 2π]</math> and <math>β ∈ [0, π]</math>. This is a two-dimensional manifold with respect to <math>\alpha</math> and <math>β</math>.<br />
* '''<math>S^2</math> Sphere''' The three dimensional surface from a 3D sphere<br />
* '''Spherical Signals''' In the paper spherical images and filters are modeled as continuous functions <math>f : s^2 → \mathbb{R}^K</math>. K is the number of channels. Such as how RGB images have 3 channels a spherical signal can have numerous channels describing the data. Examples of channels which were used can be found in the experiments section.<br />
* '''Rotations - SO(3)''' The group of 3D rotations on an <math>S^2</math> sphere. Sometimes called the "special orthogonal group". In this paper the ZYZ-Euler parameterization is used to represent SO(3) rotations with <math>\alpha, \beta</math>, and <math>\gamma</math>. Any rotation can be broken down into first a rotation (<math>\alpha</math>) about the Z-axis, then a rotation (<math>\beta</math>) about the new Y-axis (Y'), followed by a rotation (<math>\gamma</math>) about the new Z axis (Z"). [In the rest of this paper, to integrate functions on SO(3), the authors use a rotationally invariant probability measure on the Borel subsets of SO(3). This measure is an example of a Haar measure. Haar measures generalize the idea of rotationally invariant probability measures to general topological groups. For more on Haar measures, see (Feldman 2002) ]<br />
<br />
= Related Work =<br />
It is well understood that the power of CNNs stems in large part from their ability to exploit<br />
(translational) symmetries though a combination of weight sharing and translation equivariance.<br />
It thus becomes natural to consider generalizations that exploit larger groups of symmetries, and<br />
indeed this has been the subject of several recent papers by Gens and Domingos (2014); Olah (2014);<br />
Dieleman et al. (2015; 2016); Cohen and Welling (2016); Ravanbakhsh et al. (2017); Zaheer et al.<br />
(2017b); Guttenberg et al. (2016); Cohen and Welling (2017). With the exception of SO(2)-steerable<br />
networks (Worrall et al., 2017; Weiler et al., 2017), these networks are all limited to discrete groups,<br />
such as discrete rotations acting on planar images or permutations acting on point clouds. Other very<br />
recent work is concerned with the analysis of spherical images, but does not define an equivariant<br />
architecture (Su and Grauman, 2017; Boomsma and Frellsen, 2017). Our work is the first to achieve<br />
equivariance to a continuous, non-commutative group (SO(3)), and the first to use the generalized<br />
Fourier transform for fast group correlation. A preliminary version of this work appeared as Cohen<br />
et al. (2017).<br />
To efficiently perform cross-correlations on the sphere and rotation group, we use generalized FFT<br />
algorithms. Generalized Fourier analysis, sometimes called abstract- or noncommutative harmonic<br />
analysis, has a long history in mathematics and many books have been written on the subject<br />
(Sugiura, 1990; Taylor, 1986; Folland, 1995). For a good engineering-oriented treatment which<br />
covers generalized FFT algorithms, see (Chirikjian and Kyatkin, 2001). Other important works<br />
include (Driscoll and Healy, 1994; Healy et al., 2003; Potts et al., 1998; Kunis and Potts, 2003; Drake<br />
et al., 2008; Maslen, 1998; Rockmore, 2004; Kostelec and Rockmore, 2007; 2008; Potts et al., 2009;<br />
Makadia et al., 2007; Gutman et al., 2008).<br />
<br />
= Correlations on the Sphere and Rotation Group =<br />
Spherical correlation is like planar correlation except instead of translation, there is rotation. The definitions for each are provided as follows:<br />
<br />
'''Planar correlation''' The value of the output feature map at translation <math>\small x ∈ Z^2</math> is computed as an inner product between the input feature map and a filter, shifted by <math>\small x</math>.<br />
<br />
'''The unit sphere''' <math>S^2</math> can be defined as the set of points <math>x ∈ R^3</math> with norm 1. It is a two-dimensional manifold, which can be parameterized by spherical coordinates α ∈ [0, 2π] and β ∈ [0, π]. <br />
<br />
'''Spherical Signals''' We model spherical images and filters as continuous functions f : <math>S^2</math> → <math>R^K</math>, where K is the number of channels.<br />
<br />
'''Rotations''' The set of rotations in three dimensions is called SO(3), the “special orthogonal group”. Rotations can be represented by 3 × 3 matrices that preserve distance (i.e. ||Rx|| = ||x||) and orientation (det(R) = +1). If we represent points on the sphere as 3D unit vectors x, we can perform a rotation using the matrix-vector product Rx. The rotation group SO(3) is a three-dimensional manifold, and can be parameterized by ZYZ-Euler angles α ∈ [0, 2π], β ∈ [0, π], and γ ∈ [0, 2π].<br />
<br />
'''Spherical correlation''' The value of the output feature map evaluated at rotation <math>\small R ∈ SO(3)</math> is computed as an inner product between the input feature map and a filter, rotated by <math>\small R</math>.<br />
<br />
'''Rotation of Spherical Signals''' The paper introduces the rotation operator <math>L_R</math>. The rotation operator simply rotates a function (which allows us to rotate the the spherical filters) by <math>R^{-1}</math>. With this definition we have the property that <math>L_{RR'} = L_R L_{R'}</math>.<br />
<br />
'''Inner Products''' The inner product of spherical signals is simply the integral summation on the vector space over the entire sphere.<br />
<br />
<math>\langle\psi , f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (x)f_k (x)dx</math><br />
<br />
<math>dx</math> here is SO(3) rotation invariant and is equivalent to <math>d \alpha sin(\beta) d \beta / 4 \pi </math> in spherical coordinates. This comes from the ZYZ-Euler paramaterization where any rotation can be broken down into first a rotation about the Z-axis, then a rotation about the new Y-axis (Y'), followed by a rotation about the new Z axis (Z"). More details on this are given in Appendix A in the paper.<br />
<br />
By this definition, the invariance of the inner product is then guaranteed for any rotation <math>R ∈ SO(3)</math>. In other words, when subjected to rotations, the volume under a spherical heightmap does not change. The following equations show that <math>L_R</math> has a distinct adjoint (<math>L_{R^{-1}}</math>) and that <math>L_R</math> is unitary and thus preserves orthogonality and distances.<br />
<br />
<math>\langle L_R \psi \,, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (R^{-1} x)f_k (x)dx</math><br />
<br />
::::<math>= \int_{S^2} \sum_{k=1}^K \psi_k (x)f_k (Rx)dx</math><br />
<br />
::::<math>= \langle \psi , L_{R^{-1}} f \rangle</math><br />
<br />
'''Spherical Correlation''' With the above knowledge the definition of spherical correlation of two signals <math>f</math> and <math>\psi</math> is:<br />
<br />
<math>[\psi \star f](R) = \langle L_R \psi \,, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (R^{-1} x)f_k (x)dx</math><br />
<br />
The output of the above equation is a function on SO(3). This can be thought of as for each rotation combination of <math>\alpha , \beta , \gamma </math> there is a different volume under the correlation. The authors make a point of noting that previous work by Driscoll and Healey only ensures circular symmetries about the Z axis and their new formulation ensures symmetry about any rotation.<br />
<br />
'''Rotation of SO(3) Signals''' The first layer of Spherical CNNs take a function on the sphere (<math>S^2</math>) and output a function on SO(3). Therefore, if a Spherical CNN with more than one layer is going to be built there needs to be a way to find the correlation between two signals on SO(3). The authors then generalize the rotation operator (<math>L_R</math>) to encompass acting on signals from SO(3). This new definition of <math>L_R</math> is as follows: (where <math>R^{-1}Q</math> is a composition of rotations, i.e. multiplication of rotation matrices)<br />
<br />
<math>[L_Rf](Q)=f(R^{-1} Q)</math><br />
<br />
'''Rotation Group Correlation''' The correlation of two signals (<math>f,\psi</math>) on SO(3) with K channels is defined as the following:<br />
<br />
<math>[\psi \star f](R) = \langle L_R \psi , f \rangle = \int_{SO(3)} \sum_{k=1}^K \psi_k (R^{-1} Q)f_k (Q)dQ</math><br />
<br />
where dQ represents the ZYZ-Euler angles <math>d \alpha sin(\beta) d \beta d \gamma / 8 \pi^2 </math>. A complete derivation of this can be found in Appendix A.<br />
<br />
'''Equivariance''' The equivariance for the rotation group correlation is similarly demonstrated. A layer is equivariant if for some operator <math>T_R</math>, <math>\Phi \circ L_R = T_R \circ \Phi</math>, and: <br />
<br />
<math>[\psi \star [L_Qf]](R) = \langle L_R \psi , L_Qf \rangle = \langle L_{Q^{-1} R} \psi , f \rangle = [\psi \star f](Q^{-1}R) = [L_Q[\psi \star f]](R) </math>.<br />
<br />
= Implementation with GFFT =<br />
The authors leverage the Generalized Fourier Transform (GFT) and Generalized Fast Fourier Transform (GFFT) algorithms to compute the correlations outlined in the previous section. The Fast Fourier Transform (FFT) can compute correlations and convolutions efficiently by means of the Fourier theorem. The Fourier theorem states that a continuous periodic function can be expressed as a sum of a series of sine or cosine terms (called Fourier coefficients). The FT can be generalized to <math>S^2</math> and SO(3) and is then called the GFT. The GFT is a linear projection of a function onto orthogonal basis functions. The basis functions are a set of irreducible unitary representations for a group (such as for <math>S^2</math> or SO(3)). For <math>S^2</math> the basis functions are the spherical harmonics <math>Y_m^l(x)</math>. For SO(3) these basis functions are called the Wigner D-functions <math>D_{mn}^l(R)</math>. For both sets of functions the indices are restricted to <math>l\geq0</math> and <math>-l \leq m,n \geq l</math>. The Wigner D-functions are also orthogonal so the Fourier coefficients can be computed by the inner product with the Wigner D-functions (See Appendix C for complete proof). The Wigner D-functions are complete which means that any function (which is well behaved) on SO(3) can be expressed as a linear combination of the Wigner D-functions. The GFT of a function on SO(3) is thus:<br />
<br />
<math>\hat{f^l} = \int_X f(x) D^l(x)dx</math><br />
<br />
where <math>\hat{f}</math> represents the Fourier coefficients. For <math>S^2</math> we have the same equation but with the basis functions <math>Y^l</math>.<br />
<br />
The inverse SO(3) Fourier transform is:<br />
<br />
<math>f(R)=[\mathcal{F}^{-1} \hat{f}](R) = \sum_{l=0}^b (2l + 1) \sum_{m=-l}^l \sum_{n=-l}^l \hat{f_{mn}^l} D_{mn}^l(R) </math><br />
<br />
The bandwidth b represents the maximum frequency and is related to the resolution of the spatial grid. Kostelec and Rockmore are referenced for more knowledge on this topic.<br />
<br />
The authors give proofs (Appendix D) that the SO(3) correlation satisfies the Fourier theorem and the <math>S^2</math> correlation of spherical signals can be computed by the outer products of the <math>S^2</math>-FTs (Shown in Figure 2).<br />
<br />
[[File:paper26-fig2.png|center]]<br />
<br />
A high-level, approximately-correct, somewhat intuitive explanation of the above figure is that the spherical signal <math> f </math> parameterized over <math> \alpha </math> and <math> \beta </math> having <math> k </math> channels is being correlated with a single filter <math> \psi </math> with the end result being a 3-D feature map on SO(3) (parameterized by Euler angles). The size in <math> \alpha </math> and <math> \beta </math> is the kernel size. The index <math> l </math> going from 0 to 3 correspond the degree of the basis functions used in the Fourier transform. As the degree goes up, so does the dimensionality of vector-valued (for spheres) basis functions. The signals involved are discrete, so the maximum degree (analogous to number of Fourier coefficients) depends on the resolution of the signal. The SO(3) basis functions are matrix-valued, but because <math> S^2 = SO(3)/SO(2) </math>, it ends up that the sphere basis functions correspond to one column in the matrix-valued SO(3) basis functions, which is why the outer product in the figure works.<br />
<br />
The GFFT algorithm details are taken from Kostelec and Rockmore. The authors claim they have the first automatically differentiable implementation of the GFT for <math>S^2</math> and SO(3). The authors do not provide any run time comparisons for real time applications (they just mentioned that FFT can be computed in <math>O(n\mathrm{log}n)</math> time as opposed to <math>O(n^2)</math> for FT) or any comparisons on training times with/without GFFT. However, they do provide the source code of their implementation at: https://github.com/jonas-koehler/s2cnn.<br />
<br />
= Experiments =<br />
The authors provide several experiments. The first set of experiments are designed to show the numerical stability and accuracy of the outlined methods. The second group of experiments demonstrates how the algorithms can be applied to current problem domains.<br />
<br />
==Equivariance Error==<br />
In this experiment the authors try to show experimentally that their theory of equivariance holds. They express that they had doubts about the equivariance in practice due to potential discretization artifacts since equivariance was proven for the continuous case, with the potential consequence of equivariance not holding being that the weight sharing scheme becomes less effective. The experiment is set up by first testing the equivariance of the SO(3) correlation at different resolutions. 500 random rotations and feature maps (with 10 channels) are sampled. They then calculate the approximation error <math>\small\Delta = \dfrac{1}{n} \sum_{i=1}^n std(L_{R_i} \Phi(f_i) - \phi(L_{R_i} f_i))/std(\Phi(f_i))</math><br />
Note: The authors do not mention what the std function is however it is likely the standard deviation function as 'std' is the command for standard deviation in MATLAB.<br />
<math>\Phi</math> is a composition of SO(3) correlation layers with filters which have been randomly initialized. The authors mention that they were expecting <math>\Delta</math> to be zero in the case of perfect equivariance. This is due to, as proven earlier, the following two terms equaling each other in the continuous case: <math>\small L_{R_i} \Phi(f_i) - \phi(L_{R_i} f_i)</math>. The results are shown in Figure 3. <br />
<br />
[[File:paper26-fig3.png|center]]<br />
<br />
<math>\Delta</math> only grows with resolution/layers when there is no activation function. With ReLU activation the error stays constant once slightly higher than 0 resolution. The authors indicate that the error must therefore be from the feature map rotation since this type of error is exact only for bandlimited functions.<br />
<br />
==MNIST Data==<br />
The experiment using MNIST data was created by projecting MNIST digits onto a sphere using stereographic projection to create the resulting images as seen in Figure 4.<br />
<br />
[[File:paper26-fig4.png|center]]<br />
<br />
The authors created two datasets, one with the projected digits and the other with the same projected digits which were then subjected to a random rotation. The spherical CNN architecture used was <math>\small S^2</math>conv-ReLU-SO(3)conv-ReLU-FC-softmax and was attempted with bandwidths of 30,10,6 and 20,40,10 channels for each layer respectively. This model was compared to a baseline CNN with layers conv-ReLU-conv-ReLU-FC-softmax with 5x5 filters, 32,64,10 channels and stride of 3. For comparison this leads to approximately 68K parameters for the baseline and 58K parameters for the spherical CNN. Results can be seen in Table 1. It is clear from the results that the spherical CNN architecture made the network rotationally invariant. Performance on the rotated set is almost identical to the non-rotated set. This is true even when trained on the non-rotated set and tested on the rotated set. Compare this to the non-spherical architecture which becomes unusable when rotating the digits.<br />
<br />
[[File:paper26-tab1.png|center]]<br />
<br />
==SHREC17==<br />
The SHREC dataset contains 3D models from the ShapeNet dataset which are classified into categories. It consists of a regularly aligned dataset and a rotated dataset. The models from the SHREC17 dataset were projected onto a sphere by means of raycasting. Different properties of the objects obtained from the raycast of the original model and the convex hull of the model make up the different channels which are input into the spherical CNN.<br />
<br />
<br />
[[File:paper26-fig5.png|center]]<br />
<br />
<br />
The network architecture used is an initial <math>\small S^2</math>conv-BN-ReLU block which is followed by two SO(3)conv-BN-ReLU blocks. The output is then fed into a MaxPool-BN block then a linear layer to the output for final classification. An important note is that the max pooling happens over the group SO(3): if <math>f_k</math> is the <math>\small k</math>-th filter in the final layer, the result of pooling is <math>max_{x \in SO(3)} f_k(x)</math>. 50 features were used for the <math>\small S^2</math> layer, while the two SO(3) layers used 70 and 350 features. Additionally, for each layer the resolution <math>\small b</math> was reduced from 128,32,22 to 7 in the final layer. The architecture for this experiment has ~1.4M parameters, far exceeding the scale of the spherical CNNs in the other experiments.<br />
<br />
This architecture achieves state of the art results on the SHREC17 tasks. The model places 2nd or 3rd in all categories but was not submitted as the SHREC17 task is closed. Table 2 shows the comparison of results with the top 3 submissions in each category. In the table, P@N stands for precision, R@N stands for recall, F1@N stands for F-score, mAP stands for mean average precision, and NDCG stands for normalized discounted cumulative gain in relevance based on whether the category and subcategory labels are predicted correctly. The authors claim the results show empirical proof of the usefulness of spherical CNNs. They elaborate that this is largely due to the fact that most architectures on the SHREC17 competition are highly specialized whereas their model is fairly general.<br />
<br />
<br />
[[File:paper26-tab2.png|center]]<br />
<br />
==Molecular Atomization==<br />
In this experiment a spherical CNN is implemented with an architecture resembling that of ResNet. They use the QM7 dataset (Blum et al. 2009) which has the task of predicting atomization energy of molecules. The QM7 dataset is a subset of GDB-13 (database of organic molecules) composed of all molecules up to 23 atoms. The positions and charges given in the dataset are projected onto the sphere using potential functions. For each atom, a sphere is defined around its position with the radius of the sphere kept uniform across all atoms. Next, the radius is chosen as the minimal radius so no intersections between atoms occur in the training set. Finally, using potential functions, a T channel spherical signal is produced for each atom in the molecule as shown in the figure below. A summary of their results is shown in Table 3 along with some of the spherical CNN architecture details. It shows the different RMSE obtained from different methods. The results from this final experiment also seem to be promising as the network the authors present achieves the second best score. They also note that the first place method grows exponentially with the number of atoms per molecule so is unlikely to scale well.<br />
<br />
[[File:paper26-tab3.png|center]]<br />
<br />
[[File:paper26-f6.png|center]]<br />
<br />
= Conclusions =<br />
This paper presents a novel architecture called Spherical CNNs and evaluate it on 2 important learning problems and introduces a trainable signal representation for spherical signals rotationally equivariant by design. The paper defines <math>\small S^2</math> and SO(3) cross correlations, shows the theory behind their rotational invariance for continuous functions, and demonstrates that the invariance also applies to the discrete case. An effective GFFT algorithm was implemented and evaluated on two very different datasets with close to state of the art results, demonstrating that there are practical applications to Spherical CNNs. The network is able to generalize across rotation and generate comparative results in the process.<br />
<br />
For future work the authors believe that improvements can be obtained by generalizing the algorithms to the SE(3) group (SE(3) simply adds translations in 3D space to the SO(3) group). The authors also briefly mention their excitement for applying Spherical CNNs to omnidirectional vision such as in drones and autonomous cars. They state that there is very little publicly available omnidirectional image data which could be why they did not conduct any experiments in this area.<br />
<br />
= Commentary =<br />
The reviews on Spherical CNNs are very positive and it is ranked in the top 1% of papers submitted to ICLR 2018. Positive points are the novelty of the architecture, the wide variety of experiments performed, and the writing. One critique of the original submission is that the related works section only lists, instead of describing, previous methods and that a description of the methods would have provided more clarity. The authors have since expanded the section however I found that it is still limited which the authors attribute to length limitations. Another critique is that the evaluation does not provide enough depth. For example, it would have been great to see an example of omnidirectional vision for spherical networks. However, this is to be expected as it is just the introduction of spherical CNNs and more work is sure to come.<br />
<br />
= Source Code =<br />
Source code is available at:<br />
https://github.com/jonas-koehler/s2cnn<br />
<br />
= Sources =<br />
* T. Cohen et al. Spherical CNNs, 2018.<br />
* J. Feldman. Haar Measure. http://www.math.ubc.ca/~feldman/m606/haar.pdf<br />
* P. Kostelec, D. Rockmore. FFTs on the Rotation Group, 2008.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&diff=36427stat946w18/Tensorized LSTMs2018-04-21T02:50:02Z<p>W285liu: /* Deep LSTMs */</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 <math>y_t</math> at each time-step t∈ {1, ..., T} given an observed input sequence <math>x_{1:t} = {x_1,x_2, ···, x_t}</math>, where <math>x_t∈R^R</math> and <math>y_t∈R^S</math> are vectors. RNNs learn how to use a hidden state vector <math>h_t ∈ R^M</math> 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 <math>h_{t-1}^{cat} ∈R^{R+M}</math> is the concatenation of the current input <math>x_t</math> and the previous hidden state <math>h_{t−1}</math>, 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 />
<math>W^h∈R^{(R+M)\times M} </math> guarantees each hidden state provided by the previous step is of dimension M. <math> a_t ∈R^M </math> is the hidden activation, and φ(·) is the element-wise hyperbolic tangent. Finally, the output <math> y_t </math> 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 <math>W^y∈R^{M×S}</math> and <math>b^y∈R^S</math>, and <math>\varphi(·)</math> can be any differentiable function. Note that the <math>\phi</math> is a non-linear, element-wise function which generates hidden output, while <math>\varphi</math> 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 <math>h_t∈R^{M}</math> to become <math>Ht∈R^{P×M}</math>, '''where P is the tensor size,''' and '''M the channel size'''. We locally-connect the first dimension of <math>H_t</math> (which is P - the tensor size) in order to share parameters, and fully-connect the second dimension of <math>H_t</math> (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 <math>H_t</math> 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 <math>x_t</math> with a (delayed) future output. In doing this, we need to ensure that the output <math>y_t</math> is separable, i.e., not influenced by any future input <math>x_{t^{'}}</math> <math>(t^{'}>t)</math>. Thus, we concatenate the projection of <math>x_t</math> to the top of the previous hidden state <math>H_{t−1}</math>, then gradually shift the input information down when the temporal computation proceeds, and finally generate <math>y_t</math> from the bottom of <math>H_{t+L−1}</math>, 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 <math>H_t</math> 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 <math>H^{cat}_{t−1}∈R^{(P+1)×M}</math> be the concatenated hidden state, and <math>p∈Z_+</math> the location at a tensor. The channel vector <math>h^{cat}_{t−1, p }∈R^M</math> at location p of <math>H^{cat}_{t−1}</math> (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 > 1 \hspace{1cm} (6)<br />
\end{align}<br />
<br />
where <math>W^x ∈ R^{R×M}</math> and <math>b^x ∈ R^M</math> (recall the dimension of input x is R). Then, the update of tensor <math>H_t</math> 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 <math>W^h∈R^{K×M^i×M^o}</math> is the kernel weight of size K, with <math>M^i =M</math> input channels and <math>M^o =M</math> output channels, <math>b^h ∈ R^{M^o}</math> is the kernel bias, <math>A_t ∈ R^{P×M^o}</math> is the hidden activation, and <math>\circledast</math> 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 <math>y_t</math> from the channel vector <math>h_{t+L−1,P}∈R^M</math> which is located at the bottom of <math>H_{t+L−1}</math>:<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 <math>W^y ∈R^{M×S}</math> and <math>b^y ∈R^S</math>. To guarantee that the receptive field of <math>y_t</math> 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. <math> W^x</math> and <math>b_x</math>''' connect input to the first hidden node<br />
<br />
'''2. <math> W^h</math> and <math>b_h</math>''' convolute between layers<br />
<br />
'''3. <math> W^y</math> and <math>b_y</math>''' 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 <math>C_{t-1}</math> 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. '''<math>\{W^h, b^h \}</math>:''' Kernel of size K <br />
<br />
2. '''<math>A^g_t, A^i_t, A^f_t, A^o_t \in \mathbb{R}^{P\times M}</math>:''' Activations for the new content <math>G_t</math><br />
<br />
3. '''<math>I_t</math>:''' Input gate<br />
<br />
4. '''<math>F_t</math>:''' Forget gate<br />
<br />
5. '''<math>O_t</math>:''' Output gate<br />
<br />
6. '''<math>C_t \in \mathbb{R}^{P\times M}</math>:''' 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 <math>{W^h, b^h}</math> has additional <K> output channels to generate the activation <math>A^q_t ∈ R^{P×<K>}</math> for the dynamic kernel bank <math>Q_t∈R^{P × <K>}</math>, <math>q_{t,p}∈R^{<K>}</math> is the vectorized adaptive kernel at the location p of <math>Q_t</math>, and <math>W^c_t(p) ∈ R^{K×1×1}</math> is the dynamic kernel of size K with a single input/output channel, which is reshaped from <math>q_{t,p}</math>. Each channel of the previous memory cell <math>C_{t-1}</math> is convolved with <math>W_t^c(p)</math> whose values vary with <math>p</math>, to form a memory cell convolution, which produces a convolved memory cell <math>C_{t-1}^{conv} \in \mathbb{R}^{P\times M}</math>. 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 <math>C_{t-1}</math> 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 <math>Q_t</math>. 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. [3] 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 <math>\mathbf{Z}</math>, <math>\mathbf{\hat{Z}}</math>, <math>\mathbf{\Gamma}</math>, <math>\mathbf{B} \in \mathbb{R}^{P \times M^z}</math> are the original tensor, normalized tensor, gain parameter and bias parameter. The <math>m^z</math>-th channel of <math>\mathbf{Z}</math> 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 <math>z^{\mu}</math>, <math>z^{\sigma} \in \mathbb{R}^P</math> are the mean and standard deviation along the channel dimension of <math>\mathbf{Z}</math>, and <math>\hat{z_{m^z}} \in \mathbb{R}^P</math> is the <math>m^z</math>-th channel <math>\mathbf{\hat{Z}}</math>. 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 [35] 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 />
= Related work =<br />
=== Convolutional LSTMs ===<br />
<br />
Convolutional LSTMs are proposed to parallelize the computation of LSTMs when the input at each timestep is structured<br />
[[File:clstm.png|150px|center||Figure 1: Example of Convolutional LSTMs]]<br />
<br />
=== Deep LSTMs ===<br />
Deep LSTMs. Deep LSTMs (dLSTMs) extend sLSTMs by making them deeper (see Fig. 7(b)-(d)).<br />
To keep the parameter number small and ease training, Graves [22], Kalchbrenner et al. [30], Mujika<br />
et al. [38], Zilly et al. [54] apply another RNN/LSTM along the depth direction of dLSTMs, which,<br />
however, multiplies the runtime. Though there are implementations to accelerate the deep computation<br />
[1, 16], they generally aim at simple architectures such sLSTMs. Compared with dLSTMs, tLSTM<br />
performs the deep computation with little additional runtime, and employs a cross-layer convolution to<br />
enable the feedback mechanism. Moreover, the capacity of tLSTM can be increased more efficiently<br />
by using higher-dimensional tensors, whereas in dLSTM all hidden layers as a whole only equal to a<br />
2D tensor (i.e., a stack of hidden vectors), the dimensionality of which is fixed.<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. <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> (2017)<br />
#Ali Ghodsi, <Deep Learning: STAT 946 - Winter 2018><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>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=PointNet%2B%2B:_Deep_Hierarchical_Feature_Learning_on_Point_Sets_in_a_Metric_Space&diff=36420PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space2018-04-21T02:47:05Z<p>W285liu: </p>
<hr />
<div>= Introduction =<br />
This paper builds off of ideas from PointNet (Qi et al., 2017). The name PointNet is derived from the network's input - a point cloud. A point cloud is a set of three dimensional points that each have coordinates <math> (x,y,z) </math>. These coordinates usually represent the surface of an object. For example, a point cloud describing the shape of a torus is shown below.<br />
<br />
[[File:Point cloud torus.gif|thumb|center|Point cloud torus]]<br />
<br />
<br />
Processing point clouds is important in applications such as autonomous driving where point clouds are collected from an onboard LiDAR sensor. These point clouds can then be used for object detection. However, point clouds are challenging to process because:<br />
<br />
# They are unordered. If <math> N </math> is the number of points in a point cloud, then there are <math> N! </math> permutations that the point cloud can be represented.<br />
# The spatial arrangement of the points contains useful information, thus it needs to be encoded.<br />
# The function processing the point cloud needs to be invariant to transformations such as rotation and translations of all points. <br />
<br />
Previously, typical point cloud processing methods handled the challenges of point clouds by transforming the data with a 3D voxel grid or by representing the point cloud with multiple 2D images. When PointNet was introduced, it was novel because it directly took points as its input. PointNet++ improves on PointNet by using a hierarchical method to better capture local structures of the point cloud. <br />
<br />
[[File:point_cloud.png | 400px|thumb|center|Examples of point clouds and their associated task. Classification (left), part segmentation (centre), scene segmentation (right) ]]<br />
<br />
= Related Work =<br />
<br />
This paper provides a new network to understand point clouds. CNNs are the most prominent deep networks which find features in images/videos. However, the convolution process is not applicable to point clouds as point clouds contain unordered set of points with distance metric. Some techniques apply deep learning to unordered sets however they don't account for the distance metric in their model and are sensitive to translation and global normalization. Techniques like volumetric grids and geometric graphs work on the 3D metric space but the problem of non-uniform sampling density hasn't been considered.<br />
<br />
= Review of PointNet =<br />
<br />
The PointNet architecture is shown below. The input of the network is <math> n </math> points, which each have <math> (x,y,z) </math> coordinates. Each point is processed individually through a multi-layer perceptron (MLP). This network creates an encoding for each point; in the diagram, each point is represented by a 1024 dimension vector. Then, using a max pool layer a vector is created that represents the "global signature" of a point cloud. If classification is the task, this global signature is processed by another MLP to compute the classification scores. If segmentation is the task, this global signature is appended to to each point from the "nx64" layer, and these points are processed by a MLP to compute a semantic category score for each point.<br />
<br />
The core idea of the network is to learn a symmetric function on transformed points. Through the T-Nets and the MLP network, a transformation is learned with the hopes of making points invariant to point cloud transformations. Learning a symmetric function solves the challenge imposed by having unordered points; a symmetric function will produce the same value no matter the order of the input. This symmetric function is represented by the max pool layer.<br />
<br />
[[File:pointnet_arch.png | 700px|thumb|center|PointNet architecture. The blue highlighted region is when it is used for classification, and the beige highlighted region is when it is used for segmentation.]]<br />
<br />
= PointNet++ =<br />
<br />
The motivation for PointNet++ is that PointNet does not capture local, fine-grained details. Since PointNet performs a max pool layer over all of its points, information such as the local interaction between points is lost. Max-pooling reduces the dimensionality of the network in a very cheap way (with no parameters), but is only good some tasks since it only provides translatioinal variance. And by translational variance we mean that the same object with slightly change in orientation or posiution might not fire up the neuron that is supposed to recognize that object.<br />
<br />
== Problem Statement ==<br />
<br />
There is a metric space <math> X = (M,d) </math> where <math>d</math> is the metric from a Euclidean space <math>\pmb{\mathbb{R}}^n</math> and <math> M \subseteq \pmb{\mathbb{R}}^n </math> is the set of points. The goal is to learn functions <math>f</math> that take <math>X</math> as the input and produce information of semantic interest about it. In practice, <math>f</math> can often be a classification function that outputs a class label or a segmentation function that outputs a per point label for each member of <math>M</math>.<br />
<br />
== Method ==<br />
<br />
=== High Level Overview ===<br />
[[File:point_net++.png | 700px|thumb|right|PointNet++ architecture]]<br />
<br />
The PointNet++ architecture is shown on the right. The core idea is that a hierarchical architecture is used and at each level of the hierarchy a set of points is processed and abstracted to a new set with less points, i.e.,<br />
<br />
\begin{aligned}<br />
\text{Input at each level: } N \times (d + c) \text{ matrix}<br />
\end{aligned}<br />
<br />
where <math>N</math> is the number of points, <math>d</math> is the coordinate points <math>(x,y,z)</math> and <math>c</math> is the feature representation of each point, and<br />
<br />
\begin{aligned}<br />
\text{Output at each level: } N' \times (d + c') \text{ matrix}<br />
\end{aligned}<br />
<br />
where <math>N'</math> is the new number (smaller) of points and <math>c'</math> is the new feature vector.<br />
<br />
<br />
Each level has three layers: Sampling, Grouping, and PointNet. The Sampling layer selects points that will act as centroids of local regions within the point cloud. The Grouping layer then finds points near these centroids. Lastly, the PointNet layer performs PointNet on each group to encode local information.<br />
<br />
=== Sampling Layer ===<br />
<br />
The input of this layer is a set of points <math>{\{x_1,x_2,...,x_n}\}</math>. The goal of this layer is to select a subset of these points <math>{\{\hat{x}_1, \hat{x}_2,...,\hat{x}_m\}} </math> that will define the centroid of local regions.<br />
<br />
To select these points farthest point sampling is used. This is where <math>\hat{x}_j</math> is the most distant point with regards to <math>{\{\hat{x}_1, \hat{x}_2,...,\hat{x}_{j-1}\}}</math>. This ensures coverage of the entire point cloud opposed to random sampling.<br />
<br />
=== Grouping Layer ===<br />
<br />
The objective of the grouping layer is to form local regions around each centroid by grouping points near the selected centroids. The input is a point set of size <math>N \times (d + c)</math> and the coordinates of the centroids <math>N' \times d</math>. The output is the groups of points within each region <math>N' \times k \times (d+c)</math> where <math>k</math> is the number of points in each region.<br />
<br />
Note that <math>k</math> can vary per group. Later, the PointNet layer creates a feature vector that is the same size for all regions at a hierarchical level.<br />
<br />
To determine which points belong to a group a ball query is used; all points within a radius of the centroid are grouped. This is advantageous over nearest neighbour because it guarantees a fixed region space, which is important when learning local structure.<br />
<br />
=== PointNet Layer ===<br />
<br />
After grouping, PointNet is applied to the points. However, first the coordinates of points in a local region are converted to a local coordinate frame by <math> x_i = x_i - \bar{x}</math> where <math>\bar{x}</math> is the coordinates of the centroid.<br />
<br />
=== Robust Feature Learning under Non-Uniform Sampling Density ===<br />
<br />
The previous description of grouping uses a single scale. This is not optimal because the density varies per section of the point cloud. At each level, it would be better if the PointNet layer was applied to adaptively sized groups depending on the point cloud density.<br />
<br />
The two grouping methods the authors propose are shown in the diagram below. Multi-scale grouping (MSG) applies PointNet at various scales per group. The features from the various scales are concatenated to form a multi-scale feature. To train the network to learn an optimal strategy for combining the multi-scale features, the authors proposed random input dropout, which involves randomly dropping input points with a random probability for each training point set. Each input point has a dropout probability <math>\theta</math>. The authors used a <math>\theta</math> value of 0.95. As shown in the experiments section below, dropout provides robustness to input point density variations. During testing stage all points are used. MSG, however, is computationally expensive because for each region it always applies PointNet at large scale neighborhoods to all points. <br />
<br />
On the other hand, multi-resolution grouping (MRG) is less computationally expensive but still adaptively collects features. As shown in the diagram, features of a region from a certain level is a concatenation of two vectors. The left vector is obtained by applying PointNet to three points, and these three points obtained information from three groups. This vector is then concatenated by a vector that is created by using PointNet on all the points in the level below. The second vector can be weighed more heavily if the first vector contains a sparse amount of points, since the first vector is based on subregions that would be even more sparse and suffer from sampling deficiency. On the other hand, when the density of a local region is high, the first vector can be weighted more heavily as it allows for inspecting at higher resolutions in the lower levels to obtain finer details. <br />
<br />
[[File:grouping.png | 300px|thumb|center|Example of the two ways to perform grouping]]<br />
<br />
== Point Cloud Segmentation ==<br />
<br />
If the task is segmentation, the architecture is slightly modified since we want a semantic score for each point. To achieve this, distance-based interpolation and skip-connections are used.<br />
<br />
=== Distance-based Interpolation ===<br />
<br />
Here, point features from <math>N_l \times (d + C)</math> points are propagated to <math>N_{l-1} \times (d + C)</math> points where <math>N_{l-1}</math> is greater than <math>N_l</math>.<br />
<br />
To propagate features an inverse distance weighted average based on <math>k</math> nearest neighbors is used. The <math>p=2</math> and <math>k=3</math>.<br />
<br />
[[File:prop_feature.png | 500px|thumb|center|Feature interpolation during segmentation]]<br />
<br />
=== Skip-connections ===<br />
<br />
In addition, skip connections are used (see the PointNet++ architecture diagram). The features from the the skip layers are concatenated with the interpolated features. Next, a "unit-wise" PointNet is applied, which the authors describe as similar to a one-by-one convolution.<br />
<br />
== Experiments ==<br />
To validate the effectiveness of PointNet++, experiments in three areas were performed - classification in Euclidean metric space, semantic scene labeling, and classification in non-Euclidean space.<br />
<br />
=== Point Set Classification in Euclidean Metric Space ===<br />
<br />
The digit dataset, MNIST, was converted to a 2D point cloud. Pixel intensities were normalized in the range of <math>[0, 1]</math>, and only pixels with intensities larger than 0.5 were considered. The coordinate system was set at the centre of the image. PointNet++ achieved a classification error of 0.51%. The original PointNet had 0.78% classification error. The table below compares these results to the state-of-the-art.<br />
<br />
[[File:mnist_results.png | 300px|thumb|center|MNIST classification results.]]<br />
<br />
In addition, the ModelNet40 dataset was used. This dataset consists of CAD models. Three dimensional point clouds were sampled from mesh surfaces of the ModelNet40 shapes. The classification results from this dataset are shown below. The last row in the table below, "Ours (with normal)" used face normals (normal is the same for the entire face, regardless of the point picked on that face) as additional point features as well as additional points <math>(N = 5000)</math> to boost performance. All these points are normalized to have zero mean and be within one unit ball. The network contains three hierarchical levels with three fully connected layers.<br />
<br />
[[File:modelnet40.png | 300px|thumb|center|ModelNet40 classification results.]]<br />
<br />
An experiment was performed to show how the accuracy was affected by the number of points used. With PointNet++ using multi-scale grouping and dropout, the performance decreased by less than 1% when 1024 test points were reduced to 256. On the other hand, PointNet's performance was impacted by the decrease in points. This is not surprising because the dropout feature of PointNet++ ensures that the model is trained specifically to be robust to loss of points. <br />
<br />
[[File:paper28_fig4_chair.png | 300px|thumb|center|An example showing the reduction of points visually. At 256 points, the points making up the object is very spare, however the accuracy is only reduced by 1%]][[File:num_points_acc.png | 300px|thumb|center|Relationship between accuracy and the number of points used for classification.]]<br />
<br />
=== Semantic Scene Labelling ===<br />
<br />
The ScanNet dataset was used for experiments in semantic scene labelling. This dataset consists of laser scans of indoor scenes where the goal is to predict a semantic label for each point. Example results are shown below.<br />
<br />
[[File:scannet.png | 300px|thumb|center|Example ScanNet semantic segmentation results.]]<br />
<br />
To compare to other methods, the authors convert their point labels to a voxel format, and accuracy is determined on a per voxel basis. The accuracy compared to other methods is shown below.<br />
<br />
[[File:scannet_acc.png | 500px|thumb|center|ScanNet semantic segmentation classification comparison to other methods.]]<br />
<br />
To test how the trained model performed on scans with non-uniform sampling density, virtual scans of ScanNet scenes were synthesized and the network was evaluated on this data. It can be seen from the above figures that SSG performance greatly falls due to the sampling density shift. MRG network, on the other hand, is more robust to the sampling density shift since it is able to automatically switch to features depicting coarser granularity when the sampling is sparse. This proves the effectiveness of the proposed density adaptive layer design.<br />
<br />
=== Classification in Non-Euclidean Metric Space ===<br />
<br />
[[File:shrec.png | 300px|thumb|right|Example of shapes from the SHREC15 dataset.]]<br />
<br />
Lastly, experiments were performed on the SHREC15 dataset. This dataset contains shapes that have different poses. This experiment shows that PointNet++ is able to generalize to non-Euclidean spaces. Results from this dataset are provided below.<br />
<br />
[[File:shrec15_results.png | 500px|thumb|center|Results from the SHREC15 dataset.]]<br />
<br />
=== Feature Visualization ===<br />
The figure below visualizes what is learned by just the first layer kernels of the network. The model is trained on a dataset the mostly consisted of furniture which explains the lines, corners, and planes visible in the visualization. Visualization is performed by creating a voxel grid in space and only aggregating point sets that activate specific neurons the most.<br />
<br />
[[File:26_8.PNG | 800px|thumb|center|Point clouds learned from first layer kernels (red is near, blue is far)]]<br />
<br />
=== Time and Space Complexity ===<br />
To evaluate the time and space complexity between PointNet++ and PointNet the authors recorded the model size and inference time for several point set deep learning approaches. Inference time is measured with a GTX 1080 and batch size 8. The PointNet inference times are significantly better however the model sizes for PointNet++ are comparable. The table below outlines the specifics for each method.<br />
<br />
Worth noting is that ours MSG, while it has good performance in non-uniformly sampled data, it’s twice as expensive as then SSG version due the multi-scale region feature extraction. Compared with MSG,<br />
MRG is more efficient since it uses regions across layers.<br />
<br />
[[File:pointnet_complexity.PNG | 700px|thumb|center|Comparison of model size and inference time between PointNet and PointNet++]]<br />
<br />
== Critique ==<br />
<br />
It seems clear that PointNet is lacking capturing local context between points. PointNet++ seems to be an important extension, but the improvements in the experimental results seem small. Some computational efficiency experiments would have been nice. For example, the processing speed of the network, and the computational efficiency of MRG over MRG.<br />
<br />
It may be useful to note that not all raw point clouds coming from sensors are completely unordered. For example, the points from a LiDAR scanner are ordered by the specific angles of the laser scans, which can be seen as rings in the point cloud (shown in the figure below). The discontinuities along each ring could be used to provide trackable feature information for SLAM algorithms, and would be harder to recover if the point cloud is represented in an unordered manner.<br />
<br />
[[File:point_net_pp_lidar_scan.png | 500px|thumb|center|Example LiDAR point cloud]]<br />
<br />
== Code ==<br />
<br />
Code for PointNet++ can be found at: https://github.com/charlesq34/pointnet2<br />
<br />
= Conclusion =<br />
In this work, we propose PointNet++, a powerful neural network architecture for processing point<br />
sets sampled in a metric space. PointNet++ recursively functions on a nested partitioning of the<br />
input point set, and is effective in learning hierarchical features with respect to the distance metric.<br />
To handle the non uniform point sampling issue, we propose two novel set abstraction layers that<br />
intelligently aggregate multi-scale information according to local point densities. These contributions<br />
enable us to achieve state-of-the-art performance on challenging benchmarks of 3D point clouds.<br />
In the future, it’s worthwhile thinking how to accelerate inference speed of our proposed network<br />
especially for MSG and MRG layers by sharing more computation in each local regions. It’s also<br />
interesting to find applications in higher dimensional metric spaces where CNN based method would<br />
be computationally unfeasible while our method can scale well.<br />
<br />
=Sources=<br />
# Charles R. Qi, Li Yi, Hao Su, Leonidas J. Guibas. PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space, 2017<br />
# Charles R. Qi, Hao Su, Kaichun Mo, Leonidas J. Guibas. PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation, 2017<br />
# Charles R. Qi, “charlesq34/pointnet2.” GitHub, 25 Feb. 2018, github.com/charlesq34/pointnet2.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=36418Multi-scale Dense Networks for Resource Efficient Image Classification2018-04-21T02:45:03Z<p>W285liu: /* CONCLUSION */</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 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. For example, the winner of the COCO 2016 competition was an [http://image-net.org/challenges/talks/2016/GRMI-COCO-slidedeck.pdf ensemble of CNNs], which are likely far too resource-heavy to be used in any resource-limited application.<br />
<br />
Note: <br />
* There are two kinds of efficiency in this context, computational efficiency and resource efficiency.<br />
* There are multiple cases for hard examples, such as large number of classification label, randomly blocked or zoomed images, or even complicated background that makes image recognition even more difficult. <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. (For example, a Branchynet (Teerapittayanon et al., 2016) is a deeply supervised network explicitly designed for efficiency. A Branchynet has multiple exit branches at various depths, each leading to a softmax classifier. At test time, if a classifier on an early exit branch makes a confident prediction, the rest of network need not be evaluated. However, unlike in MSDnets, in Branchynets early classifiers to not have access to low-resolution features. )<br />
* The feature concatenation method from DenseNets(Dense net is CNN with shorter connections close to input and output) 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 & Bagnell, 2012), there is a finite computational budget <math>B > 0</math> available for each test example <math>x</math>. 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 />
They assume that the budget is drawn from some joint distribution <math>P(x,B)</math>. They denote the loss of a model <math>f(x)</math> that has to produce a prediction for instance x with a budget of <math>B</math> by <math>L(f(x),B)</math>. The goal of the anytime learner is to minimize the expected loss under the budget distribution <math>L(f)=\mathop{\mathbb{E}}[L(f(x),B)]_{P(x,B)}</math>.<br />
<br />
== Budgeted Batch Classification ==<br />
In the budgeted batch classification setting, the model needs to classify a set of examples <math>D_{test} = {x_1, . . . , x_M}</math> within a finite computational budget <math>B > 0</math> that is known in advance. The learner aims to minimize the loss across all examples in the <math>D_{test}</math>, within a cumulative cost bounded by <math>B</math>, which is denoted as <math>L(f(D_{test}),B)</math> for some suitable loss function <math>L</math>.<br />
<br />
= Multi-Scale Dense Networks =<br />
Two solutions to the problems mentioned above: <br />
<br />
* Train multiple networks of increasing capacity, and evaluate them at test time.<br />
**Anytime setting: the evaluation can be stopped at any time point and return the most recent prediction<br />
**Batch setting: the evaluation is stopped with no continuous training when the network is good enough.<br />
* Build a deep network with a cascade of classifiers operating on the features of internal layers.<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 />
'''Concern:''' 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 />
'''Solution:''' 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 />
'''Concern:''' The results in figure 3 shows that the introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its accuracy by up to 7%. That may because 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 />
'''Solution:''' MSDNets use dense connectivity to avoid this issue. DenseNet suffers much less from this effect. Dense connectivity connects each layer with all subsequent layers and allows later layers to bypass features optimized for the short-term, to maintain the high accuracy of the final classifier. 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. Which means If an earlier layer collapses information to generate short-term features, the lost information can be recovered through the direct connection to its preceding layer. The final classifier’s performance becomes (more or less) independent of the location of the intermediate classifier.<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 consists of two convolutional layers, an average pooling layer and a linear layer and 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 />
<math>\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) </math><br />
<br />
Here <math>w_i</math> represents the weights and <math>L(f_k)</math> 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, <math>|D_{test}|\sum_{k}(q_k C_k) \leq B </math> must be true. Where <math>|D_{test}|</math> is the total number of test samples, <math>C_k</math> is the computational requirement to get an output from the <math>k</math>th classifier, and <math>q_k </math> is the probability that a sample exits at the <math>k</math>th classifier. Assuming that all classifiers have the same base probability, <math>q</math>, then <math>q_k</math> 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 <math>S</math> blocks along the depth dimension and keeping the <math>(S-i+1)</math> scales in the <math>i^{\text{th}}</math> block.Whenever a scale is removed, a transition layer merges the concatenated features using 1x1 convolution and feeds the fine grained features to coarser scales.<br />
# Remove unnecessary computations: Group the computation in "diagonal blocks"; this propagates the example along paths that are required for the evaluation of the next classifier.<br />
<br />
The strategy of minimizing unnecessary computations when the computational budget is over is known as the ''lazy evaluation''.<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 "easy" and "hard" 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 "hard" 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 "hard"/"easy" 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 />
= Conclusion =<br />
We presented the MSDNet, a novel convolutional network architecture, optimized to incorporate<br />
CPU budgets at test-time. Our design is based on two high-level design principles, to generate and<br />
maintain coarse level features throughout the network and to inter-connect the layers with dense<br />
connectivity. The former allows us to introduce intermediate classifiers even at early layers and<br />
the latter ensures that these classifiers do not interfere with each other. The final design is a two<br />
dimensional array of horizontal and vertical layers, which decouples depth and feature coarseness.<br />
Whereas in traditional convolutional networks features only become coarser with increasing depth,<br />
the MSDNet generates features of all resolutions from the first layer on and maintains them throughout.<br />
The result is an architecture with an unprecedented range of efficiency. A single network can<br />
outperform all competitive baselines on an impressive range of budgets ranging from highly limited<br />
CPU constraints to almost unconstrained settings.<br />
As future work we plan to investigate the use of cost-aware deep architectures beyond object classification,<br />
e.g. image segmentation (Long et al., 2015). Further, we intend to explore approaches that<br />
combine MSDNets with model compression (Chen et al., 2015; Han et al., 2015), spatially adaptive<br />
computation (Figurnov et al., 2016) and more efficient convolution operations (Chollet, 2016;<br />
Howard et al., 2017) to further improve computational efficiency.<br />
<br />
= Sources =<br />
# Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., & 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. "Optimal brain damage." Advances in neural information processing systems. 1990.<br />
# Hassibi, Babak, David G. Stork, and Gregory J. Wolff. "Optimal brain surgeon and general network pruning." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Li, Hao, et al. "Pruning filters for efficient convnets." arXiv preprint arXiv:1608.08710 (2016).<br />
# Hubara, Itay, et al. "Binarized neural networks." Advances in neural information processing systems. 2016.<br />
# Rastegari, Mohammad, et al. "Xnor-net: Imagenet classification using binary convolutional neural networks." 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.<br />
# Teerapittayanon, Surat, Bradley McDanel, and H. T. Kung. "Branchynet: Fast inference via early exiting from deep neural networks." Pattern Recognition (ICPR), 2016 23rd International Conference on. IEEE, 2016.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=36417Multi-scale Dense Networks for Resource Efficient Image Classification2018-04-21T02:44:46Z<p>W285liu: </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 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. For example, the winner of the COCO 2016 competition was an [http://image-net.org/challenges/talks/2016/GRMI-COCO-slidedeck.pdf ensemble of CNNs], which are likely far too resource-heavy to be used in any resource-limited application.<br />
<br />
Note: <br />
* There are two kinds of efficiency in this context, computational efficiency and resource efficiency.<br />
* There are multiple cases for hard examples, such as large number of classification label, randomly blocked or zoomed images, or even complicated background that makes image recognition even more difficult. <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. (For example, a Branchynet (Teerapittayanon et al., 2016) is a deeply supervised network explicitly designed for efficiency. A Branchynet has multiple exit branches at various depths, each leading to a softmax classifier. At test time, if a classifier on an early exit branch makes a confident prediction, the rest of network need not be evaluated. However, unlike in MSDnets, in Branchynets early classifiers to not have access to low-resolution features. )<br />
* The feature concatenation method from DenseNets(Dense net is CNN with shorter connections close to input and output) 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 & Bagnell, 2012), there is a finite computational budget <math>B > 0</math> available for each test example <math>x</math>. 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 />
They assume that the budget is drawn from some joint distribution <math>P(x,B)</math>. They denote the loss of a model <math>f(x)</math> that has to produce a prediction for instance x with a budget of <math>B</math> by <math>L(f(x),B)</math>. The goal of the anytime learner is to minimize the expected loss under the budget distribution <math>L(f)=\mathop{\mathbb{E}}[L(f(x),B)]_{P(x,B)}</math>.<br />
<br />
== Budgeted Batch Classification ==<br />
In the budgeted batch classification setting, the model needs to classify a set of examples <math>D_{test} = {x_1, . . . , x_M}</math> within a finite computational budget <math>B > 0</math> that is known in advance. The learner aims to minimize the loss across all examples in the <math>D_{test}</math>, within a cumulative cost bounded by <math>B</math>, which is denoted as <math>L(f(D_{test}),B)</math> for some suitable loss function <math>L</math>.<br />
<br />
= Multi-Scale Dense Networks =<br />
Two solutions to the problems mentioned above: <br />
<br />
* Train multiple networks of increasing capacity, and evaluate them at test time.<br />
**Anytime setting: the evaluation can be stopped at any time point and return the most recent prediction<br />
**Batch setting: the evaluation is stopped with no continuous training when the network is good enough.<br />
* Build a deep network with a cascade of classifiers operating on the features of internal layers.<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 />
'''Concern:''' 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 />
'''Solution:''' 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 />
'''Concern:''' The results in figure 3 shows that the introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its accuracy by up to 7%. That may because 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 />
'''Solution:''' MSDNets use dense connectivity to avoid this issue. DenseNet suffers much less from this effect. Dense connectivity connects each layer with all subsequent layers and allows later layers to bypass features optimized for the short-term, to maintain the high accuracy of the final classifier. 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. Which means If an earlier layer collapses information to generate short-term features, the lost information can be recovered through the direct connection to its preceding layer. The final classifier’s performance becomes (more or less) independent of the location of the intermediate classifier.<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 consists of two convolutional layers, an average pooling layer and a linear layer and 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 />
<math>\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) </math><br />
<br />
Here <math>w_i</math> represents the weights and <math>L(f_k)</math> 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, <math>|D_{test}|\sum_{k}(q_k C_k) \leq B </math> must be true. Where <math>|D_{test}|</math> is the total number of test samples, <math>C_k</math> is the computational requirement to get an output from the <math>k</math>th classifier, and <math>q_k </math> is the probability that a sample exits at the <math>k</math>th classifier. Assuming that all classifiers have the same base probability, <math>q</math>, then <math>q_k</math> 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 <math>S</math> blocks along the depth dimension and keeping the <math>(S-i+1)</math> scales in the <math>i^{\text{th}}</math> block.Whenever a scale is removed, a transition layer merges the concatenated features using 1x1 convolution and feeds the fine grained features to coarser scales.<br />
# Remove unnecessary computations: Group the computation in "diagonal blocks"; this propagates the example along paths that are required for the evaluation of the next classifier.<br />
<br />
The strategy of minimizing unnecessary computations when the computational budget is over is known as the ''lazy evaluation''.<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 "easy" and "hard" 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 "hard" 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 "hard"/"easy" 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 />
= CONCLUSION =<br />
We presented the MSDNet, a novel convolutional network architecture, optimized to incorporate<br />
CPU budgets at test-time. Our design is based on two high-level design principles, to generate and<br />
maintain coarse level features throughout the network and to inter-connect the layers with dense<br />
connectivity. The former allows us to introduce intermediate classifiers even at early layers and<br />
the latter ensures that these classifiers do not interfere with each other. The final design is a two<br />
dimensional array of horizontal and vertical layers, which decouples depth and feature coarseness.<br />
Whereas in traditional convolutional networks features only become coarser with increasing depth,<br />
the MSDNet generates features of all resolutions from the first layer on and maintains them throughout.<br />
The result is an architecture with an unprecedented range of efficiency. A single network can<br />
outperform all competitive baselines on an impressive range of budgets ranging from highly limited<br />
CPU constraints to almost unconstrained settings.<br />
As future work we plan to investigate the use of cost-aware deep architectures beyond object classification,<br />
e.g. image segmentation (Long et al., 2015). Further, we intend to explore approaches that<br />
combine MSDNets with model compression (Chen et al., 2015; Han et al., 2015), spatially adaptive<br />
computation (Figurnov et al., 2016) and more efficient convolution operations (Chollet, 2016;<br />
Howard et al., 2017) to further improve computational efficiency.<br />
<br />
= Sources =<br />
# Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., & 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. "Optimal brain damage." Advances in neural information processing systems. 1990.<br />
# Hassibi, Babak, David G. Stork, and Gregory J. Wolff. "Optimal brain surgeon and general network pruning." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Li, Hao, et al. "Pruning filters for efficient convnets." arXiv preprint arXiv:1608.08710 (2016).<br />
# Hubara, Itay, et al. "Binarized neural networks." Advances in neural information processing systems. 2016.<br />
# Rastegari, Mohammad, et al. "Xnor-net: Imagenet classification using binary convolutional neural networks." 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.<br />
# Teerapittayanon, Surat, Bradley McDanel, and H. T. Kung. "Branchynet: Fast inference via early exiting from deep neural networks." Pattern Recognition (ICPR), 2016 23rd International Conference on. IEEE, 2016.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=36415MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-04-21T02:42:47Z<p>W285liu: /* Conclusion */</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 re-projection consistency between the 3D shape and the estimated sketch. 2.5D is the construction of a 3D environment using 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 />
=== Notes: 2.5D === <br />
<br />
Two and a half dimensional (shortened to 2.5D, known alternatively as three-quarter perspective and pseudo-3D) is a term used to describe either 2D graphical projections and similar techniques used to cause images to simulate the appearance of being three-dimensional (3D) when in fact they are not, or gameplay in an otherwise three-dimensional video game that is restricted to a two-dimensional plane or has a virtual camera with fixed angle.<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[10] 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, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> 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 <math>v_{x, y}, d_{x, y}</math> 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 />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0 when there is no intersection between the line and its shape, referred as the silhouette criterion.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> 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 <math>z = d_{x, y}</math>:<br />
<br />
<math><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 />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><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 />
</math><br />
and<br />
<math><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 />
</math><br />
<br />
Gradients along y are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x, y-1, z+\frac{n_b}{n_c}}} = 2(v_{x, y-1, z+\frac{n_b}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x, y+1, z-\frac{n_b}{n_c}}} = 2(v_{x, y+1, z-\frac{n_b}{n_c}}-1)<br />
</math><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 />
The data is based on synthesized images of ShapeNet chairs [Chang et al., 2015]. From the SUN database [Xiao et al., 2010], they combine the chars with random backgrounds and use a physics-based renderer by Jakob to render the corresponding RGB, depth, surface normal, and silhouette images.<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 following the training paradigm defined previously but 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. Specifically, the 2.5D sketch estimator is trained using ground truth depth, normal and silhouette images and a L2 reconstruction loss. The 3D shape estimation module takes in the masked ground truth depth and normal images as input, and predicts 3D voxels of size 128×128×128 with a binary cross entropy loss.<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 />
Also followed the paradigm described and train each module separately on the ShapeNet dataset. Then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. Unlike previous works this model requires no silhouettes as input during fine-tuning; it instead estimates silhouette jointly. As an ablation study, the author compare three variants of our model: The first is trained using ShapeNet data only without 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. Therefore, future work should address more "difficult" shapes and forms; it should be more difficult to generalize shapes that are more complex than furniture.<br />
<br />
Also there is ambiguity in terms of how the aforementioned self-supervision can work as the authors claim that the model can be fine-tuned using a single image itself. If the parameters are constrained to a single image, then it means it will not generalize well. It is not clearly explained as to what can be fine-tuned.<br />
<br />
The paper does not propose or implement a baseline model to which MarrNet should be compared.<br />
<br />
The model uses information from a single image. 3D shape estimation in biological agents incorporates information from multiple images or even video. A logical next step for improving this model would be to include images of the object from multiple angles.<br />
<br />
= Conclusion =<br />
We proposed MarrNet, a novel model that explicitly models 2.5D sketches for single image 3D<br />
shape reconstruction. The use of 2.5D sketches enhanced the model’s performance, and made it<br />
easily adaptive to images across domains or even categories. We also developed differentiable loss<br />
functions for the consistency between 3D shape and 2.5D sketches, so that MarrNet can be end-to-end<br />
fine-tuned on real images without annotations. Experiments demonstrated that our model performs<br />
well, and is preferred by human annotators over competitors.<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.<br />
# Rohit Girdhar, David F. Fouhey, Mikel Rodriguez and Abhinav Gupta, Learning a Predictable and Generative Vector Representation for Objects, in ECCV 2016<br />
#Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al. Shapenet: An information-rich 3d model repository. arXiv:1512.03012, 2015. <br />
#Jianxiong Xiao, James Hays, Krista A Ehinger, Aude Oliva, and Antonio Torralba. Sun database: Large-scale scene recognition from abbey to zoo. In CVPR, 2010. <br />
#Wenzel Jakob. Mitsuba renderer, 2010. http://www.mitsuba-renderer.org.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36413Dynamic Routing Between Capsules STAT9462018-04-21T02:40:14Z<p>W285liu: /* Introduction */</p>
<hr />
<div>= Presented by =<br />
<br />
Yang, Tong(Richard)<br />
= Introduction = <br />
A capsule is a group of neurons whose activity vector represents the instantiation<br />
parameters of a specific type of entity such as an object or an object part. We use<br />
the length of the activity vector to represent the probability that the entity exists and<br />
its orientation to represent the instantiation parameters. Active capsules at one level<br />
make predictions, via transformation matrices, for the instantiation parameters of<br />
higher-level capsules. When multiple predictions agree, a higher level capsule<br />
becomes active. We show that a discrimininatively trained, multi-layer capsule<br />
system achieves state-of-the-art performance on MNIST and is considerably better<br />
than a convolutional net at recognizing highly overlapping digits. To achieve these<br />
results we use an iterative routing-by-agreement mechanism: A lower-level capsule<br />
prefers to send its output to higher level capsules whose activity vectors have a big<br />
scalar product with the prediction coming from the lower-level capsule.<br />
<br />
= Contributions =<br />
<br />
This paper introduces the concept of "capsules" 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 [4], 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 />
<math>b_{ij}</math> 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 <math>b_{ij}</math> for all possible pairs of capsules are set to 0. In the very first routing iteration, <math>b_{ij}</math> equals to zero. For each routing iteration, <math>b_{ij}</math> gets updated by the value of agreement, which will be explained later.<br />
<br />
=== Coupling Coefficient === <br />
<br />
<math>c_{ij}</math> represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability <math>b_{ij}</math>. 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 />
<math>c_{ij}</math> 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 />
<math>u_{i}</math> are the output vector from capsule i in the lower layer, and <math>\hat{u}_{j|i}</math> are the input vector for capsule j, which are the "prediction vectors" from the capsules in the layer below. <math>\hat{u}_{j|i}</math> is produced by multiplying <math>u_{i}</math> by a weight matrix <math>W_{ij}</math>, such as the following:<br />
<br />
\begin{align}<br />
\hat{u}_{j|i} = W_{ij}u_i<br />
\end{align}<br />
<br />
where <math>W_{ij}</math> 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 <math>s_j</math> 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 "squashing" 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 "squashing" 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 "routing by agreement" 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 <math>a_{ij}</math>, 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 "squashing" section, the length of <math>v_{j}</math> is either close to 0 or close to 1, which will effect the magnitude of <math>a_{ij}</math> in this case. Therefore, the magnitude of <math>a_{ij}</math> 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, <math>b_{ij}</math> 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 "squashing" process, we will eventually end up with a capsule j with its <math>v_{j}</math> close to 1 while all other capsules with its <math>v_{j}</math> 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 <math>L_k</math> 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 <math>m^+ = 0.9</math>, <math>m^- = 0.1</math>, and <math>\lambda = 0.5</math>.<br />
<br />
<math>T_k</math> 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, <math>\lambda</math> 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: <math>28 \times 28</math> pixels.<br />
* Kernel size: <math>9 \times 9</math>.<br />
* Number of Kernels: 256.<br />
* Activation function: ReLU.<br />
* Output: <math>20 \times 20 \times 256</math> 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 <math>9 \times 9</math> kernel and a stride of 2. Each capsule will take a <math>20 \times 20 \times 256</math> tensor from Conv1 and produce an output of a <math>6 \times 6 \times 8</math> tensor.<br />
<br />
* Layer Type: Convolutional Layer<br />
* Input: <math>20 \times 20 \times 256</math> tensor.<br />
* Number of capsules: 32.<br />
* Number of convolutional units in each capsule: 8.<br />
* Size of each convolutional unit: <math>6 \times 6</math>.<br />
* Output: <math>6 \times 6 \times 8</math> 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, <math>W_{ij}</math> is a <math>8 \times 16</math> matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform "squashing" to get the output. <br />
<br />
* Layer Type: Fully connected layer.<br />
* Input: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
* Output: <math>16 \times 10 </math> 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 "crowding" 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. Also, it's not clear why "agreement" is a good criteria for routing.<br />
* Capsule network suffers a perceptual phenomenon called "crowding", 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 />
* Capsule networks have not been able to produce results on data sets such as CIFAR10, smallNORB and SVHN that are comparable with the state of the art. This is likely due to the fact that Capsule nets have a hard time dealing with background image information.<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. "What is wrong with convolutional neural nets?", https://youtu.be/rTawFwUvnLE?t=612</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36412Dynamic Routing Between Capsules STAT9462018-04-21T02:39:22Z<p>W285liu: </p>
<hr />
<div>= Presented by =<br />
<br />
Yang, Tong(Richard)<br />
= Introduction = <br />
Human vision ignores irrelevant details by using a carefully determined sequence of fixation points<br />
to ensure that only a tiny fraction of the optic array is ever processed at the highest resolution.<br />
Introspection is a poor guide to understanding how much of our knowledge of a scene comes from<br />
the sequence of fixations and how much we glean from a single fixation, but in this paper we will<br />
assume that a single fixation gives us much more than just a single identified object and its properties.<br />
We assume that our multi-layer visual system creates a parse tree-like structure on each fixation, and<br />
we ignore the issue of how these single-fixation parse trees are coordinated over multiple fixations.<br />
Parse trees are generally constructed on the fly by dynamically allocating memory. Following Hinton<br />
et al. [2000], however, we shall assume that, for a single fixation, a parse tree is carved out of a fixed<br />
multilayer neural network like a sculpture is carved from a rock. Each layer will be divided into many<br />
small groups of neurons called “capsules” (Hinton et al. [2011]) and each node in the parse tree will<br />
correspond to an active capsule. Using an iterative routing process, each active capsule will choose a<br />
capsule in the layer above to be its parent in the tree. For the higher levels of a visual system, this<br />
iterative process will be solving the problem of assigning parts to wholes.<br />
The activities of the neurons within an active capsule represent the various properties of a particular<br />
entity that is present in the image. These properties can include many different types of instantiation<br />
parameter such as pose (position, size, orientation), deformation, velocity, albedo, hue, texture, etc.<br />
One very special property is the existence of the instantiated entity in the image. An obvious way to<br />
represent existence is by using a separate logistic unit whose output is the probability that the entity<br />
exists. In this paper we explore an interesting alternative which is to use the overall length of the<br />
vector of instantiation parameters to represent the existence of the entity and to force the orientation<br />
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.<br />
of the vector to represent the properties of the entity1 . We ensure that the length of the vector output<br />
of a capsule cannot exceed 1 by applying a non-linearity that leaves the orientation of the vector<br />
unchanged but scales down its magnitude. The fact that the output of a capsule is a vector makes it possible to use a powerful dynamic routing mechanism to ensure that the output of the capsule gets sent to an appropriate parent in the layer<br />
above. Initially, the output is routed to all possible parents but is scaled down by coupling coefficients<br />
that sum to 1. For each possible parent, the capsule computes a “prediction vector” by multiplying its<br />
own output by a weight matrix. If this prediction vector has a large scalar product with the output of<br />
a possible parent, there is top-down feedback which increases the coupling coefficient for that parent<br />
and decreasing it for other parents. This increases the contribution that the capsule makes to that<br />
parent thus further increasing the scalar product of the capsule’s prediction with the parent’s output.<br />
This type of “routing-by-agreement” should be far more effective than the very primitive form of<br />
routing implemented by max-pooling, which allows neurons in one layer to ignore all but the most<br />
active feature detector in a local pool in the layer below. We demonstrate that our dynamic routing<br />
mechanism is an effective way to implement the “explaining away” that is needed for segmenting<br />
highly overlapping objects.<br />
Convolutional neural networks (CNNs) use translated replicas of learned feature detectors. This<br />
allows them to translate knowledge about good weight values acquired at one position in an image<br />
to other positions. This has proven extremely helpful in image interpretation. Even though we are<br />
replacing the scalar-output feature detectors of CNNs with vector-output capsules and max-pooling<br />
with routing-by-agreement, we would still like to replicate learned knowledge across space. To<br />
achieve this, we make all but the last layer of capsules be convolutional. As with CNNs, we make<br />
higher-level capsules cover larger regions of the image. Unlike max-pooling however, we do not throw<br />
away information about the precise position of the entity within the region. For low level capsules,<br />
location information is “place-coded” by which capsule is active. As we ascend the hierarchy,<br />
more and more of the positional information is “rate-coded” in the real-valued components of the<br />
output vector of a capsule. This shift from place-coding to rate-coding combined with the fact that<br />
higher-level capsules represent more complex entities with more degrees of freedom suggests that the<br />
dimensionality of capsules should increase as we ascend the hierarchy.<br />
<br />
= Contributions =<br />
<br />
This paper introduces the concept of "capsules" 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 [4], 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 />
<math>b_{ij}</math> 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 <math>b_{ij}</math> for all possible pairs of capsules are set to 0. In the very first routing iteration, <math>b_{ij}</math> equals to zero. For each routing iteration, <math>b_{ij}</math> gets updated by the value of agreement, which will be explained later.<br />
<br />
=== Coupling Coefficient === <br />
<br />
<math>c_{ij}</math> represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability <math>b_{ij}</math>. 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 />
<math>c_{ij}</math> 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 />
<math>u_{i}</math> are the output vector from capsule i in the lower layer, and <math>\hat{u}_{j|i}</math> are the input vector for capsule j, which are the "prediction vectors" from the capsules in the layer below. <math>\hat{u}_{j|i}</math> is produced by multiplying <math>u_{i}</math> by a weight matrix <math>W_{ij}</math>, such as the following:<br />
<br />
\begin{align}<br />
\hat{u}_{j|i} = W_{ij}u_i<br />
\end{align}<br />
<br />
where <math>W_{ij}</math> 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 <math>s_j</math> 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 "squashing" 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 "squashing" 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 "routing by agreement" 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 <math>a_{ij}</math>, 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 "squashing" section, the length of <math>v_{j}</math> is either close to 0 or close to 1, which will effect the magnitude of <math>a_{ij}</math> in this case. Therefore, the magnitude of <math>a_{ij}</math> 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, <math>b_{ij}</math> 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 "squashing" process, we will eventually end up with a capsule j with its <math>v_{j}</math> close to 1 while all other capsules with its <math>v_{j}</math> 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 <math>L_k</math> 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 <math>m^+ = 0.9</math>, <math>m^- = 0.1</math>, and <math>\lambda = 0.5</math>.<br />
<br />
<math>T_k</math> 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, <math>\lambda</math> 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: <math>28 \times 28</math> pixels.<br />
* Kernel size: <math>9 \times 9</math>.<br />
* Number of Kernels: 256.<br />
* Activation function: ReLU.<br />
* Output: <math>20 \times 20 \times 256</math> 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 <math>9 \times 9</math> kernel and a stride of 2. Each capsule will take a <math>20 \times 20 \times 256</math> tensor from Conv1 and produce an output of a <math>6 \times 6 \times 8</math> tensor.<br />
<br />
* Layer Type: Convolutional Layer<br />
* Input: <math>20 \times 20 \times 256</math> tensor.<br />
* Number of capsules: 32.<br />
* Number of convolutional units in each capsule: 8.<br />
* Size of each convolutional unit: <math>6 \times 6</math>.<br />
* Output: <math>6 \times 6 \times 8</math> 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, <math>W_{ij}</math> is a <math>8 \times 16</math> matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform "squashing" to get the output. <br />
<br />
* Layer Type: Fully connected layer.<br />
* Input: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
* Output: <math>16 \times 10 </math> 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 "crowding" 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. Also, it's not clear why "agreement" is a good criteria for routing.<br />
* Capsule network suffers a perceptual phenomenon called "crowding", 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 />
* Capsule networks have not been able to produce results on data sets such as CIFAR10, smallNORB and SVHN that are comparable with the state of the art. This is likely due to the fact that Capsule nets have a hard time dealing with background image information.<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. "What is wrong with convolutional neural nets?", https://youtu.be/rTawFwUvnLE?t=612</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wasserstein_Auto-Encoders&diff=36410Wasserstein Auto-Encoders2018-04-21T02:28:15Z<p>W285liu: /* Introduction */</p>
<hr />
<div><br />
= Introduction =<br />
The field of representation learning was initially driven by supervised approaches, with impressive results using large labelled datasets. Unsupervised generative modeling, in contrast, used to be a domain governed by probabilistic approaches focusing on low-dimensional data.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 <math>\Pi(p_r, p_g)</math> is the set of all joint probability distributions with marginals <math>p_r</math> and <math>p_g</math>. Here the distribution <math>\gamma</math> is called a transport plan because its 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 />
Which means that the total amount of dirt moved to point <math>y</math> is <math>p_g(y)</math>. Similarly, we have:<br />
<br />
\begin{align}<br />
\int\gamma(x, y)dy = p_r(x)<br />
\end{align}<br />
Which means that the total amount of dirt moved out of point <math>x</math> is <math>p_r(x)</math><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. <math>\small {\mathcal{X}}</math>, are used for sets, capital letters, i.e. <math>\small X</math>, are used for random variables and lower case letters, i.e. <math>\small x</math>, for their values. Probability distributions are denoted with capital letters, i.e. <math>\small P(X)</math>, and corresponding densities with lower case letters, i.e. <math>\small p(x)</math>.<br />
<br />
This work aims to minimize OT <math>\small W_c(P_X, P_G)</math> between the true (but unknown) data distribution <math>\small P_X</math> and a latent variable model <math>\small P_G</math> specified by the prior distribution <math>\small P_Z</math> of latent codes <math>\small Z \in \pmb{\mathbb{Z}}</math> and the generative model <math>\small P_G(X|Z)</math> of the data points <math>\small X \in \pmb{\mathbb{X}}</math> given <math>\small Z</math>. <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 <math>\small c(x,y)</math> is any measurable cost function and <math>\small {\mathcal{P}(X \sim P_X,Y \sim P_G)}</math> is a set of all joint distributions of <math>\small (X,Y)</math> with marginals <math>\small P_X</math> and <math>\small P_G</math>. When <math>\small c(x,y)=d(x,y)</math>, the following Kantorovich-Rubinstein duality holds for the <math>\small 1^{st}</math> root of <math>\small W_c</math>:<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 <math>\small {\mathcal{F_L}}</math> 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 <math>\small P_G </math> defined by a two step procedure, where first a code <math>\small Z</math> is sampled from a fixed prior distribution <math>\small P_Z</math> on a latent space <math>\small {\mathcal{Z}}</math> and then <math>\small Z</math> is mapped to the image <math>\small X \in {\mathcal{X}}</math> 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 <math>\small Z </math> to <math>\small X = G(Z) </math>, then the OT cost takes a much simpler form as stated below by Theorem 1.<br />
<br />
'''Theorem 1''' For any function <math>\small G:{\mathcal{Z}} \rightarrow {\mathcal{X}}</math>, where <math>\small Q(Z) </math> is the marginal distribution of <math>\small Z </math> when <math>\small X \sim P_X </math> and <math>\small Z \sim Q(Z|X) </math>,<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 <math>\small \Gamma </math> between two random variables living in the <math>\small {\mathcal{X}} </math> space, one distributed according to <math>\small P_X </math> and the other one according to <math>\small P_G </math>, it is sufficient to find a conditional distribution <math>\small Q(Z|X) </math> such that its <math>\small Z </math> marginal <math>\small Q_Z(Z) := {\pmb{\mathbb{E}}_{X \sim P_X}[Q(Z|X)]} </math> is identical to the prior distribution <math>\small P_Z </math>. In order to implement a numerical solution to Theorem 1, the constraints on <math>\small Q(Z|X) </math> and <math>\small P_Z </math> 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 <math>\small Q </math> is any non-parametric set of probabilistic encoders, <math>\small {\mathcal{D}}_Z </math> is an arbitrary divergence between <br />
<math>\small Q_Z </math> and <math>\small P_Z </math>, and <math>\small \lambda > 0 </math> is a hyperparameter. The authors propose two different penalties <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) </math> 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 <math>\small \lambda \mathbb{E}(\parallel \nabla f(X) \parallel - 1)^2 </math><br />
<br />
===WAE-GAN: GAN-based===<br />
The first option is to choose <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) = D_{JS}(Q_Z,P_Z)</math>, where <math>\small D_{JS} </math> is the Jensen-Shannon divergence metric, and use adversarial training to estimate it. Specifically a discriminator is introduced in the latent space <math>\small {\mathcal{Z}} </math> trying to separate true points sampled from <math>\small P_Z </math> from fake ones sampled from <math>\small Q_Z </math>. 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 <math>\small k: {\mathcal{Z}} \times {\mathcal{Z}} \rightarrow {\mathcal{R}}</math>, 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 <math>\mathcal{H}_k</math> is the reproducing kernel Hilbert space of real-valued functions mapping <math>\mathcal{Z}</math> to <math>\mathcal{R}</math>. This can be used as a divergence measure and the authors propose to use <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) = MMD_k(P_Z,Q_Z) </math>, 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 <math>\small P_Z</math>. This however does not guarantee that the overall encoded distribution <math>\small {{\pmb{\mathbb{E}}_{P_X}}}[Q(Z|X)]</math> matches <math>\small P_Z</math>. 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 <math>\small c(x,y)={\parallel x-y \parallel}_2^2</math>, WAE-GAN is equivalent to adversarial autoencoders (AAE). Thus the theory suggests that AAE minimize the 2-Wasserstein distance between <math>\small P_X</math> and <math>\small P_G</math>.<br />
<br />
==OT, W-GAN and WAE==<br />
Literature on OT address computing the OT cost in large scale using SGD and sampling. They approach this task either through the dual formulation, or via a regularized version of the primal. They do not discuss any implications for generative modeling. The author's approach is based on the primal form of OT, we arrive at regularizers which are very different, and our main focus is on generative modeling.<br />
The Wasserstein GAN (W-GAN) minimizes the 1-Wasserstein distance <math>\small W_1(P_X,P_G)</math> for generative modeling. The W-GAN formulation is approached from the dual form and thus it cannot be applied to another other cost <math>\small W_c</math> as the neat form of the Kantorovich-Rubinstein duality holds only for <math>\small W_1</math>. WAE approaches the same problem from the primal form, can be applied to any cost function <math>\small c</math> 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 />
There have been many other approaches trying to blend the adversarial training of GANs with auto-encoder architectures. The approach is perhaps the most relevant to the purpose of the model. Some approaches suggest a workaround they propose to include an additional reconstruction term to the objective. Which means WAE does not necessarily lead to a min-max game, uses a different penalty, and has a clear theoretical foundation. Several works used reproducing kernels in context of GANs. A method called WAE-MMD uses MMD to match QZ to the prior PZ in the latent space Z. Typically Z has no more than 100 dimensions and PZ is Gaussian, which allows us to use regular mini-batch sizes to accurately estimate MMD.<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 <math> \small P_Z</math> and squared cost function <math> \small c(x,y)</math> 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, <math>(x,y)</math> from the test set are Auto-encoded (separately), to produce <math>(z_x, z_y)</math> 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 <math> \small D </math> 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 <math> \small k(x,y)=C/(C+\parallel{x-y}_2^2\parallel) </math>. 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 <math> (m,C) </math> denote the mean vector and covariance matrix of the features of the inception network (Szegedy et al. 2017) applied to model samples. Let <math>(m_w,C_w) </math> 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 <math> ||m-m_w||^2 +\mathrm{tr}(C+C_w-2(CC_w)^{\frac{1}{2}} )\,</math> (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 "test reconstructions" (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|center|Qualitative Assessment of Images]]<br />
<br />
'''Network structures:'''<br />
<br />
The Encoder, Decoder, and Adversary architectures used for the MNIST and CelebA datasets are as sown in the following two images:<br />
<br />
[[File:WAE_MNIST.png|700px|thumb|center|Network architectures used to evaluate on the MNIST dataset.]]<br />
<br />
[[File:WAE_CelebA.png|700px|thumb|center|Network architectures used to evaluate on the CelebA dataset.]]<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. "Gans trained by a two time-scale update rule converge to a local nash equilibrium." Advances in Neural Information Processing Systems. 2017.<br />
<br />
3. Christian Szegedy et al. "Inception-v4, inception-resnet and the impact of residual connections on learning." 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>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning&diff=36385One-Shot Imitation Learning2018-04-21T02:04:26Z<p>W285liu: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
We are interested in robotic systems that are able to perform a variety of complex useful tasks. 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: is an object categorization problem in computer vision. Whereas most machine learning based object categorization algorithms require training on hundreds or thousands of images and very large datasets, one-shot learning aims to learn information about object categories from one, or only a few , training images.<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 <math>H </math>, 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. For block stacking project, the demonstrations can span hundreds to thousands of time<br />
steps, and training with such long sequences can be demanding in both time and memory usage. Hence, the author randomly discard a subset of time steps during training, such operation is called "temporal dropout". Denote p as the proportion of time steps that are thrown away (in this case p = 95%).<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 <math>q</math>, a list of context vectors <math>\{c_j\}</math>, and a list of memory vectors <math>\{m_j\}</math> 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 "context embedding" 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 "adversarial" 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 [2].<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 [2].<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. "Neural machine translation by jointly learning to align and translate." 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. "One-shot imitation learning." 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. "Model-agnostic meta-learning for fast adaptation of deep networks." arXiv preprint arXiv:1703.03400 (2017).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=36381stat946w18/AmbientGAN: Generative Models from Lossy Measurements2018-04-21T01:54:27Z<p>W285liu: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
Generative models are powerful tools to concisely represent the structure in large datasets. Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often, we only have access to noisy or partial observations, which will, from here on, be referred to as measurements of the true data. If we know the measurement function and would like to train a generative model for the true data, there are several ways to continue which have varying degrees of success. We will use noisy MNIST data as an illustrative example, and show the results of 1. ignoring the problem, 2. trying to recover the lost information, and 3. using AmbientGAN as a way to recover the true data distribution. Suppose we only see MNIST data that has been run through a Gaussian kernel (blurred) with some noise from a <math>N(0, 0.5^2)</math> distribution added to each pixel:<br />
<br />
<gallery mode="packed"><br />
File:mnist.png| True Data (Unobserved)<br />
File:mnistmeasured.png| Measured Data (Observed)<br />
</gallery><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 inverted.<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. AmbientGAN works by training a generator which attempts to have the measurements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. This paper is published in ICLR 2018.<br />
<br />
== Contributions ==<br />
The paper makes the following contributions: <br />
<br />
=== Theoretical Contribution ===<br />
The authors show that the distribution of measured images uniquely determines the distribution of original images. This implies that a pure Nash equilibrium for the GAN game must find a generative model that matches the true distribution. They show similar results for a dropout measurement model, where each pixel is set to zero with some probability p, and a random projection measurement model, where they observe the inner product of the image with a random Gaussian vector.<br />
<br />
Also, the author listed a few theorems to support assumptions satisfied under Gaussian-Projection, Convolve+Noise and Block-Pixels measurement models, thus showing that that we can recover the true underlying distribution with the AmbientGAN framework. For example, the Gaussian theorem guarantees the uniqueness of underlying distribution. Finally by showing that this assumption is satisfied under Gaussian-Projection, Convolve+Noise and Block-Pixels measurement models, the author finally proved that can recover the true underlying distribution with the AmbientGAN framework.<br />
<br />
=== Empirical Contribution ===<br />
The authors consider CelebA and MNIST dataset for which the measurement model is unknown and show that Ambient GAN recovers a lot of the underlying structure.<br />
<br />
= Related Work = <br />
Currently there exist two distinct approaches for constructing neural network based generative models; they are autoregressive [4,5] and adversarial [6] based methods. The adversarial model has shown to be very successful in modeling complex data distributions such as images, 3D models, state action distributions and many more. This paper is related to the work in [7] where the authors create 3D object shapes from a dataset of 2D projections. This paper states that the work in [7] is a special case of the AmbientGAN framework where the measurement process creates 2D projections using weighted sums of voxel occupancies.<br />
<br />
= Datasets and Model Architectures=<br />
We used three datasets for our experiments: MNIST, CelebA and CIFAR-10 datasets We briefly describe the generative models used for the experiments. For the MNIST dataset, we use two GAN models. The first model is a conditional DCGAN, while the second model is an unconditional Wasserstein GAN with gradient penalty (WGANGP). For the CelebA dataset, we use an unconditional DCGAN. For the CIFAR-10 dataset, we use an Auxiliary Classifier Wasserstein GAN with gradient penalty (ACWGANGP). For measurements with 2D outputs, i.e. Block-Pixels, Block-Patch, Keep-Patch, Extract-Patch, and Convolve+Noise, we use the same discriminator architectures as in the original work. For 1D projections, i.e. Pad-Rotate-Project, Pad-Rotate-Project-θ, we use fully connected discriminators. The architecture of the fully connected discriminator used for the MNIST dataset was 25-25-1 and for the CelebA dataset was 100-100-1.<br />
<br />
= Model =<br />
For the following variables superscript <math>r</math> represents the true distributions while superscript <math>g</math> represents the generated distributions. Let <math>x</math>, represent the underlying space and <math>y</math> for the measurement.<br />
<br />
Thus, <math>p_x^r</math> is the real underlying distribution over <math>\mathbb{R}^n</math> that we are interested in. However if we assume that our (known) measurement functions, <math>f_\theta: \mathbb{R}^n \to \mathbb{R}^m</math> are parameterized by <math>\Theta \sim p_\theta</math>, we can then observe <math>Y = f_\theta(x) \sim p_y^r</math> where <math>p_y^r</math> is a distribution over the measurements <math>y</math>.<br />
<br />
Mirroring the standard GAN setup we let <math>Z \in \mathbb{R}^k, Z \sim p_z</math> and <math>\Theta \sim p_\theta</math> be random variables coming from a distribution that is easy to sample. <br />
<br />
If we have a generator <math>G: \mathbb{R}^k \to \mathbb{R}^n</math> then we can generate <math>X^g = G(Z)</math> which has distribution <math>p_x^g</math> a measurement <math>Y^g = f_\Theta(G(Z))</math> which has distribution <math>p_y^g</math>. <br />
<br />
Unfortunately, we do not observe any <math>X^g \sim p_x</math> so we cannot use the discriminator directly on <math>G(Z)</math> to train the generator. Instead we will use the discriminator to distinguish between the <math>Y^g -<br />
f_\Theta(G(Z))</math> and <math>Y^r</math>. That is, we train the discriminator, <math>D: \mathbb{R}^m \to \mathbb{R}</math> to detect if a measurement came from <math>p_y^r</math> or <math>p_y^g</math>.<br />
<br />
AmbientGAN has the objective function:<br />
<br />
\begin{align}<br />
\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))))]<br />
\end{align}<br />
<br />
where <math>q(.)</math> is the quality function; for the standard GAN <math>q(x) = log(x)</math> and for Wasserstein GAN <math>q(x) = x</math>.<br />
<br />
As a technical limitation we require <math>f_\theta</math> to be differentiable with respect to each input for all values of <math>\theta</math>.<br />
<br />
With this set up we sample <math>Z \sim p_z</math>, <math>\Theta \sim p_\theta</math>, and <math>Y^r \sim U\{y_1, \cdots, y_s\}</math> each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating <math>G</math> and updating <math>D</math>.<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 />
=== Convolve + Noise ===<br />
Additional results with the convolve + noise case with the celebA dataset. The AmbientGAN is compared to the baseline results with Wiener deconvolution. It is clear that AmbientGAN has superior performance in this case. The measurement is created using a Gaussian kernel and IID Gaussian noise, with <math>f_{\Theta}(x) = k*x + \Theta</math>, where <math>*</math> is the convolution operation, <math>k</math> is the convolution kernel, and <math>\Theta \sim p_{\theta}</math> is the noise distribution.<br />
<br />
[[File:paper7_fig3.png]]<br />
<br />
Images undergone convolve + noise transformations (left). Results with Wiener deconvolution (middle). Results with AmbientGAN (right).<br />
<br />
=== Block-Pixels ===<br />
With the block-pixels measurement function each pixel is independently set to 0 with probability <math>p</math>.<br />
<br />
[[File:block-pixels.png]]<br />
<br />
Measurements from the celebA dataset with <math>p=0.95</math> (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-<math>\theta</math> ===<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 <math>\theta</math>, and projects it on to the x axis. For each measurement the value of <math>\theta</math> 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 />
For the Keep-Patch measurement model, no pixels outside a box are known and thus inpainting methods are not suitable. For the Pad-Rotate-Project-θ measurements, a conventional technique is to sample many angles, and use techniques for inverting the Radon transform . However, since only a few projections are observed at a time, these methods aren’t readily applicable hence it is unclear how to obtain an approximate inverse function shown below. <br />
<br />
[[File:keep-patch.png]]<br />
<br />
=== Explanation of Inception Score ===<br />
To evaluate GAN performance, the authors make use of the inception score, a metric introduced by Salimans et al.(2016). To evaluate the inception score on a datapoint, a pre-trained inception classification model (Szegedy et al. 2016) is applied to that datapoint, and the KL divergence between its label distribution conditional on the datapoint and its marginal label distribution is computed. This KL divergence is the inception score. The idea is that meaningful images should be recognized by the inception model as belonging to some class, and so the conditional distribution should have low entropy, while the model should produce a variety of images, so the marginal should have high entropy. Thus an effective GAN should have a high inception score.<br />
<br />
=== MNIST Inception ===<br />
<br />
[[File:MNIST-inception.png]]<br />
<br />
AmbientGAN was compared with baselines through training several models with different probability <math>p</math> of blocking pixels. The plot on the left shows that the inception scores change as the block probability <math>p</math> changes. All four models are similar when no pixels are blocked <math>(p=0)</math>. By the increase of the blocking probability, AmbientGAN models present a relatively stable performance and perform better than the baseline models. Therefore, AmbientGAN is more robust than all other baseline models.<br />
<br />
The plot on the right reveals the changes in inception scores while the standard deviation of the additive Gaussian noise increased. Baselines perform better when the noise is small. By the increase of the variance, AmbientGAN models present a much better performance compare to the baseline models. Further AmbientGAN retains high inception scores as measurements become more and more lossy.<br />
<br />
For 1D projection, Pad-Rotate-Project model achieved an inception score of 4.18. Pad-Rotate-Project-θ model achieved an inception score of 8.12, which is close to the score of vanilla GAN 8.99.<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. Similar trends were observed on the CIFAR-10 data, and AmbientGAN maintains relatively stable inception score as the block probability was increased.<br />
<br />
=== Robustness To Measurement Model ===<br />
<br />
In order to empirically gauge robustness to measurement modelling error, the authors used the block-pixels measurement model: the image dataset was computed with <math> p^* = 0.5 </math>, and several versions of the model were trained, each using different values of blocking probability <math> p </math>. The inception scores were calculated and plotted as a function of <math> p </math>. This is shown on the left below:<br />
<br />
[[File:robustnessambientgan.png | 800px]]<br />
<br />
The authors observe that the inception score peaks when the model uses the correct probability, but decreases smoothly as the probability moves away, demonstrating some robustness.<br />
<br />
=== Compressed Sensing ===<br />
<br />
As described in Bora et al. (2017), generative models were found to outperform sparsity-based approaches in sensing. Using this knowledge, the generator from AmbientGAN can be tested against Lasso to determine the required measurements to minimize the reconstruction error. As shown on the right of Figure 16, AmbientGAN outperforms Lasso in a fraction of the number of measurements<br />
<br />
= Theoretical Results =<br />
<br />
The theoretical results in the paper prove the true underlying distribution of <math>p_x^r</math> 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 <math>p_y^r</math> corresponds to a unique distribution <math>p_x^r</math>. Thus even when the measurement itself is non-invertible the effect of the measurement on the distribution <math>p_x^r</math> 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 <math>p_x^r</math> be the true data distribution, and <math>p_\theta</math> be the distributions over the parameters of the measurement function. Let <math>p_y^r</math> be the induced measurement distribution. <br />
<br />
Assume for <math>p_\theta</math> there is a unique probability distribution <math>p_x^r</math> that induces <math>p_y^r</math>. <br />
<br />
Then for the standard GAN model if the discriminator <math>D</math> is optimal such that <math>D(\cdot) = \frac{p_y^r(\cdot)}{p_y^r(\cdot) + p_y^g(\cdot)}</math>, then a generator <math>G</math> is optimal if and only if <math>p_x^g = p_x^r</math>. <br />
<br />
=== Theorems 5.2===<br />
For the Gussian-Projection measurement model, there is a unique underlying distribution <math>p_x^{r} </math> that can induce the observed measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.3===<br />
Let <math> \mathcal{F} (\cdot) </math> denote the Fourier transform and let <math>supp (\cdot) </math> be the support of a function. Consider the Convolve+Noise measurement model with the convolution kernel <math> k </math>and additive noise distribution <math>p_\theta </math>. If <math> supp( \mathcal{F} (k))^{c}=\phi </math> and <math> supp( \mathcal{F} (p_\theta))^{c}=\phi </math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.4===<br />
Assume that each image pixel takes values in a finite set P. Thus <math>x \in P^n \subset \mathbb{R}^{n} </math>. Assume <math>0 \in P </math>, and consider the Block-Pixels measurement model with <math>p </math> being the probability of blocking a pixel. If <math>p <1</math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>. Further, for any <math> \epsilon > 0, \delta \in (0, 1] </math>, given a dataset of<br />
\begin{equation}<br />
s=\Omega \left( \frac{|P|^{2n}}{(1-p)^{2n} \epsilon^{2}} log \left( \frac{|P|^{n}}{\delta} \right) \right)<br />
\end{equation}<br />
IID measurement samples from pry , if the discriminator D is optimal, then with probability <math> \geq 1 - \delta </math> over the dataset, any optimal generator G must satisfy <math> d_{TV} \left( p^g_x , p^r_x \right) \leq \epsilon </math>, where <math> d_{TV} \left( \cdot, \cdot \right) </math> is the total variation distance.<br />
<br />
= Conclusion =<br />
Generative models are powerful tools, but constructing a generative model requires a large, high quality dataset of the distribution of interest. The authors show how to relax this requirement, by learning a distribution from a dataset that only contains incomplete, noisy measurements of the distribution. This allows for the construction of new generative models of distributions for which no high quality dataset exists.<br />
<br />
= Future Research =<br />
<br />
One critical weakness of AmbientGAN is the assumption that the measurement model is known and that this <math>f_theta</math> is also differentiable. In fact, when the measurement model is known, there's no obvious reason not to invert the noisy measurement first(as illustrated in the second approach). 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, or at the very least to remove the differentiability restriction from <math>f_theta</math>.<br />
<br />
A related piece of work is [https://arxiv.org/abs/1802.01284 here]. In particular, Algorithm 2 in the paper excluding the discriminator is similar to AmbientGAN.<br />
<br />
=Open Source Code=<br />
An implementation of Ambient GAN can be found here: https://github.com/AshishBora/ambient-gan.<br />
<br />
= References =<br />
# https://openreview.net/forum?id=Hy7fDog0b<br />
# Salimans, Tim, et al. "Improved techniques for training gans." Advances in Neural Information Processing Systems. 2016.<br />
# Szegedy, Christian, et al. "Rethinking the inception architecture for computer vision." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.<br />
# Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv:1312.6114, 2013.<br />
# Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016a.<br />
# Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural infor- mation processing systems, pp. 2672–2680, 2014.<br />
# Matheus Gadelha, Subhransu Maji, and Rui Wang. 3d shape induction from 2d views of multiple objects. arXiv preprint arXiv:1612.05872, 2016.<br />
# Ashish Bora, Ajil Jalal, Eric Price, and Alexandros G Dimakis. Compressed sensing using generative models. arXiv preprint arXiv:1703.03208, 2017.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=36380A Neural Representation of Sketch Drawings2018-04-21T01:48:31Z<p>W285liu: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
<br />
There have been many recent advances in neural generative models for low resolution pixel-based images.Generative Adversarial Networks (GANs), Variational Inference(VI), and Autoregressive (AR) models have become popular tools in this fast growing area 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 <math>(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})</math>. For each vector, <math>\Delta x</math> and <math>\Delta y</math> 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; <math>p_{1}</math> indicates that the pen is down and a line should be drawn between the current point and the next point, <math>p_{2}</math> indicates that the pen is up and no line should be drawn between the current point and the next point, and <math>p_{3}</math> 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 <math>S</math>) in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state <math>h</math>. <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 <math>h</math> is then projected into two vectors <math>\mu</math> and <math>\hat{\sigma}</math> each of size <math>N_{z}</math> using a fully connected layer. <math>\hat{\sigma}</math> is then converted into a non-negative standard deviation parameter <math>\sigma</math> using an exponential operator. These two parameters <math>\mu</math> and <math>\sigma</math> are then used along with an IID Gaussian vector distributed as <math>\mathcal{N}(0, I)</math> of size <math>N_{z}</math> to construct a random vector <math>z \in ℝ^{N_{z}}</math>, 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 <math>z</math>. The initial hidden states of each recurrent neuron are determined using <math>[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})</math>. Each step of the decoder RNN accepts the previous point <math>S_{i-1}</math> and the latent vector <math>z</math> 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 <math>S_{i}</math>. Outputs <math>\Delta x</math> and <math>\Delta y</math> are modeled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states <math>(q_{1}, q_{2}, q_{3})</math> 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 />
<math>N(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})</math> is the probability distribution function for a bivariate normal distribution.<br />
For each of the M distributions in the GMM, parameters <math>\mu</math> and <math>\sigma</math> 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 <math>\rho_{xy}</math> signifying correlation of each bivariate normal distribution. An additional vector <math>\Pi</math> is an output giving the mixture weights for the GMM. The output <math>S_{i}</math> 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 <math>N_{max}</math> 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 <math>N_{max}</math>, 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 <math>N_{max}</math>. The authors also introduce a “temperature” parameter <math>\tau</math> 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 <math>\tau</math> 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 <math>z</math>. Figure 3 above shows different sketches that are sampled from the network by only varying the temperature parameter <math>\tau</math> 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 <math>L_{R}</math> and KL Divergence Loss <math>L_{KL}</math>. The reconstruction loss term is composed of two terms; <math>L_{s}</math>, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data <math>S</math> and <math>L_{p}</math> 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 <math>L_{KL}</math> measures the difference between the latent vector <math>z</math> and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions <math>N_{z}</math> 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 <math>w_{KL}</math> has the effect that as <math>w_{KL} \rightarrow 0</math>, 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 <math>S</math> and how closely the latent vector <math>z</math> follows a standard normal distribution) - smaller values of <math>w_{KL}</math> lead to better <math>L_R</math> and worse <math>L_{KL}</math> compared to bigger values of <math>w_{KL}</math>. Also for unconditional generation, the model is a standalone decoder, so there will be no <math>L_{KL}</math> term as only <math>L_{R}</math> is optimized for. This trade-off is illustrated in Figure 4 showing different settings of <math>w_{KL}</math> and the resulting <math>L_{KL}</math> and <math>L_{R}</math>, as well as just <math>L_{R}</math> in the case of unconditional generation with only a standalone decoder.<br />
<br />
[[File:paper15_fig4.png|600px]]<br />
<br />
In practice however it was found that annealing the KL term improves the training of the network. While the original loss can be used for testing and validation, when training the following variation on the loss is used:<br />
\begin{align}<br />
\eta_{step} = 1 - (1 - \eta_{min})R^{step}<br />
\end{align}<br />
\begin{align}<br />
Loss_{train} = L_{R} + w_{KL} \eta_{step} max(L_{KL},KL_{min})<br />
\end{align}<br />
<br />
As can be seen above, the <math>\eta_{step}</math> term will start at some preset <math>\eta_{min}</math> value. As <math>R</math> is a value slightly smaller than 1, the <math>R^{step}</math> term will converge to 0, and thus <math>\eta_{step}</math> will converge to 1. This will have the affect of focusing the training loss in the early stages on the reconstruction loss <math>L_{R}</math>, but to a more balanced loss in the later steps. Additionally, in practice it was found that when <math>L_{KL}</math> got too low, the network would cease learning. To combat this a floor value for the KL loss was implemented by the <math>max(.)</math> function.<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 <math>w_{KL}</math> and recorded the different <math>L_{R}</math> and <math>L_{KL}</math> 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 <math display="inline"> z </math>. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector <math display="inline"> z </math>. 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 <math>S’</math> given input sketch <math>S</math> using different values for the temperature hyperparameter <math>\tau</math>. The figure above shows the results for different values of <math>\tau</math> 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 <math>z</math> 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 <math>w_{KL}</math> 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 <math>w_{KL}</math> 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 <math>\tau</math> 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 <math>z</math> by manipulating the <math>\tau</math> 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 <math>\mu</math> and <math>\sigma</math>, 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 states 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., & 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<br />
# David Ha, Andrew Dai, Quoc V. Le. HyperNetworks. (2016) arXiv:1609.09106</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=36379stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-04-21T01:29:46Z<p>W285liu: /* Future Work */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems are usually trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora does not exist<br />
# To provide a strong lower bound that any semi-supervised machine translation system is supposed to yield<br />
<br />
<br />
=== Note: What is a corpus (plural corpora)? ===<br />
<br />
In linguistics, a corpus (plural corpora) or text corpus and structured set of texts (nowadays usually electronically stored and processed). They are used to do statistical analysis and hypothesis testing, checking occurrences or validating linguistic rules within a specific language territory. A corpus may contain texts in a single language (monolingual corpus) or text data in multiple language (multilingual corpus).<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
This paper investigates whether it is possible to train a general machine translation system without any form of supervision whatsoever. Based on the assumption that there exists a monolingual corpus (explained earlier) on each language. This set up is interesting for two reasons. <br />
<br />
* First, this is applicable whenever we encounter a new language pair for which we have no annotation. <br />
<br />
* Second, it provides a strong lower bound performance on what any good semi-supervised approach is expected to yield.<br />
<br />
[[File:paper4_fig1.png|frame|none|alt=Alt text|A toy example of illustrating the training process which guides the design of the objective function. The key idea here is to build a common latent space between languages. On the left, the model is trained to reconstruct a sentence from a noisy version of it in the same language. x is the target, C(x) is the noisy input, <math> \hat{x} </math> is the reconstruction. On the right, the model is trained to reconstruct a sentence given the same sentence but in another language.]]<br />
<br />
==Notation==<br />
Let <math>S</math> denote the set of words in the source language, and let <math>T</math> denote the set of words in the target language. Let <math>H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math>S'</math> and <math>T'</math> denote the sets of finite sequences of words in the source and target language, and let <math>H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math>\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -g(\text{homme}) +g(\text{femme})\approx g(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
[[File:paper4_fig2.png|700px|]]<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturbed a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
The reason why we need to de-noise is because: during the training process of an auto-encoder of sentences, if the sequence-to-sequence model is provided<br />
with an attention mechanism. Then without any constraint, the auto-encoder tempts to merely copy every input word one by one. Resulting in perfectly copy sequences of random words, suggesting that the model does not learn any useful structure in the data.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
<br />
It is interesting to observe that while the intuition justifying the use of the latent space suggests that the latent space representation of a sentence should be language-independent, this is not actually true: if two sentences are translations of one another, but have different lengths, their latent-space representations will necessarily be different, since a a sentence's latent space representation has the same length as the sentence itself.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, so they do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
As justification, they show empirically that the score generated by applying BLEU on back-and-forth translation is correlated with applying BLEU using parallel corpora.<br />
[[File:paper4fig3.png]]<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using the same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentences each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
The discriminator is a MLP with 3 hidden layers of size 1024, Leaky-ReLU activation functions and an output logistic unit. The encoder and the decoder are trained using Adam with<br />
a learning rate of 0.0003, and a mini-batch size of 32. The discriminator is trained using RMSProp with a learning rate of 0.0005.<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, for two reasons. Firstly, Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. Secondly, because the same encoder is used to read sentences of both languages, the encoder cannot adapt to the unique word-order properties of either language. This would become a problem for language pairs with very different grammar. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method likely suffers.<br />
<br />
<br />
<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seems promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
Of course, adding more parallel examples allows the supervised approach to outperform our method,<br />
but the good performance of our unsupervised method suggests that it could be very effective for<br />
low-resources languages where no parallel data are available. Moreover, these results open the<br />
door to the development of semi-supervised translation models, which will be the focus of future<br />
investigation. With a phrase-based machine translation system, we obtain 21.6 and 22.4 BLEU<br />
on WMT en-fr and fr-en, which is better than the supervised NMT baseline we report for that<br />
same amount of parallel sentences, which is 16.8 and 16.4 respectively. However, if we train the<br />
same supervised NMT model with BPE (Sennrich et al., 2015b), we obtain 22.6 BLEU for en-fr,<br />
suggesting that our results on unsupervised machine translation could also be improved by using<br />
BPE, as this removes unknown words (about 9% of the words in de-en are replaced by the unknown<br />
token otherwise).<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36378Training And Inference with Integers in Deep Neural Networks2018-04-21T01:23:48Z<p>W285liu: /* Limitation */</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 will be 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 & 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="wikitable"<br />
|+Rough Energy Costs in 45nm 0.9V<sup>[[#References|[1]]]</sup><br />
!<br />
! colspan="2" |Energy(pJ)<br />
! colspan="2" |Area(<math>\mu m^2</math>)<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 <sup>[[#References|[2]]]</sup> 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 <sup>[[#References|[11]]]</sup> 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) <sup>[[#References|[3]]]</sup> and Trained ternary quantization (TTQ)<sup>[[#References|[9]]]</sup> 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.<sup>[[#References|[14]]]</sup> 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<sup>[[#References|[13]]]</sup>; 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 />
<br />
As can be observed from the graph, the author extended the original definition of errors to multi-layer: error e is the gradient of activation a for the perspective of each convolution or fully-connected layer, while gradient g particularly refers to the gradient accumulation of weight W. Considering the i-th layer of a feed-forward network. In this sense, the error and gradient are defined as:<br />
<br />
<math>e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}</math><br />
<br />
where L is the loss function.<br />
<br />
The precision in bits of the errors, activations, gradients, and weights are <math>k_E</math>, <math>k_A</math>, <math>k_G</math>, and <math>k_W</math> 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 <math>k</math> with a uniform spacing of<br />
<br />
<math>\sigma(k) = 2^{1-k}, k \in Z_+ </math><br />
With this, the full quantization function is<br />
<br />
<math>Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}</math>, <br />
<br />
where <math>round</math> approximates continuous values to their nearest discrete state, and <math>Clip</math> is the saturation function that clips unbounded values to <math>[-1 + \sigma, 1 - \sigma]</math>. 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 />
<math>Shift(x) = 2^{round(log_2(x))}</math><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 <math>P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil</math> denotes probability, vector, floor and ceil, respectively. <math>Shift(\cdot)</math> 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<sup>[[#References|[4]]]</sup>.<br />
<br />
<math>W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma</math><br />
<br />
<math>n_{in}</math> is the layer fan-in number, <math>U</math> denotes uniform distribution. The original initialization method for <math>\eta</math> is modified by adding the condition that the distribution width should be at least <math>\beta \sigma</math>, where <math>\beta</math> is a constant greater than 1 and <math>\sigma</math> 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 <math>Q_W(\cdot)</math> ====<br />
<math>W_q = Q_W(W) = Q(W, k_W)</math><br />
<br />
The quantization operator is simply the quantization function previously introduced. <br />
<br />
==== Activation <math>Q_A(\cdot)</math> ====<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 <math>\alpha</math>. Notice that it is constant for each layer.<br />
<br />
<math>\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}</math><br />
<br />
The quantization operator is then<br />
<br />
<math>a_q = Q_A(a) = Q(a/\alpha, k_A)</math><br />
<br />
The scaling factor approximates batch normalization.<br />
<br />
==== Error <math>Q_E(\cdot)</math> ====<br />
The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net<sup>[[#References|[5]]]</sup>) solves the issue by using an affine transform to map the error to the range <math>[-1, 1]</math>, 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 <math>\left [ -\sqrt2, \sqrt2 \right ]</math> and quantise:<br />
<br />
<math>e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)</math><br />
<br />
Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br />
<br />
==== Gradient <math>Q_G(\cdot)</math> ====<br />
Similar to the activations and errors, the gradients are rescaled:<br />
<br />
<math>g_s = \eta \cdot g/Shift(max\{|g|\})</math><br />
<br />
<math> \eta </math> is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes <math> \sigma(k) </math>. 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 />
<math>\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 \}</math><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 />
<math>W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}</math><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<sup>[[#References|[6]]]</sup> with 32C5-MP2-64C5-MP2-512FC-10SSE.<br />
<br />
SVHN & CIFAR10: VGG variant<sup>[[#References|[7]]]</sup> 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)<sup>[[#References|[10]]]</sup> for training.<br />
<br />
ImageNet: AlexNet variant<sup>[[#References|[8]]]</sup> on ILSVRC12 dataset.<br />
{| class="wikitable"<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 />
!<math>k_W</math><br />
!<math>k_A</math><br />
!<math>k_G</math><br />
!<math>k_E</math><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 <math>k_E</math>]]<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 <math>k_G</math> for gradients using the CIFAR10 dataset. <br />
<br />
{| class="wikitable"<br />
|+Test error rates (%) on CIFAR10 with different <math>k_G</math><br />
!<math>k_G</math><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 <math>k_E</math>.<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="wikitable"<br />
|+Top-5 error rates (%) on ImageNet with different <math>k_G</math>and <math>k_E</math><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 <math>k_G</math> for gradient accumulation than if it has high-resolution orientation <math>k_E</math>. The authors also noted that batch normalization and <math>k_G</math> 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 />
[[File:DAIMA.jpg|center|thumb|800px|]]<br />
== Limitation == <br />
<br />
* The paper states the advantages in energy costs as well as convergence speed , but is there any limitation or trade-off by selecting integer instead of float-point-operation? What is a good situation for such implementation? The authors should explain more on this.<br />
<br />
== References ==<br />
# Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 "Efficient Processing of Deep Neural Networks: A Tutorial and Survey"]. arXiv:1703.09039 [cs].<br />
# Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 "BinaryConnect: Training Deep Neural Networks with binary weights during propagations"]. arXiv:1511.00363 [cs].<br />
# Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 "Ternary Weight Networks"]. arXiv:1605.04711 [cs].<br />
# He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification"]. 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 "DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients"]. arXiv:1606.06160 [cs].<br />
# Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true "Gradient-based learning applied to document recognition"]. 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 "Very Deep Convolutional Networks for Large-Scale Image Recognition"]. 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., & 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. "How to train a compact binary neural network with high accuracy?." AAAI. 2017.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36377Training And Inference with Integers in Deep Neural Networks2018-04-21T01:23:27Z<p>W285liu: /* Limitation */</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 will be 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 & 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="wikitable"<br />
|+Rough Energy Costs in 45nm 0.9V<sup>[[#References|[1]]]</sup><br />
!<br />
! colspan="2" |Energy(pJ)<br />
! colspan="2" |Area(<math>\mu m^2</math>)<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 <sup>[[#References|[2]]]</sup> 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 <sup>[[#References|[11]]]</sup> 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) <sup>[[#References|[3]]]</sup> and Trained ternary quantization (TTQ)<sup>[[#References|[9]]]</sup> 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.<sup>[[#References|[14]]]</sup> 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<sup>[[#References|[13]]]</sup>; 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 />
<br />
As can be observed from the graph, the author extended the original definition of errors to multi-layer: error e is the gradient of activation a for the perspective of each convolution or fully-connected layer, while gradient g particularly refers to the gradient accumulation of weight W. Considering the i-th layer of a feed-forward network. In this sense, the error and gradient are defined as:<br />
<br />
<math>e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}</math><br />
<br />
where L is the loss function.<br />
<br />
The precision in bits of the errors, activations, gradients, and weights are <math>k_E</math>, <math>k_A</math>, <math>k_G</math>, and <math>k_W</math> 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 <math>k</math> with a uniform spacing of<br />
<br />
<math>\sigma(k) = 2^{1-k}, k \in Z_+ </math><br />
With this, the full quantization function is<br />
<br />
<math>Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}</math>, <br />
<br />
where <math>round</math> approximates continuous values to their nearest discrete state, and <math>Clip</math> is the saturation function that clips unbounded values to <math>[-1 + \sigma, 1 - \sigma]</math>. 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 />
<math>Shift(x) = 2^{round(log_2(x))}</math><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 <math>P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil</math> denotes probability, vector, floor and ceil, respectively. <math>Shift(\cdot)</math> 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<sup>[[#References|[4]]]</sup>.<br />
<br />
<math>W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma</math><br />
<br />
<math>n_{in}</math> is the layer fan-in number, <math>U</math> denotes uniform distribution. The original initialization method for <math>\eta</math> is modified by adding the condition that the distribution width should be at least <math>\beta \sigma</math>, where <math>\beta</math> is a constant greater than 1 and <math>\sigma</math> 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 <math>Q_W(\cdot)</math> ====<br />
<math>W_q = Q_W(W) = Q(W, k_W)</math><br />
<br />
The quantization operator is simply the quantization function previously introduced. <br />
<br />
==== Activation <math>Q_A(\cdot)</math> ====<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 <math>\alpha</math>. Notice that it is constant for each layer.<br />
<br />
<math>\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}</math><br />
<br />
The quantization operator is then<br />
<br />
<math>a_q = Q_A(a) = Q(a/\alpha, k_A)</math><br />
<br />
The scaling factor approximates batch normalization.<br />
<br />
==== Error <math>Q_E(\cdot)</math> ====<br />
The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net<sup>[[#References|[5]]]</sup>) solves the issue by using an affine transform to map the error to the range <math>[-1, 1]</math>, 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 <math>\left [ -\sqrt2, \sqrt2 \right ]</math> and quantise:<br />
<br />
<math>e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)</math><br />
<br />
Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br />
<br />
==== Gradient <math>Q_G(\cdot)</math> ====<br />
Similar to the activations and errors, the gradients are rescaled:<br />
<br />
<math>g_s = \eta \cdot g/Shift(max\{|g|\})</math><br />
<br />
<math> \eta </math> is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes <math> \sigma(k) </math>. 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 />
<math>\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 \}</math><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 />
<math>W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}</math><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<sup>[[#References|[6]]]</sup> with 32C5-MP2-64C5-MP2-512FC-10SSE.<br />
<br />
SVHN & CIFAR10: VGG variant<sup>[[#References|[7]]]</sup> 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)<sup>[[#References|[10]]]</sup> for training.<br />
<br />
ImageNet: AlexNet variant<sup>[[#References|[8]]]</sup> on ILSVRC12 dataset.<br />
{| class="wikitable"<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 />
!<math>k_W</math><br />
!<math>k_A</math><br />
!<math>k_G</math><br />
!<math>k_E</math><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 <math>k_E</math>]]<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 <math>k_G</math> for gradients using the CIFAR10 dataset. <br />
<br />
{| class="wikitable"<br />
|+Test error rates (%) on CIFAR10 with different <math>k_G</math><br />
!<math>k_G</math><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 <math>k_E</math>.<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="wikitable"<br />
|+Top-5 error rates (%) on ImageNet with different <math>k_G</math>and <math>k_E</math><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 <math>k_G</math> for gradient accumulation than if it has high-resolution orientation <math>k_E</math>. The authors also noted that batch normalization and <math>k_G</math> 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 />
[[File:DAIMA.jpg|center|thumb|800px|]]<br />
== Limitation == <br />
<br />
* The paper states the advantages in energy costs , but is there any limitation or trade-off by selecting integer instead of float-point-operation? What is a good situation for such implementation? The authors should explain more on this.<br />
<br />
== References ==<br />
# Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 "Efficient Processing of Deep Neural Networks: A Tutorial and Survey"]. arXiv:1703.09039 [cs].<br />
# Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 "BinaryConnect: Training Deep Neural Networks with binary weights during propagations"]. arXiv:1511.00363 [cs].<br />
# Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 "Ternary Weight Networks"]. arXiv:1605.04711 [cs].<br />
# He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification"]. 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 "DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients"]. arXiv:1606.06160 [cs].<br />
# Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true "Gradient-based learning applied to document recognition"]. 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 "Very Deep Convolutional Networks for Large-Scale Image Recognition"]. 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., & 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. "How to train a compact binary neural network with high accuracy?." AAAI. 2017.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36376Training And Inference with Integers in Deep Neural Networks2018-04-21T01:20:41Z<p>W285liu: /* Limitation */</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 will be 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 & 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="wikitable"<br />
|+Rough Energy Costs in 45nm 0.9V<sup>[[#References|[1]]]</sup><br />
!<br />
! colspan="2" |Energy(pJ)<br />
! colspan="2" |Area(<math>\mu m^2</math>)<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 <sup>[[#References|[2]]]</sup> 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 <sup>[[#References|[11]]]</sup> 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) <sup>[[#References|[3]]]</sup> and Trained ternary quantization (TTQ)<sup>[[#References|[9]]]</sup> 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.<sup>[[#References|[14]]]</sup> 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<sup>[[#References|[13]]]</sup>; 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 />
<br />
As can be observed from the graph, the author extended the original definition of errors to multi-layer: error e is the gradient of activation a for the perspective of each convolution or fully-connected layer, while gradient g particularly refers to the gradient accumulation of weight W. Considering the i-th layer of a feed-forward network. In this sense, the error and gradient are defined as:<br />
<br />
<math>e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}</math><br />
<br />
where L is the loss function.<br />
<br />
The precision in bits of the errors, activations, gradients, and weights are <math>k_E</math>, <math>k_A</math>, <math>k_G</math>, and <math>k_W</math> 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 <math>k</math> with a uniform spacing of<br />
<br />
<math>\sigma(k) = 2^{1-k}, k \in Z_+ </math><br />
With this, the full quantization function is<br />
<br />
<math>Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}</math>, <br />
<br />
where <math>round</math> approximates continuous values to their nearest discrete state, and <math>Clip</math> is the saturation function that clips unbounded values to <math>[-1 + \sigma, 1 - \sigma]</math>. 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 />
<math>Shift(x) = 2^{round(log_2(x))}</math><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 <math>P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil</math> denotes probability, vector, floor and ceil, respectively. <math>Shift(\cdot)</math> 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<sup>[[#References|[4]]]</sup>.<br />
<br />
<math>W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma</math><br />
<br />
<math>n_{in}</math> is the layer fan-in number, <math>U</math> denotes uniform distribution. The original initialization method for <math>\eta</math> is modified by adding the condition that the distribution width should be at least <math>\beta \sigma</math>, where <math>\beta</math> is a constant greater than 1 and <math>\sigma</math> 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 <math>Q_W(\cdot)</math> ====<br />
<math>W_q = Q_W(W) = Q(W, k_W)</math><br />
<br />
The quantization operator is simply the quantization function previously introduced. <br />
<br />
==== Activation <math>Q_A(\cdot)</math> ====<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 <math>\alpha</math>. Notice that it is constant for each layer.<br />
<br />
<math>\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}</math><br />
<br />
The quantization operator is then<br />
<br />
<math>a_q = Q_A(a) = Q(a/\alpha, k_A)</math><br />
<br />
The scaling factor approximates batch normalization.<br />
<br />
==== Error <math>Q_E(\cdot)</math> ====<br />
The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net<sup>[[#References|[5]]]</sup>) solves the issue by using an affine transform to map the error to the range <math>[-1, 1]</math>, 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 <math>\left [ -\sqrt2, \sqrt2 \right ]</math> and quantise:<br />
<br />
<math>e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)</math><br />
<br />
Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br />
<br />
==== Gradient <math>Q_G(\cdot)</math> ====<br />
Similar to the activations and errors, the gradients are rescaled:<br />
<br />
<math>g_s = \eta \cdot g/Shift(max\{|g|\})</math><br />
<br />
<math> \eta </math> is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes <math> \sigma(k) </math>. 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 />
<math>\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 \}</math><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 />
<math>W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}</math><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<sup>[[#References|[6]]]</sup> with 32C5-MP2-64C5-MP2-512FC-10SSE.<br />
<br />
SVHN & CIFAR10: VGG variant<sup>[[#References|[7]]]</sup> 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)<sup>[[#References|[10]]]</sup> for training.<br />
<br />
ImageNet: AlexNet variant<sup>[[#References|[8]]]</sup> on ILSVRC12 dataset.<br />
{| class="wikitable"<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 />
!<math>k_W</math><br />
!<math>k_A</math><br />
!<math>k_G</math><br />
!<math>k_E</math><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 <math>k_E</math>]]<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 <math>k_G</math> for gradients using the CIFAR10 dataset. <br />
<br />
{| class="wikitable"<br />
|+Test error rates (%) on CIFAR10 with different <math>k_G</math><br />
!<math>k_G</math><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 <math>k_E</math>.<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="wikitable"<br />
|+Top-5 error rates (%) on ImageNet with different <math>k_G</math>and <math>k_E</math><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 <math>k_G</math> for gradient accumulation than if it has high-resolution orientation <math>k_E</math>. The authors also noted that batch normalization and <math>k_G</math> 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 />
[[File:DAIMA.jpg|center|thumb|800px|]]<br />
== Limitation == <br />
<br />
* The paper states the advantages in energy costs as well as convergence speed, but is there any limitation or trade-off by selecting integer instead of float-point-operation? What is a good situation for such implementation? The authors should explain more on this.<br />
<br />
== References ==<br />
# Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 "Efficient Processing of Deep Neural Networks: A Tutorial and Survey"]. arXiv:1703.09039 [cs].<br />
# Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 "BinaryConnect: Training Deep Neural Networks with binary weights during propagations"]. arXiv:1511.00363 [cs].<br />
# Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 "Ternary Weight Networks"]. arXiv:1605.04711 [cs].<br />
# He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification"]. 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 "DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients"]. arXiv:1606.06160 [cs].<br />
# Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true "Gradient-based learning applied to document recognition"]. 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 "Very Deep Convolutional Networks for Large-Scale Image Recognition"]. 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., & 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. "How to train a compact binary neural network with high accuracy?." AAAI. 2017.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=36296Word translation without parallel data2018-04-17T01:11:31Z<p>W285liu: /* Cross-Domain Similarity Local Scaling (CSLS) */</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 />
= Related Work =<br />
<br />
'''Bilingual Lexicon Induction'''<br />
<br />
Many papers have addressed this subject by using discrete word representations. Regularly however these methods need to have an initialization of prior knowledge, such as the editing distance between the input and output ground truth. This unfortunately only works for closely related languages.<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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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. Here <math>||\cdot||_F</math> is the Frobenius matrix norm which is the square root of the sum of the squared components.<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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V.<br />
<br />
<math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 which is the cosine of the angle between two vectors. Likewise, the mean similarity of a target word <math> y_t </math> to its neighborhood is denoted as <math> r_S(y_t) </math>. 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<br />
<br />
Tomas Mikolov, Kai Chen, Greg Corrado, Jeffrey Dean<br />
| Efficient Estimation of Word Representations in Vector Space, 2013<br />
| arXiv:1301.3781<br />
<br />
<br />
Xing, C., Wang, D., Liu, C., & Lin, Y. (2015). Normalized Word Embedding and Orthogonal Transform for Bilingual Word Translation. HLT-NAACL.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=36295Word translation without parallel data2018-04-17T01:10:50Z<p>W285liu: /* Cross-Domain Similarity Local Scaling (CSLS) */</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 />
= Related Work =<br />
<br />
'''Bilingual Lexicon Induction'''<br />
<br />
Many papers have addressed this subject by using discrete word representations. Regularly however these methods need to have an initialization of prior knowledge, such as the editing distance between the input and output ground truth. This unfortunately only works for closely related languages.<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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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. Here <math>||\cdot||_F</math> is the Frobenius matrix norm which is the square root of the sum of the squared components.<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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets {\displaystyle U} U and {\displaystyle V} V such that every edge connects a vertex in {\displaystyle U} U to one in {\displaystyle V} V.<br />
<br />
<math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 which is the cosine of the angle between two vectors. Likewise, the mean similarity of a target word <math> y_t </math> to its neighborhood is denoted as <math> r_S(y_t) </math>. 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<br />
<br />
Tomas Mikolov, Kai Chen, Greg Corrado, Jeffrey Dean<br />
| Efficient Estimation of Word Representations in Vector Space, 2013<br />
| arXiv:1301.3781<br />
<br />
<br />
Xing, C., Wang, D., Liu, C., & Lin, Y. (2015). Normalized Word Embedding and Orthogonal Transform for Bilingual Word Translation. HLT-NAACL.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=35417stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-24T23:42:30Z<p>W285liu: /* Methods */</p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victims is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their physical position accurately. This can happen for instance when the caller is disoriented, held hostage, or a child is calling on behalf of the victim. GPS sensors on smartphones can provide the rescuers with the geographic location. However GPS fails to give an accurate floor level inside a tall building. Previous work have explored using Wi-Fi signals or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. <br />
Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any external prior knowledge.<br />
<br />
= Data Description =<br />
The authors developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings were recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoor or outdoor and detects the instances of transition between them. When an outdoor to indoor transition event occurs, the elevation of the user is saved using an estimation from the cellphone barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimation is applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true labels and the predictions.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoors or outdoors is that it learns a pattern of how the walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor/outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances measures the similarity of two sets and is calculated with the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance between <math>V_{1}</math> and sub-sequence <math> s_i </math> is greater or equal to the threshold 0.4, it means there was a transition from indoors to outdoors in the vicinity of the 20 second range of the vector mask. Similarly, a distance of to 0.4 or greater to <math>V_{2}</math> indicates a transition from outdoors to indoors. Sets of transition windows are merged together if they occur close in time to each other, with the average transition time of both windows being used as the new transition time.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
In appendix B.1, the authors acknowledge that for this system to work, pressures variations due to weather or temperature must be accounted for as those variations are on the same order of magnitude or larger than the pressure variations caused by changing altitude. They suggest using a nearby reference station with known altitude to continuously measure and correct for this effect.<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitude data collected are clustered into groups. Each cluster represents the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
The authors performed evaluation on two different tasks: The indoor-outdoor classification task and the floor level prediction task. In the indoor-outdoor detection task, they compared six different models, LSTM, feedforward neural networks, logistic regression, SVM, HMM and Random Forests. In the floor level prediction task, they evaluated the full system.<br />
<br />
== Indoor-Outdoor Classification Results ==<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
== Floor Level Prediction Results ==<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
== Floor Level Clustering Results == <br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieves an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used in this paper only contains 3 layers, and the clustering is applied on one-dimensional data. This leads to the question whether deep learning is necessary and suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not specifically mentioned in the paper, but they do note that differences between outdoors and pressure-sealed buildings is a problem<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) collect data from multiple users (which is what the paper’s results are based on). However, such data collection would need to be done in accordance with local laws. Perhaps a better solution would be to use elevation reading to estimate a floor based on typical floor height. Even having a small range of floors of interest could help first responders significantly narrow down their response time.<br />
<br />
Aside from all the technical issues, if knowing the exact floor is required, would it maybe be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=35415stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-24T22:09:25Z<p>W285liu: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victims is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their physical position accurately. This can happen for instance when the caller is disoriented, held hostage, or a child is calling on behalf of the victim. GPS sensors on smartphones can provide the rescuers with the geographic location. However GPS fails to give an accurate floor level inside a tall building. Previous work have explored using Wi-Fi signals or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. <br />
Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any external prior knowledge.<br />
<br />
= Data Description =<br />
The authors developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings were recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors and detects the instances of transition between them. When a outdoor to indoor transition event occurs, the elevation of the user is saved using an estimation from the cellphone barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimation is applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoors or outdoors is that it learns a pattern of how the walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor/outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances measures the similarity of two sets and is calculated with the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance between <math>V_{1}</math> and sub-sequence <math> s_i </math> is greater or equal to the threshold 0.4, it means there was a transition from indoors to outdoors in the vicinity of the 20 second range of the vector mask. Similarly, a distance of to 0.4 or greater to <math>V_{2}</math> indicates a transition from outdoors to indoors. Sets of transition windows are merged together if they occur close in time to each other, with the average transition time of both windows being used as the new transition time.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
In appendix B.1, the authors acknowledge that for this system to work, pressures variations due to weather or temperature must be accounted for as those variations are on the same order of magnitude or larger than the pressure variations caused by changing altitude. They suggest using a nearby reference station with known altitude to continuously measure and correct for this effect.<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitude data collected are clustered into groups. Each cluster represents the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
The authors performed evaluation on two different tasks: The indoor-outdoor classification task and the floor level prediction task. In the indoor-outdoor detection task, they compared six different models, LSTM, feedforward neural networks, logistic regression, SVM, HMM and Random Forests. In the floor level prediction task, they evaluated the full system.<br />
<br />
== Indoor-Outdoor Classification Results ==<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
== Floor Level Prediction Results ==<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
== Floor Level Clustering Results == <br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieves an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used in this paper only contains 3 layers, and the clustering is applied on one-dimensional data. This leads to the question whether deep learning is necessary and suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not specifically mentioned in the paper, but they do note that differences between outdoors and pressure-sealed buildings is a problem<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) collect data from multiple users (which is what the paper’s results are based on). However, such data collection would need to be done in accordance with local laws. Perhaps a better solution would be to use elevation reading to estimate a floor based on typical floor height. Even having a small range of floors of interest could help first responders significantly narrow down their response time.<br />
<br />
Aside from all the technical issues, if knowing the exact floor is required, would it maybe be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Label-Free_Supervision_of_Neural_Networks_with_Physics_and_Domain_Knowledge&diff=35408Label-Free Supervision of Neural Networks with Physics and Domain Knowledge2018-03-24T17:18:28Z<p>W285liu: /* Tracking an object in free fall */</p>
<hr />
<div>== Introduction ==<br />
Applications of machine learning are often encumbered by the need for large amounts of labeled training data. Neural networks have made large amounts of labeled data even more crucial to success (LeCun, Bengio, and Hinton 2015[1]). Nonetheless, humans are often able to learn without direct examples, opting instead for high level instructions for how a task should be performed, or what it will look like when completed. This work explores whether a similar principle can be applied to teaching machines: can we supervise networks without individual examples by instead describing only the structure of desired outputs.<br />
<br />
[[File:c433li-1.png|300px|center]]<br />
<br />
Unsupervised learning methods such as autoencoders, also aim to uncover hidden structure in the data without having access to any label. Such systems succeed in producing highly compressed, yet informative representations of the inputs (Kingma and Welling 2013; Le 2013). However, these representations differ from ours as they are not explicitly constrained to have a particular meaning or semantics. This paper attempts to explicitly provide the semantics of the hidden variables we hope to discover, but still train without labels by learning from constraints that are known to hold according to prior domain knowledge. By training without direct examples of the values our hidden (output) variables take, several advantages are gained over traditional supervised learning, including:<br />
* a reduction in the amount of work spent labeling, <br />
* an increase in generality, as a single set of constraints can be applied to multiple data sets without relabeling.<br />
<br />
== Problem Setup ==<br />
In a traditional supervised learning setting, we are given a training set <math>D=\{(x_1, y_1), \cdots, (x_n, y_n)\}</math> of <math>n</math> training examples. Each example is a pair <math>(x_i,y_i)</math> formed by an instance <math>x_i \in X</math> and the corresponding output (label) <math>y_i \in Y</math>. The goal is to learn a function <math>f: X \rightarrow Y</math> mapping inputs to outputs. To quantify performance, a loss function <math>\ell:Y \times Y \rightarrow \mathbb{R}</math> is provided, and a mapping is found via <br />
<br />
<center><math> f^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) </math></center><br />
<br />
where the optimization is over a pre-defined class of functions <math>\mathcal{F}</math> (hypothesis class). In our case, <math>\mathcal{F}</math> will be (convolutional) neural networks parameterized by their weights. The loss could be for example <math>\ell(f(x_i),y_i) = 1[f(x_i) \neq y_i]</math>. By restricting the space of possible functions specifying the hypothesis class <math>\mathcal{F}</math>, we are leveraging prior knowledge about the specific problem we are trying to solve. Informally, the so-called No Free Lunch Theorems state that every machine learning algorithm must make such assumptions in order to work. Another common way in which a modeler incorporates prior knowledge is by specifying an a-priori preference for certain functions in <math>\mathcal{F}</math>, incorporating a regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math>, and solving for <math> f^* = argmin_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) + R(f)</math>. Typically, the regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math> specifies a preference for "simpler" functions (Occam's razor) to prevent overfitting the model on the training data.<br />
<br />
The focus is on the set of problems/domains where the problem is a complex environment having a complex representation of the output space, for example mapping an input image to the height of an object(since this leads to a complex output space) rather than simple binary classification problem.<br />
<br />
In this paper, prior knowledge on the structure of the outputs is modelled by providing a weighted constraint function <math>g:X \times Y \rightarrow \mathbb{R}</math>, used to penalize “structures” that are not consistent with our prior knowledge. And whether this weak form of supervision is sufficient to learn interesting functions is explored. While one clearly needs labels <math>y</math> to evaluate <math>f^*</math>, labels may not be necessary to discover <math>f^*</math>. If prior knowledge informs us that outputs of <math>f^*</math> have other unique properties among functions in <math>\mathcal{F}</math>, we may use these properties for training rather than direct examples <math>y</math>. <br />
<br />
Specifically, an unsupervised approach where the labels <math>y_i</math> are not provided to us is considered, where a necessary property of the output <math>g</math> is optimized instead.<br />
<center><math>\hat{f}^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n g(x_i,f(x_i))+ R(f) </math></center><br />
<br />
If the optimizing the above equation is sufficient to find <math>\hat{f}^*</math>, we can use it in replace of labels. If it's not sufficient, additional regularization terms are added. The idea is illustrated with three examples, as described in the next section.<br />
<br />
== Experiments ==<br />
=== Tracking an object in free fall ===<br />
In the first experiment, they record videos of an object being thrown across the field of view, and aim to learn the object's height in each frame. The goal is to obtain a regression network mapping from <math>{R^{\text{height} \times \text{width} \times 3}} \rightarrow \mathbb{R}</math>, where <math>\text{height}</math> and <math>\text{width}</math> are the number of vertical and horizontal pixels per frame, and each pixel has 3 color channels. This network is trained as a structured prediction problem operating on a sequence of <math>N</math> images to produce a sequence of <math>N</math> heights, <math>\left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each piece of data <math>x_i</math> will be a vector of images, <math>\mathbf{x}</math>.<br />
Rather than supervising the network with direct labels, <math>\mathbf{y} \in \mathbb{R}^N</math>, the network is instead supervised to find an object obeying the elementary physics of free falling objects. An object acting under gravity will have a fixed acceleration of <math>a = -9.8 m / s^2</math>, and the plot of the object's height over time will form a parabola:<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
<br />
The idea is, given any trajectory of <math>N</math> height predictions, <math>f(\mathbf{x})</math>, we fit a parabola with fixed curvature to those predictions, and minimize the resulting residual. Formally, if we specify <math>\mathbf{a} = [\frac{1}{2} a\Delta t^2, \frac{1}{2} a(2 \Delta t)^2, \ldots, \frac{1}{2} a(N \Delta t)^2]</math>, the prediction produced by the fitted parabola is:<br />
<center><math> \text{argmin}_{v_0, y_0}\sum_i(y_i-y_0-v_0(i\Delta_t)-\frac{1}{2}a(i\Delta_t)^2) </math></center><br />
By the solution of ordinary least square estimation: <br />
<center><math> \mathbf{\hat{y}} = \mathbf{a} + \mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T (f(\mathbf{x}) - \mathbf{a}) </math></center><br />
<br />
where<br />
<center><br />
<math><br />
\mathbf{A} = <br />
\left[ {\begin{array}{*{20}c}<br />
\Delta t & 1 \\<br />
2\Delta t & 1 \\<br />
3\Delta t & 1 \\<br />
\vdots & \vdots \\<br />
N\Delta t & 1 \\<br />
\end{array} } \right]<br />
</math><br />
</center><br />
<br />
The constraint loss is then defined as<br />
<center><math>g(\mathbf{x},f(\mathbf{x})) = g(f(\mathbf{x})) = \sum_{i=1}^{N} |\mathbf{\hat{y}}_i - f(\mathbf{x})_i|</math></center><br />
<br />
Note that <math>\hat{y}</math> is not the ground truth labels. Because <math>g</math> is differentiable almost everywhere, it can be optimized with SGD. They find that when combined with existing regularization methods for neural networks, this optimization is sufficient to recover <math>f^*</math> up to an additive constant <math>C</math> (specifying what object height corresponds to 0).<br />
<br />
[[File:c433li-2.png|650px|center]]<br />
<br />
The data set is collected on a laptop webcam running at 10 frames per second (<math>\Delta t = 0.1s</math>). The camera position is fixed and 65 diverse trajectories of the object in flight, totalling 602 images are recorded. For each trajectory, the network is trained on randomly selected intervals of <math>N=5</math> contiguous frames. Images are resized to <math>56 \times 56</math> pixels before going into a small, randomly initialized neural network with no pretraining. The network consists of 3 Conv/ReLU/MaxPool blocks followed by 2 Fully Connected/ReLU layers with probability 0.5 dropout and a single regression output.<br />
<br />
Since scaling the <math>y_0</math> and <math>v_0</math> results in the same constraint loss <math>g</math>, the authors evaluate the result by the correlation of predicted heights with ground truth pixel measurements. This method was used since the distance from the object to the camera could not be accurately recorded, and this distance is required to calculate the height in meters. This is not a bullet proof evaluation, and is discussed in further detail in the critique section. The results are compared to a supervised network trained with the labels to directly predict the height of the object in pixels. The supervised learning task is viewed as a substantially easier task. From this knowledge we can see from the table below that, under their evaluation criteria, the result is pretty satisfying.<br />
<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 12.1% || 94.5% || 90.1%<br />
|}<br />
<br />
=== Tracking the position of a walking man ===<br />
In the second experiment, they aim to detect the horizontal position of a person walking across a frame without providing direct labels <math>y \in \mathbb{R}</math> by exploiting the assumption that the person will be walking at a constant velocity over short periods of time. This is formulated as a structured prediction problem <math>f: \left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each training instances <math>x_i</math> are a vector of images, <math>\mathbf{x}</math>, being mapped to a sequence of predictions, <math>\mathbf{y}</math>. Given the similarities to the first experiment with free falling objects, we might hope to simply remove the gravity term from equation and retrain. However, in this case, that is not possible, as the constraint provides a necessary, but not sufficient, condition for convergence.<br />
<br />
Given any sequence of correct outputs, <math>(\mathbf{y}_1, \ldots, \mathbf{y}_N)</math>, the modified sequence, <math>(\lambda * \mathbf{y}_1 + C, \ldots, \lambda * \mathbf{y}_N + C)</math> (<math>\lambda, C \in \mathbb{R}</math>) will also satisfy the constant velocity constraint. In the worst case, when <math>\lambda = 0</math>, <math>f \equiv C</math>, and the network can satisfy the constraint while having no dependence on the image. The trivial output is avoided by adding two two additional loss terms.<br />
<br />
<center><math>h_1(\mathbf{x}) = -\text{std}(f(\mathbf{x}))</math></center><br />
which seeks to maximize the standard deviation of the output, and<br />
<br />
<center><br />
<math>\begin{split}<br />
h_2(\mathbf{x}) = \hphantom{'} & \text{max}(\text{ReLU}(f(\mathbf{x}) - 10)) \hphantom{\text{ }}+ \\<br />
& \text{max}(\text{ReLU}(0 - f(\mathbf{x})))<br />
\end{split}<br />
</math><br />
</center><br />
which limit the output to a fixed ranged <math>[0, 10]</math>, the final loss is thus:<br />
<br />
<center><br />
<math><br />
\begin{split}<br />
g(\mathbf{x}) = \hphantom{'} & ||(\mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T - \mathbf{I}) * f(\mathbf{x})||_1 \hphantom{\text{ }}+ \\<br />
& \gamma_1 * h_1(\mathbf{x}) <br />
\hphantom{\text{ }}+ \\<br />
& \gamma_2 * h_2(\mathbf{x})<br />
% h_2(y) & = \text{max}(\text{ReLU}(y - 10)) + \\<br />
% & \hphantom{=}\hphantom{a} \text{max}(\text{ReLU}(0 - y))<br />
\end{split}<br />
</math><br />
</center><br />
<br />
[[File:c433li-3.png|650px|center]]<br />
<br />
The data set contains 11 trajectories across 6 distinct scenes, totalling 507 images resized to <math>56 \times 56</math>. The network is trained to output linearly consistent positions on 5 strided frames from the first half of each trajectory, and is evaluated on the second half. The boundary violation penalty is set to <math>\gamma_2 = 0.8</math> and the standard deviation bonus is set to <math>\gamma_1 = 0.6</math>.<br />
<br />
As in the previous experiment, the result is evaluated by the correlation with the ground truth. The result is as follows:<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 45.9% || 80.5% || 95.4%<br />
|}<br />
Surprisingly, the approach in this paper beats the same network trained with direct labeled supervision on the test set, which can be attributed to overfitting on the small amount of training data available (as correlation on training data reached 99.8%).<br />
<br />
=== Detecting objects with causal relationships ===<br />
In the previous experiments, the authors explored options for incorporating constraints pertaining to dynamics equations in real-world phenomena, i.e., prior knowledge derived from elementary physics. In this experiment, the authors explore the possibilities of learning from logical constraints imposed on single images. More specifically, they ask whether it is possible to learn from causal phenomena.<br />
<br />
[[File:paper18_Experiment_3.png|400px|center]]<br />
<br />
Here, the authors provide images containing a stochastic collection of up to four characters: Peach, Mario, Yoshi, and Bowser, with each character having small appearance changes across frames due to rotation and reflection. Example images can be seen in Fig. (4). While the existence of objects in each frame is non-deterministic, the generating distribution encodes the underlying phenomenon that Mario will always appear whenever Peach appears. The aim is to create a pair of neural networks <math>f_1, f_2</math> for identifying Peach and Mario, respectively. The networks, <math>f_k : R^{height×width×3} → \{0, 1\}</math>, map the image to the discrete boolean variables, <math>y_1</math> and <math>y_2</math>. Rather than supervising with direct labels, the authors train the networks by constraining their outputs to have the logical relationship <math>y_1 ⇒ y_2</math>. This problem is challenging because the networks must simultaneously learn to recognize the characters and select them according to logical relationships. To avoid the trivial solution <math>y_1 \equiv 1, y_2 \equiv 1</math> on every image, three additional loss terms need to be added:<br />
<br />
<center><math> h_1(\mathbf{x}, k) = \frac{1}{M}\sum_i^M |Pr[f_k(\mathbf{x}) = 1] - Pr[f_k(\rho(\mathbf{x})) = 1]|, </math></center><br />
<br />
which forces rotational independence of the outputs in order to encourage the network to learn the existence, rather than location of objects, <br />
<br />
<center><math> h_2(\mathbf{x}, k) = -\text{std}_{i \in [1 \dots M]}(Pr[f_k(\mathbf{x}_i) = 1]), </math></center><br />
<br />
which seeks high variance outputs, and<br />
<br />
<center><br />
<math> h_3(\mathbf{x}, v) = \frac{1}{M}\sum_i^{M} (Pr[f(\mathbf{x}_i) = v] - \frac{1}{3} + (\frac{1}{3} - \mu_v))^2 \\<br />
\mu_{v} = \frac{1}{M}\sum_i^{M} \mathbb{1}\{v = \text{argmax}_{v' \in \{0, 1\}^2} Pr[f(\mathbf{x}) = v']\}. </math><br />
</center><br />
<br />
which seeks high entropy outputs. The final loss function then becomes: <br />
<br />
<center><br />
<math> \begin{split}<br />
g(\mathbf{x}) & = \mathbb{1}\{f_1(\mathbf{x}) \nRightarrow f_2(\mathbf{x})\} \hphantom{\text{ }} + \\<br />
& \sum_{k \in \{1, 2\}} \gamma_1 h_1(\mathbf{x}, k) + \gamma_2 h_2(\mathbf{x}, k) + <br />
\hspace{-0.7em} \sum_{v \neq \{1,0\}} \hspace{-0.7em} \gamma_3 * h_3(\mathbf{x}, v)<br />
\end{split}<br />
</math><br />
</center><br />
<br />
====Evaluation====<br />
<br />
The input images, shown in Fig. (4), are 56 × 56 pixels. The authors used <math>\gamma_1 = 0.65, \gamma_2 = 0.65, \gamma_3 = 0.95</math>, and trained for 4,000 iterations. This experiment demonstrates that networks can learn from constraints that operate over discrete sets with potentially complex logical rules. Removing constraints will cause learning to fail. Thus, the experiment also shows that sophisticated sufficiency conditions can be key to success when learning from constraints.<br />
<br />
== Conclusion and Critique ==<br />
This paper has introduced a method for using physics and other domain constraints to supervise neural networks. However, the approach described in this paper is not entirely new. Similar ideas are already widely used in Q learning, where the Q value are not available, and the network is supervised by the constraint, as in Deep Q learning (Mnih, Riedmiller et al. 2013[2]).<br />
<center><math>Q(s,a) = R(r,s) + \gamma \sum_{s' ~ P_{sa}}{\text{max}_{a'}Q(s',a')}</math></center><br />
<br />
<br />
Also, the paper has a mistake where they quote the free fall equation as<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + a(i\Delta t)^2</math></center><br />
which should be<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
Although in this case it doesn't affect the result.<br />
<br />
<br />
For the evaluation of the experiments, they used correlation with ground truth as the metric to avoid the fact that the output can be scaled without affecting the constraint loss. This is fine if the network gives output of the same scale. However, there's no such guarantee, and the network may give output of varying scale for different inputs, in which case, we can't say that the network has learnt the correct thing, although it may have a high correlation with ground truth. In fact, to solve the scaling issue, an obvious way is to combine the constraints introduced in this paper with some labeled training data. It's not clear why the author didn't experiment with a combination of these two losses.<br />
<br />
In regards to the free fall experiment in particular, the authors apply a fixed acceleration model to create the constraint loss, with the goal of having the network predict height. However, since they did not measure the true height of the object to create test labels, they evaluate using height in pixel space. They do not mention the accuracy of their camera calibration, nor what camera model was used to remove lens distortion. Since lens distortion tends to be worse at the extreme edges of the image, and that they tossed the pillow throughout the entire frame, it is likely that the ground truth labels were corrupted by distortion. If that is the case, it is possible the supervised network is actually performing worse, because it learning how to predict distorted (beyond a constant scaling factor) heights instead of the true height.<br />
<br />
These methods essentially boil down to generating approximate labels for training data using some knowledge of the dynamic that the labels should follow.<br />
<br />
Finally, this paper only picks examples where the constraints are easy to design, while in some more common tasks such as image classification, what kind of constraints are needed is not straightforward at all.<br />
<br />
== References ==<br />
[1] LeCun, Y.; Bengio, Y.; and Hinton, G. 2015. Deep learning. Nature 521(7553):436–444.<br />
<br />
[2] Mnih, V.; Kavukcuoglu, K.; Silver, D.; Graves, A.; Antonoglou, I.; Wierstra, D.; and Riedmiller, M. 2013. Playing Atari with Deep Reinforcement Learning. arxiv 1312.5602.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Label-Free_Supervision_of_Neural_Networks_with_Physics_and_Domain_Knowledge&diff=35407Label-Free Supervision of Neural Networks with Physics and Domain Knowledge2018-03-24T17:16:52Z<p>W285liu: /* Tracking an object in free fall */</p>
<hr />
<div>== Introduction ==<br />
Applications of machine learning are often encumbered by the need for large amounts of labeled training data. Neural networks have made large amounts of labeled data even more crucial to success (LeCun, Bengio, and Hinton 2015[1]). Nonetheless, humans are often able to learn without direct examples, opting instead for high level instructions for how a task should be performed, or what it will look like when completed. This work explores whether a similar principle can be applied to teaching machines: can we supervise networks without individual examples by instead describing only the structure of desired outputs.<br />
<br />
[[File:c433li-1.png|300px|center]]<br />
<br />
Unsupervised learning methods such as autoencoders, also aim to uncover hidden structure in the data without having access to any label. Such systems succeed in producing highly compressed, yet informative representations of the inputs (Kingma and Welling 2013; Le 2013). However, these representations differ from ours as they are not explicitly constrained to have a particular meaning or semantics. This paper attempts to explicitly provide the semantics of the hidden variables we hope to discover, but still train without labels by learning from constraints that are known to hold according to prior domain knowledge. By training without direct examples of the values our hidden (output) variables take, several advantages are gained over traditional supervised learning, including:<br />
* a reduction in the amount of work spent labeling, <br />
* an increase in generality, as a single set of constraints can be applied to multiple data sets without relabeling.<br />
<br />
== Problem Setup ==<br />
In a traditional supervised learning setting, we are given a training set <math>D=\{(x_1, y_1), \cdots, (x_n, y_n)\}</math> of <math>n</math> training examples. Each example is a pair <math>(x_i,y_i)</math> formed by an instance <math>x_i \in X</math> and the corresponding output (label) <math>y_i \in Y</math>. The goal is to learn a function <math>f: X \rightarrow Y</math> mapping inputs to outputs. To quantify performance, a loss function <math>\ell:Y \times Y \rightarrow \mathbb{R}</math> is provided, and a mapping is found via <br />
<br />
<center><math> f^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) </math></center><br />
<br />
where the optimization is over a pre-defined class of functions <math>\mathcal{F}</math> (hypothesis class). In our case, <math>\mathcal{F}</math> will be (convolutional) neural networks parameterized by their weights. The loss could be for example <math>\ell(f(x_i),y_i) = 1[f(x_i) \neq y_i]</math>. By restricting the space of possible functions specifying the hypothesis class <math>\mathcal{F}</math>, we are leveraging prior knowledge about the specific problem we are trying to solve. Informally, the so-called No Free Lunch Theorems state that every machine learning algorithm must make such assumptions in order to work. Another common way in which a modeler incorporates prior knowledge is by specifying an a-priori preference for certain functions in <math>\mathcal{F}</math>, incorporating a regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math>, and solving for <math> f^* = argmin_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) + R(f)</math>. Typically, the regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math> specifies a preference for "simpler" functions (Occam's razor) to prevent overfitting the model on the training data.<br />
<br />
The focus is on the set of problems/domains where the problem is a complex environment having a complex representation of the output space, for example mapping an input image to the height of an object(since this leads to a complex output space) rather than simple binary classification problem.<br />
<br />
In this paper, prior knowledge on the structure of the outputs is modelled by providing a weighted constraint function <math>g:X \times Y \rightarrow \mathbb{R}</math>, used to penalize “structures” that are not consistent with our prior knowledge. And whether this weak form of supervision is sufficient to learn interesting functions is explored. While one clearly needs labels <math>y</math> to evaluate <math>f^*</math>, labels may not be necessary to discover <math>f^*</math>. If prior knowledge informs us that outputs of <math>f^*</math> have other unique properties among functions in <math>\mathcal{F}</math>, we may use these properties for training rather than direct examples <math>y</math>. <br />
<br />
Specifically, an unsupervised approach where the labels <math>y_i</math> are not provided to us is considered, where a necessary property of the output <math>g</math> is optimized instead.<br />
<center><math>\hat{f}^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n g(x_i,f(x_i))+ R(f) </math></center><br />
<br />
If the optimizing the above equation is sufficient to find <math>\hat{f}^*</math>, we can use it in replace of labels. If it's not sufficient, additional regularization terms are added. The idea is illustrated with three examples, as described in the next section.<br />
<br />
== Experiments ==<br />
=== Tracking an object in free fall ===<br />
In the first experiment, they record videos of an object being thrown across the field of view, and aim to learn the object's height in each frame. The goal is to obtain a regression network mapping from <math>{R^{\text{height} \times \text{width} \times 3}} \rightarrow \mathbb{R}</math>, where <math>\text{height}</math> and <math>\text{width}</math> are the number of vertical and horizontal pixels per frame, and each pixel has 3 color channels. This network is trained as a structured prediction problem operating on a sequence of <math>N</math> images to produce a sequence of <math>N</math> heights, <math>\left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each piece of data <math>x_i</math> will be a vector of images, <math>\mathbf{x}</math>.<br />
Rather than supervising the network with direct labels, <math>\mathbf{y} \in \mathbb{R}^N</math>, the network is instead supervised to find an object obeying the elementary physics of free falling objects. An object acting under gravity will have a fixed acceleration of <math>a = -9.8 m / s^2</math>, and the plot of the object's height over time will form a parabola:<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
<br />
The idea is, given any trajectory of <math>N</math> height predictions, <math>f(\mathbf{x})</math>, we fit a parabola with fixed curvature to those predictions, and minimize the resulting residual. Formally, if we specify <math>\mathbf{a} = [\frac{1}{2} a\Delta t^2, \frac{1}{2} a(2 \Delta t)^2, \ldots, \frac{1}{2} a(N \Delta t)^2]</math>, the prediction produced by the fitted parabola is:<br />
<center><math> \text{argmin}_{v_0, y_0}\sum_i(y_i-y_0-v_0(i\delta_t)-\frac{1}{2}a(i\delta_t)^2) </math></center><br />
<center><math> \mathbf{\hat{y}} = \mathbf{a} + \mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T (f(\mathbf{x}) - \mathbf{a}) </math></center><br />
<br />
where<br />
<center><br />
<math><br />
\mathbf{A} = <br />
\left[ {\begin{array}{*{20}c}<br />
\Delta t & 1 \\<br />
2\Delta t & 1 \\<br />
3\Delta t & 1 \\<br />
\vdots & \vdots \\<br />
N\Delta t & 1 \\<br />
\end{array} } \right]<br />
</math><br />
</center><br />
<br />
The constraint loss is then defined as<br />
<center><math>g(\mathbf{x},f(\mathbf{x})) = g(f(\mathbf{x})) = \sum_{i=1}^{N} |\mathbf{\hat{y}}_i - f(\mathbf{x})_i|</math></center><br />
<br />
Note that <math>\hat{y}</math> is not the ground truth labels. Because <math>g</math> is differentiable almost everywhere, it can be optimized with SGD. They find that when combined with existing regularization methods for neural networks, this optimization is sufficient to recover <math>f^*</math> up to an additive constant <math>C</math> (specifying what object height corresponds to 0).<br />
<br />
[[File:c433li-2.png|650px|center]]<br />
<br />
The data set is collected on a laptop webcam running at 10 frames per second (<math>\Delta t = 0.1s</math>). The camera position is fixed and 65 diverse trajectories of the object in flight, totalling 602 images are recorded. For each trajectory, the network is trained on randomly selected intervals of <math>N=5</math> contiguous frames. Images are resized to <math>56 \times 56</math> pixels before going into a small, randomly initialized neural network with no pretraining. The network consists of 3 Conv/ReLU/MaxPool blocks followed by 2 Fully Connected/ReLU layers with probability 0.5 dropout and a single regression output.<br />
<br />
Since scaling the <math>y_0</math> and <math>v_0</math> results in the same constraint loss <math>g</math>, the authors evaluate the result by the correlation of predicted heights with ground truth pixel measurements. This method was used since the distance from the object to the camera could not be accurately recorded, and this distance is required to calculate the height in meters. This is not a bullet proof evaluation, and is discussed in further detail in the critique section. The results are compared to a supervised network trained with the labels to directly predict the height of the object in pixels. The supervised learning task is viewed as a substantially easier task. From this knowledge we can see from the table below that, under their evaluation criteria, the result is pretty satisfying.<br />
<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 12.1% || 94.5% || 90.1%<br />
|}<br />
<br />
=== Tracking the position of a walking man ===<br />
In the second experiment, they aim to detect the horizontal position of a person walking across a frame without providing direct labels <math>y \in \mathbb{R}</math> by exploiting the assumption that the person will be walking at a constant velocity over short periods of time. This is formulated as a structured prediction problem <math>f: \left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each training instances <math>x_i</math> are a vector of images, <math>\mathbf{x}</math>, being mapped to a sequence of predictions, <math>\mathbf{y}</math>. Given the similarities to the first experiment with free falling objects, we might hope to simply remove the gravity term from equation and retrain. However, in this case, that is not possible, as the constraint provides a necessary, but not sufficient, condition for convergence.<br />
<br />
Given any sequence of correct outputs, <math>(\mathbf{y}_1, \ldots, \mathbf{y}_N)</math>, the modified sequence, <math>(\lambda * \mathbf{y}_1 + C, \ldots, \lambda * \mathbf{y}_N + C)</math> (<math>\lambda, C \in \mathbb{R}</math>) will also satisfy the constant velocity constraint. In the worst case, when <math>\lambda = 0</math>, <math>f \equiv C</math>, and the network can satisfy the constraint while having no dependence on the image. The trivial output is avoided by adding two two additional loss terms.<br />
<br />
<center><math>h_1(\mathbf{x}) = -\text{std}(f(\mathbf{x}))</math></center><br />
which seeks to maximize the standard deviation of the output, and<br />
<br />
<center><br />
<math>\begin{split}<br />
h_2(\mathbf{x}) = \hphantom{'} & \text{max}(\text{ReLU}(f(\mathbf{x}) - 10)) \hphantom{\text{ }}+ \\<br />
& \text{max}(\text{ReLU}(0 - f(\mathbf{x})))<br />
\end{split}<br />
</math><br />
</center><br />
which limit the output to a fixed ranged <math>[0, 10]</math>, the final loss is thus:<br />
<br />
<center><br />
<math><br />
\begin{split}<br />
g(\mathbf{x}) = \hphantom{'} & ||(\mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T - \mathbf{I}) * f(\mathbf{x})||_1 \hphantom{\text{ }}+ \\<br />
& \gamma_1 * h_1(\mathbf{x}) <br />
\hphantom{\text{ }}+ \\<br />
& \gamma_2 * h_2(\mathbf{x})<br />
% h_2(y) & = \text{max}(\text{ReLU}(y - 10)) + \\<br />
% & \hphantom{=}\hphantom{a} \text{max}(\text{ReLU}(0 - y))<br />
\end{split}<br />
</math><br />
</center><br />
<br />
[[File:c433li-3.png|650px|center]]<br />
<br />
The data set contains 11 trajectories across 6 distinct scenes, totalling 507 images resized to <math>56 \times 56</math>. The network is trained to output linearly consistent positions on 5 strided frames from the first half of each trajectory, and is evaluated on the second half. The boundary violation penalty is set to <math>\gamma_2 = 0.8</math> and the standard deviation bonus is set to <math>\gamma_1 = 0.6</math>.<br />
<br />
As in the previous experiment, the result is evaluated by the correlation with the ground truth. The result is as follows:<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 45.9% || 80.5% || 95.4%<br />
|}<br />
Surprisingly, the approach in this paper beats the same network trained with direct labeled supervision on the test set, which can be attributed to overfitting on the small amount of training data available (as correlation on training data reached 99.8%).<br />
<br />
=== Detecting objects with causal relationships ===<br />
In the previous experiments, the authors explored options for incorporating constraints pertaining to dynamics equations in real-world phenomena, i.e., prior knowledge derived from elementary physics. In this experiment, the authors explore the possibilities of learning from logical constraints imposed on single images. More specifically, they ask whether it is possible to learn from causal phenomena.<br />
<br />
[[File:paper18_Experiment_3.png|400px|center]]<br />
<br />
Here, the authors provide images containing a stochastic collection of up to four characters: Peach, Mario, Yoshi, and Bowser, with each character having small appearance changes across frames due to rotation and reflection. Example images can be seen in Fig. (4). While the existence of objects in each frame is non-deterministic, the generating distribution encodes the underlying phenomenon that Mario will always appear whenever Peach appears. The aim is to create a pair of neural networks <math>f_1, f_2</math> for identifying Peach and Mario, respectively. The networks, <math>f_k : R^{height×width×3} → \{0, 1\}</math>, map the image to the discrete boolean variables, <math>y_1</math> and <math>y_2</math>. Rather than supervising with direct labels, the authors train the networks by constraining their outputs to have the logical relationship <math>y_1 ⇒ y_2</math>. This problem is challenging because the networks must simultaneously learn to recognize the characters and select them according to logical relationships. To avoid the trivial solution <math>y_1 \equiv 1, y_2 \equiv 1</math> on every image, three additional loss terms need to be added:<br />
<br />
<center><math> h_1(\mathbf{x}, k) = \frac{1}{M}\sum_i^M |Pr[f_k(\mathbf{x}) = 1] - Pr[f_k(\rho(\mathbf{x})) = 1]|, </math></center><br />
<br />
which forces rotational independence of the outputs in order to encourage the network to learn the existence, rather than location of objects, <br />
<br />
<center><math> h_2(\mathbf{x}, k) = -\text{std}_{i \in [1 \dots M]}(Pr[f_k(\mathbf{x}_i) = 1]), </math></center><br />
<br />
which seeks high variance outputs, and<br />
<br />
<center><br />
<math> h_3(\mathbf{x}, v) = \frac{1}{M}\sum_i^{M} (Pr[f(\mathbf{x}_i) = v] - \frac{1}{3} + (\frac{1}{3} - \mu_v))^2 \\<br />
\mu_{v} = \frac{1}{M}\sum_i^{M} \mathbb{1}\{v = \text{argmax}_{v' \in \{0, 1\}^2} Pr[f(\mathbf{x}) = v']\}. </math><br />
</center><br />
<br />
which seeks high entropy outputs. The final loss function then becomes: <br />
<br />
<center><br />
<math> \begin{split}<br />
g(\mathbf{x}) & = \mathbb{1}\{f_1(\mathbf{x}) \nRightarrow f_2(\mathbf{x})\} \hphantom{\text{ }} + \\<br />
& \sum_{k \in \{1, 2\}} \gamma_1 h_1(\mathbf{x}, k) + \gamma_2 h_2(\mathbf{x}, k) + <br />
\hspace{-0.7em} \sum_{v \neq \{1,0\}} \hspace{-0.7em} \gamma_3 * h_3(\mathbf{x}, v)<br />
\end{split}<br />
</math><br />
</center><br />
<br />
====Evaluation====<br />
<br />
The input images, shown in Fig. (4), are 56 × 56 pixels. The authors used <math>\gamma_1 = 0.65, \gamma_2 = 0.65, \gamma_3 = 0.95</math>, and trained for 4,000 iterations. This experiment demonstrates that networks can learn from constraints that operate over discrete sets with potentially complex logical rules. Removing constraints will cause learning to fail. Thus, the experiment also shows that sophisticated sufficiency conditions can be key to success when learning from constraints.<br />
<br />
== Conclusion and Critique ==<br />
This paper has introduced a method for using physics and other domain constraints to supervise neural networks. However, the approach described in this paper is not entirely new. Similar ideas are already widely used in Q learning, where the Q value are not available, and the network is supervised by the constraint, as in Deep Q learning (Mnih, Riedmiller et al. 2013[2]).<br />
<center><math>Q(s,a) = R(r,s) + \gamma \sum_{s' ~ P_{sa}}{\text{max}_{a'}Q(s',a')}</math></center><br />
<br />
<br />
Also, the paper has a mistake where they quote the free fall equation as<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + a(i\Delta t)^2</math></center><br />
which should be<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
Although in this case it doesn't affect the result.<br />
<br />
<br />
For the evaluation of the experiments, they used correlation with ground truth as the metric to avoid the fact that the output can be scaled without affecting the constraint loss. This is fine if the network gives output of the same scale. However, there's no such guarantee, and the network may give output of varying scale for different inputs, in which case, we can't say that the network has learnt the correct thing, although it may have a high correlation with ground truth. In fact, to solve the scaling issue, an obvious way is to combine the constraints introduced in this paper with some labeled training data. It's not clear why the author didn't experiment with a combination of these two losses.<br />
<br />
In regards to the free fall experiment in particular, the authors apply a fixed acceleration model to create the constraint loss, with the goal of having the network predict height. However, since they did not measure the true height of the object to create test labels, they evaluate using height in pixel space. They do not mention the accuracy of their camera calibration, nor what camera model was used to remove lens distortion. Since lens distortion tends to be worse at the extreme edges of the image, and that they tossed the pillow throughout the entire frame, it is likely that the ground truth labels were corrupted by distortion. If that is the case, it is possible the supervised network is actually performing worse, because it learning how to predict distorted (beyond a constant scaling factor) heights instead of the true height.<br />
<br />
These methods essentially boil down to generating approximate labels for training data using some knowledge of the dynamic that the labels should follow.<br />
<br />
Finally, this paper only picks examples where the constraints are easy to design, while in some more common tasks such as image classification, what kind of constraints are needed is not straightforward at all.<br />
<br />
== References ==<br />
[1] LeCun, Y.; Bengio, Y.; and Hinton, G. 2015. Deep learning. Nature 521(7553):436–444.<br />
<br />
[2] Mnih, V.; Kavukcuoglu, K.; Silver, D.; Graves, A.; Antonoglou, I.; Wierstra, D.; and Riedmiller, M. 2013. Playing Atari with Deep Reinforcement Learning. arxiv 1312.5602.</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35406Word translation without parallel data2018-03-24T16:25:23Z<p>W285liu: /* Orthogonality */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35405Word translation without parallel data2018-03-24T16:25:03Z<p>W285liu: /* Orthogonality */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35404Word translation without parallel data2018-03-24T16:23:59Z<p>W285liu: /* Orthogonality */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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)<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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35403Word translation without parallel data2018-03-24T16:16:23Z<p>W285liu: /* Orthogonality */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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) = ||W^TW-I||_F = \nabla\text{Tr}(W^TW-I)(W^TW-I)=2\text{Tr}(\nabla(W^TW-I))(W^TW-I)<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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35402Word translation without parallel data2018-03-24T16:03:36Z<p>W285liu: /* Estimation of Word Representations in Vector Space */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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 />
<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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35401Word translation without parallel data2018-03-24T16:03:08Z<p>W285liu: /* Estimation of Word Representations in Vector Space */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{Cauchy-Swarz inequality}\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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 />
<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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=35400Word translation without parallel data2018-03-24T15:53:08Z<p>W285liu: /* Estimation of Word Representations in Vector Space */</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 />
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. 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. 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). 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 />
= 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 <math> \{x_i,y_i\}_{i\in{1,n}} </math>. 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, <math> M_d(R) </math> 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 <math> YX^T </math> :<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 />
<br />
This can be proven as follows. First note that <br />
\begin{align}<br />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\\<br />
&= \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 learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve 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 <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> 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 <math> WX={Wx_1,...,Wx_n} </math> 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 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 <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> 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 <math> L_D </math> and <math> L_W </math><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. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> 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 <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. 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, it 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 />
<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 to generate 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 increase even more 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. Figure 2 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 />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also 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 was 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 />
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 />
= 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</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32306stat946w18/Spectral normalization for generative adversial network2018-02-25T22:24:31Z<p>W285liu: /* Optimization */</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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_{dis} </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> 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 <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><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) ==</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32198stat946w18/Spectral normalization for generative adversial network2018-02-24T19:24:40Z<p>W285liu: /* Conclusions */</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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_dis </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> 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 <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><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) ==</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32197stat946w18/Spectral normalization for generative adversial network2018-02-24T19:20:01Z<p>W285liu: </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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_dis </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> 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 <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><br />
<br />
== Conclusions ==<br />
<br />
== Critique(to be edited) ==</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32196stat946w18/Spectral normalization for generative adversial network2018-02-24T19:19:10Z<p>W285liu: /* Conclusions */</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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_dis </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> 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 <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><br />
<br />
= Conclusions =</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32195stat946w18/Spectral normalization for generative adversial network2018-02-24T19:18:49Z<p>W285liu: </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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_dis </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> 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 <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><br />
<br />
= Conclusions =<br />
[[Media:https://openreview.net/pdf?id=B1QRgziT-]]<br />
<br />
== Critique (to be edited) ==</div>W285liuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32194stat946w18/Spectral normalization for generative adversial network2018-02-24T19:15:04Z<p>W285liu: /* Source */</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 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 & 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. In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x: <math> f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots)))) </math> where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function. The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. The conventional form of <math>V(G,D) </math> is given by <math>E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')</math> where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> 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 <math>V(G,D) </math> is given by <math> D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x))</math>.<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <math> \arg\max_{||f||_{Lip}\le k}V(G,D)</math>, where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
<math> ||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} </math><br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
<math> \bar{W_{SN}}:= W/\sigma(W) </math><br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
<math> \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} </math><br />
<br />
<math> \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}})</math><br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma(2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\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 </math> which means that <math> \bar{W_{WN}} </math> 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 <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte 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 />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are 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 <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> 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 <math> ||\nabla_{\hat{x}} f ||_2 </math> 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 />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> 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)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> 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)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_dis </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> 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 CIFRA-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 Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<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 ILRSVRC2012 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 th iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> with a random vector (sampled from isotropic distribution) <br />
* For each update and each layer l:<br />
*