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diff --git a/parse/train/BydrOIcle/BydrOIcle.md b/parse/train/BydrOIcle/BydrOIcle.md
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+# UNROLLED GENERATIVE ADVERSARIAL NETWORKS
+
+Luke Metz∗   
+Google Brain   
+lmetz@google.com   
+Ben Poole†   
+Stanford University   
+poole@cs.stanford.edu
+
+David Pfau Google DeepMind pfau@google.com
+
+Jascha Sohl-Dickstein Google Brain jaschasd@google.com
+
+# ABSTRACT
+
+We introduce a method to stabilize Generative Adversarial Networks (GANs) by defining the generator objective with respect to an unrolled optimization of the discriminator. This allows training to be adjusted between using the optimal discriminator in the generator’s objective, which is ideal but infeasible in practice, and using the current value of the discriminator, which is often unstable and leads to poor solutions. We show how this technique solves the common problem of mode collapse, stabilizes training of GANs with complex recurrent generators, and increases diversity and coverage of the data distribution by the generator.
+
+# 1 INTRODUCTION
+
+The use of deep neural networks as generative models for complex data has made great advances in recent years. This success has been achieved through a surprising diversity of training losses and model architectures, including denoising autoencoders (Vincent et al., 2010), variational autoencoders (Kingma & Welling, 2013; Rezende et al., 2014; Gregor et al., 2015; Kulkarni et al., 2015; Burda et al., 2015; Kingma et al., 2016), generative stochastic networks (Alain et al., 2015), diffusion probabilistic models (Sohl-Dickstein et al., 2015), autoregressive models (Theis & Bethge, 2015; van den Oord et al., 2016a;b), real non-volume preserving transformations (Dinh et al., 2014; 2016), Helmholtz machines (Dayan et al., 1995; Bornschein et al., 2015), and Generative Adversarial Networks (GANs) (Goodfellow et al., 2014).
+
+# 1.1 GENERATIVE ADVERSARIAL NETWORKS
+
+While most deep generative models are trained by maximizing log likelihood or a lower bound on log likelihood, GANs take a radically different approach that does not require inference or explicit calculation of the data likelihood. Instead, two models are used to solve a minimax game: a generator which samples data, and a discriminator which classifies the data as real or generated. In theory these models are capable of modeling an arbitrarily complex probability distribution. When using the optimal discriminator for a given class of generators, the original GAN proposed by Goodfellow et al. minimizes the Jensen-Shannon divergence between the data distribution and the generator, and extensions generalize this to a wider class of divergences (Nowozin et al., 2016; Sonderby et al., 2016; Poole et al., 2016).
+
+The ability to train extremely flexible generating functions, without explicitly computing likelihoods or performing inference, and while targeting more mode-seeking divergences as made GANs extremely successful in image generation (Odena et al., 2016; Salimans et al., 2016; Radford et al., 2015), and image super resolution (Ledig et al., 2016). The flexibility of the GAN framework has also enabled a number of successful extensions of the technique, for instance for structured prediction (Reed et al., 2016a;b; Odena et al., 2016), training energy based models (Zhao et al., 2016), and combining the GAN loss with a mutual information loss (Chen et al., 2016).
+
+In practice, however, GANs suffer from many issues, particularly during training. One common failure mode involves the generator collapsing to produce only a single sample or a small family of very similar samples. Another involves the generator and discriminator oscillating during training, rather than converging to a fixed point. In addition, if one agent becomes much more powerful than the other, the learning signal to the other agent becomes useless, and the system does not learn. To train GANs many tricks must be employed, such as careful selection of architectures (Radford et al., 2015), minibatch discrimination (Salimans et al., 2016), and noise injection (Salimans et al., 2016; Sonderby et al., 2016). Even with these tricks the set of hyperparameters for which training is successful is generally very small in practice.
+
+Once converged, the generative models produced by the GAN training procedure normally do not cover the whole distribution (Dumoulin et al., 2016; Che et al., 2016), even when targeting a modecovering divergence such as KL. Additionally, because it is intractable to compute the GAN training loss, and because approximate measures of performance such as Parzen window estimates suffer from major flaws (Theis et al., 2016), evaluation of GAN performance is challenging. Currently, human judgement of sample quality is one of the leading metrics for evaluating GANs. In practice this metric does not take into account mode dropping if the number of modes is greater than the number of samples one is visualizing. In fact, the mode dropping problem generally helps visual sample quality as the model can choose to focus on only the most common modes. These common modes correspond, by definition, to more typical samples. Additionally, the generative model is able to allocate more expressive power to the modes it does cover than it would if it attempted to cover all modes.
+
+# 1.2 DIFFERENTIATING THROUGH OPTIMIZATION
+
+Many optimization schemes, including SGD, RMSProp (Tieleman & Hinton, 2012), and Adam (Kingma & Ba, 2014), consist of a sequence of differentiable updates to parameters. Gradients can be backpropagated through unrolled optimization updates in a similar fashion to backpropagation through a recurrent neural network. The parameters output by the optimizer can thus be included, in a differentiable way, in another objective (Maclaurin et al., 2015). This idea was first suggested for minimax problems in (Pearlmutter & Siskind, 2008), while (Zhang & Lesser, 2010) provided a theoretical analysis and experimental results on differentiating through a single step of gradient ascent for simple matrix games. Differentiating through unrolled optimization was first scaled to deep networks in (Maclaurin et al., 2015), where it was used for hyperparameter optimization. More recently, (Belanger & McCallum, 2015; Han et al., 2016; Andrychowicz et al., 2016) backpropagate through optimization procedures in contexts unrelated to GANs or minimax games.
+
+In this work we address the challenges of unstable optimization and mode collapse in GANs by unrolling optimization of the discriminator objective during training.
+
+# 2 METHOD
+
+# 2.1 GENERATIVE ADVERSARIAL NETWORKS
+
+The GAN learning problem is to find the optimal parameters $\theta _ { G } ^ { * }$ for a generator function $G \left( z ; \theta _ { G } \right)$ in a minimax objective,
+
+$$
+\begin{array} { c } { { \theta _ { G } ^ { * } = \underset { \theta _ { G } } { \mathrm { a r g m i n } } \underset { \theta _ { D } } { \mathrm { m a x } } f \left( \theta _ { G } , \theta _ { D } \right) } } \\ { { \ \mathrm { ~ } } } \\ { { \displaystyle \qquad = \underset { \theta _ { G } } { \mathrm { a r g m i n } } f \left( \theta _ { G } , \theta _ { D } ^ { * } \left( \theta _ { G } \right) \right) } } \\ { { \theta _ { D } ^ { * } \left( \theta _ { G } \right) = \underset { \theta _ { D } } { \mathrm { a r g m a x } } f \left( \theta _ { G } , \theta _ { D } \right) , } } \end{array}
+$$
+
+where $f$ is commonly chosen to be
+
+$$
+f \left( \theta _ { G } , \theta _ { D } \right) = \mathbb { E } _ { x \sim p _ { d a t a } } \left[ \log \left( D \left( x ; \theta _ { D } \right) \right) \right] + \mathbb { E } _ { z \sim N ( 0 , I ) } \left[ \log \left( 1 - D \left( G \left( z ; \theta _ { G } \right) ; \theta _ { D } \right) \right) \right] .
+$$
+
+Here $x \in \mathcal { X }$ is the data variable, $z \in { \mathcal { Z } }$ is the latent variable, $p _ { d a t a }$ is the data distribution, the discriminator $D ( \cdot ; \theta _ { D } ) : \mathcal { X }  [ 0 , 1 ]$ outputs the estimated probability that a sample $x$ comes from the data distribution, $\theta _ { D }$ and $\theta _ { G }$ are the discriminator and generator parameters, and the generator function $G ( \cdot ; \theta _ { G } ) : \mathcal { Z }  \mathcal { X }$ transforms a sample in the latent space into a sample in the data space.
+
+For the minimax loss in Eq. 4, the optimal discriminator $D ^ { \ast } \left( x \right)$ is a known smooth function of the generator probability $p _ { G } \left( x \right)$ (Goodfellow et al., 2014),
+
+$$
+D ^ { * } \left( x \right) = \frac { p _ { d a t a } \left( x \right) } { p _ { d a t a } \left( x \right) + p _ { G } \left( x \right) } .
+$$
+
+When the generator loss in Eq. 2 is rewritten directly in terms of $p _ { G } \left( x \right)$ and Eq. 5 rather than $\theta _ { G }$ and $\theta _ { D } ^ { * } \left( \theta _ { G } ^ { - } \right)$ , then it is similarly a smooth function of $p _ { G } \left( x \right)$ . These smoothness guarantees are typically lost when $D \left( x ; \theta _ { D } \right)$ and $G \left( z ; \theta _ { G } \right)$ are drawn from parametric families. They nonetheless suggest that the true generator objective in Eq. 2 will often be well behaved, and is a desirable target for direct optimization.
+
+Explicitly solving for the optimal discriminator parameters $\theta _ { D } ^ { * } \left( \theta _ { G } \right) $ for every update step of the generator $G$ is computationally infeasible for discriminators based on neural networks. Therefore this minimax optimization problem is typically solved by alternating gradient descent on $\theta _ { G }$ and ascent on $\theta _ { D }$ .
+
+The optimal solution $\theta ^ { * } = \{ \theta _ { G } ^ { * } , \theta _ { D } ^ { * } \}$ is a fixed point of these iterative learning dynamics. Additionally, if $f \left( { \theta } _ { G } , { \theta } _ { D } \right)$ is convex in $\theta _ { G }$ and concave in $\theta _ { D }$ , then alternating gradient descent (ascent) trust region updates are guaranteed to converge to the fixed point, under certain additional weak assumptions (Juditsky et al., 2011). However in practice $f \left( { \theta } _ { G } , { \theta } _ { D } \right)$ is typically very far from convex in $\theta _ { G }$ and concave in $\theta _ { D }$ , and updates are not constrained in an appropriate way. As a result GAN training suffers from mode collapse, undamped oscillations, and other problems detailed in Section 1.1. In order to address these difficulties, we will introduce a surrogate objective function $f _ { K } \left( \theta _ { G } , \theta _ { D } \right)$ for training the generator which more closely resembles the true generator objective $f \left( \theta _ { G } , \theta _ { D } ^ { * } \left( \theta _ { G } \right) \right)$ .
+
+# 2.2 UNROLLING GANS
+
+A local optimum of the discriminator parameters $\theta _ { D } ^ { * }$ can be expressed as the fixed point of an iterative optimization procedure,
+
+$$
+\begin{array} { c } { { \theta _ { D } ^ { 0 } = \theta _ { D } } } \\ { { \displaystyle } } \\ { { \theta _ { D } ^ { k + 1 } = \theta _ { D } ^ { k } + \eta ^ { k } \displaystyle \frac { \mathrm { d } f ( \theta _ { G } , \theta _ { D } ^ { k } ) } { \mathrm { d } \theta _ { D } ^ { k } } } } \\ { { \displaystyle } } \\ { { \theta _ { D } ^ { * } ( \theta _ { G } ) = \displaystyle \operatorname* { l i m } _ { k  \infty } \theta _ { D } ^ { k } , } } \end{array}
+$$
+
+where $\eta ^ { k }$ is the learning rate schedule. For clarity, we have expressed Eq. 7 as a full batch steepest gradient ascent equation. More sophisticated optimizers can be similarly unrolled. In our experiments we unroll Adam (Kingma & Ba, 2014).
+
+By unrolling for $K$ steps, we create a surrogate objective for the update of the generator,
+
+$$
+f _ { K } \left( \theta _ { G } , \theta _ { D } \right) = f \left( \theta _ { G } , \theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) \right) .
+$$
+
+When $K = 0$ this objective corresponds exactly to the standard GAN objective, while as $K  \infty$ it corresponds to the true generator objective function $f \left( \theta _ { G } , \theta _ { D } ^ { * } \left( G \right) \right)$ . By adjusting the number of unrolling steps $K$ , we are thus able to interpolate between standard GAN training dynamics with their associated pathologies, and more costly gradient descent on the true generator loss.
+
+# 2.3 PARAMETER UPDATES
+
+The generator and discriminator parameter updates using this surrogate loss are
+
+$$
+\begin{array} { r l } & { \theta _ { G }  \theta _ { G } - \eta \frac { \mathrm { d } f _ { K } ( \theta _ { G } , \theta _ { D } ) } { \mathrm { d } \theta _ { G } } } \\ & { \theta _ { D }  \theta _ { D } + \eta \frac { \mathrm { d } f ( \theta _ { G } , \theta _ { D } ) } { \mathrm { d } \theta _ { D } } . } \end{array}
+$$
+
+For clarity we use full batch steepest gradient descent (ascent) with stepsize $\eta$ above, while in experiments we instead use minibatch Adam for both updates. The gradient in Eq. 10 requires backpropagating through the optimization process in Eq. 7. A clear description of differentiation through gradient descent is given as Algorithm 2 in (Maclaurin et al., 2015), though in practice the use of an automatic differentiation package means this step does not need to be programmed explicitly. A pictorial representation of these updates is provided in Figure 1.
+
+![](images/b8d41d44f95393092b08110ba25482b02bb1bcb0a2930f383731e4e370791ca3.jpg)  
+Figure 1: An illustration of the computation graph for an unrolled GAN with 3 unrolling steps. The generator update in Equation 10 involves backpropagating the generator gradient (blue arrows) through the unrolled optimization. Each step $k$ in the unrolled optimization uses the gradients of $f _ { k }$ with respect to $\theta _ { D } ^ { k }$ , as described in Equation 7 and indicated by the green arrows. The discriminator update in Equation 11 does not depend on the unrolled optimization (red arrow).
+
+It is important to distinguish this from an approach suggested in (Goodfellow et al., 2014), that several update steps of the discriminator parameters should be run before each single update step for the generator. In that approach, the update steps for both models are still gradient descent (ascent) with respect to fixed values of the other model parameters, rather than the surrogate loss we describe in Eq. 9. Performing $K$ steps of discriminator update between each single step of generator update corresponds to updating the generator parameters $\theta _ { G }$ using only the first term in Eq. 12 below.
+
+# 2.4 THE MISSING GRADIENT TERM
+
+To better understand the behavior of the surrogate loss $f _ { K } \left( \theta _ { G } , \theta _ { D } \right)$ , we examine its gradient with respect to the generator parameters $\theta _ { G }$ ,
+
+$$
+\frac { \mathrm { d } f _ { K } \left( \theta _ { G } , \theta _ { D } \right) } { \mathrm { d } \theta _ { G } } = \frac { \partial f \left( \theta _ { G } , \theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) \right) } { \partial \theta _ { G } } + \frac { \partial f \left( \theta _ { G } , \theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) \right) } { \partial \theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) } \frac { \mathrm { d } \theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) } { \mathrm { d } \theta _ { G } } .
+$$
+
+Standard GAN training corresponds exactly to updating the generator parameters using only the first term in this gradient, with $\theta _ { D } ^ { K } \left( \theta _ { G } , \theta _ { D } \right) $ being the parameters resulting from the discriminator update step. An optimal generator for any fixed discriminator is a delta function at the $x$ to which the discriminator assigns highest data probability. Therefore, in standard GAN training, each generator update step is a partial collapse towards a delta function.
+
+The second term captures how the discriminator would react to a change in the generator. It reduces the tendency of the generator to engage in mode collapse. For instance, the second term reflects that as the generator collapses towards a delta function, the discriminator reacts and assigns lower probability to that state, increasing the generator loss. It therefore discourages the generator from collapsing, and may improve stability.
+
+As $K  \infty$ , $\theta _ { D } ^ { K }$ goes to a local optimum of $f$ , where $\frac { \partial f } { \partial \theta _ { D } ^ { K } } = 0$ , and therefore the second term in Eq. 12 goes to 0 (Danskin, 1967). The gradient of the unrolled surrogate loss $f _ { K } \left( \theta _ { G } , \theta _ { D } \right)$ with respect to $\theta _ { G }$ is thus identical to the gradient of the standard GAN loss $f \bar { ( \theta _ { G } , \theta _ { D } ) }$ both when $K = 0$ and when $K  \infty$ , where we take $K  \infty$ to imply that in the standard GAN the discriminator is also fully optimized between each generator update. Between these two extremes, $f _ { K } \left( \theta _ { G } , \theta _ { D } \right)$ captures additional information about the response of the discriminator to changes in the generator.
+
+# 2.5 CONSEQUENCES OF THE SURROGATE LOSS
+
+GANs can be thought of as a game between the discriminator $( D )$ and the generator $( G )$ . The agents take turns taking actions and updating their parameters until a Nash equilibrium is reached. The optimal action for $D$ is to evaluate the probability ratio $\frac { p _ { d a t a } ( x ) } { p _ { G } ( x ) + p _ { d a t a } ( x ) }$ for the generator’s move $x$ (Eq. 5). The optimal generator action is to move its mass to maximize this ratio.
+
+The initial move for $G$ will be to move as much mass as its parametric family and update step permits to the single point that maximizes the ratio of probability densities. The action $D$ will then take is quite simple. It will track that point, and to the extent allowed by its own parametric family and update step assign low data probability to it, and uniform probability everywhere else. This cycle of $G$ moving and $D$ following will repeat forever or converge depending on the rate of change of the two agents. This is similar to the situation in simple matrix games like rock-paper-scissors and matching pennies, where alternating gradient descent (ascent) with a fixed learning rate is known not to converge (Singh et al., 2000; Bowling & Veloso, 2002).
+
+In the unrolled case, however, this undesirable behavior no longer occurs. Now $G$ ’s actions take into account how $D$ will respond. In particular, $G$ will try to make steps that $D$ will have a hard time responding to. This extra information helps the generator spread its mass to make the next $D$ step less effective instead of collapsing to a point.
+
+In principle, a surrogate loss function could be used for both $D$ and $G$ . In the case of 1-step unrolled optimization this is known to lead to convergence for games in which gradient descent (ascent) fails (Zhang & Lesser, 2010). However, the motivation for using the surrogate generator loss in Section 2.2, of unrolling the inner of two nested min and max functions, does not apply to using a surrogate discriminator loss. Additionally, it is more common for the discriminator to overpower the generator than vice-versa when training a GAN. Giving more information to $G$ by allowing it to ‘see into the future’ may thus help the two models be more balanced.
+
+# 3 EXPERIMENTS
+
+In this section we demonstrate improved mode coverage and stability by applying this technique to five datasets of increasing complexity. Evaluation of generative models is a notoriously hard problem (Theis et al., 2016). As such the de facto standard in GAN literature has become sample quality as evaluated by a human and/or evaluated by a heuristic (Inception score for example, (Salimans et al., 2016)). While these evaluation metrics do a reasonable job capturing sample quality, they fail to capture sample diversity. In our first 2 experiments diversity is easily evaluated via visual inspection. In our later experiments this is not the case, and we will use a variety of methods to quantify coverage of samples. Our measures are individually strongly suggestive of unrolling reducing mode-collapse and improving stability, but none of them alone are conclusive. We believe that taken together however, they provide extremely compelling evidence for the advantages of unrolling.
+
+When doing stochastic optimization, we must choose which minibatches to use in the unrolling updates in Eq. 7. We experimented with both a fixed minibatch and re-sampled minibatches for each unrolling step, and found it did not significantly impact the result. We use fixed minibatches for all experiments in this section.
+
+We provide a reference implementation of this technique at github.com/poolio/unrolled gan.
+
+# 3.1 MIXTURE OF GAUSSIANS DATASET
+
+To illustrate the impact of discriminator unrolling, we train a simple GAN architecture on a 2D mixture of 8 Gaussians arranged in a circle. For a detailed list of architecture and hyperparameters see Appendix A. Figure 2 shows the dynamics of this model through time. Without unrolling the generator rotates around the valid modes of the data distribution but is never able to spread out mass. When adding in unrolling steps $\mathbf { G }$ quickly learns to spread probability mass and the system converges to the data distribution.
+
+In Appendix B we perform further experiments on this toy dataset. We explore how unrolling compares to historical averaging, and compares to using the unrolled discriminator to update the generator, but without backpropagating through the generator. In both cases we find that the unrolled objective performs better.
+
+![](images/a4459b06fa7e9fcdf667d911bf408b5c5ccfe73af1d14e29428d98326008c54a.jpg)  
+Figure 2: Unrolling the discriminator stabilizes GAN training on a toy 2D mixture of Gaussians dataset. Columns show a heatmap of the generator distribution after increasing numbers of training steps. The final column shows the data distribution. The top row shows training for a GAN with 10 unrolling steps. Its generator quickly spreads out and converges to the target distribution. The bottom row shows standard GAN training. The generator rotates through the modes of the data distribution. It never converges to a fixed distribution, and only ever assigns significant probability mass to a single data mode at once.
+
+![](images/ff67f1c079a9b83431ee462581bfb81893ea4eadc9da332a9fff1968becf8c08.jpg)  
+Figure 3: Unrolled GAN training increases stability for an RNN generator and convolutional discriminator trained on MNIST. The top row was run with 20 unrolling steps. The bottom row is a standard GAN, with 0 unrolling steps. Images are samples from the generator after the indicated number of training steps.
+
+# 3.2 PATHOLOGICAL MODEL WITH MISMATCHED GENERATOR AND DISCRIMINATOR
+
+To evaluate the ability of this approach to improve trainability, we look to a traditionally challenging family of models to train – recurrent neural networks (RNNs). In this experiment we try to generate MNIST samples using an LSTM (Hochreiter & Schmidhuber, 1997). MNIST digits are $2 8 \mathbf { x } 2 8$ pixel images. At each timestep of the generator LSTM, it outputs one column of this image, so that after 28 timesteps it has output the entire sample. We use a convolutional neural network as the discriminator. See Appendix C for the full model and training details. Unlike in all previously successful GAN models, there is no symmetry between the generator and the discriminator in this task, resulting in a more complex power balance. Results can be seen in Figure 3. Once again, without unrolling the model quickly collapses, and rotates through a sequence of single modes. Instead of rotating spatially, it cycles through proto-digit like blobs. When running with unrolling steps the generator disperses and appears to cover the whole data distribution, as in the 2D example.
+
+<table><tr><td rowspan=1 colspan=1>Discriminator Size</td><td rowspan=1 colspan=1>Unrolling steps</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>1/4 size of D compared to G</td><td rowspan=1 colspan=1>Modes generated</td><td rowspan=1 colspan=1>30.6± 20.73</td><td rowspan=1 colspan=1>65.4 ± 34.75</td><td rowspan=1 colspan=1>236.4 ± 63.30</td><td rowspan=1 colspan=1>327.2 ± 74.67</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>KL(model||data)</td><td rowspan=1 colspan=1>5.99± 0.42</td><td rowspan=1 colspan=1>5.911 ± 0.14</td><td rowspan=1 colspan=1>4.67 ± 0.43</td><td rowspan=1 colspan=1>4.66 ± 0.46</td></tr><tr><td rowspan=1 colspan=1>1/2 size of D compared to G</td><td rowspan=1 colspan=1>Modes generated</td><td rowspan=1 colspan=1>628.0± 140.9</td><td rowspan=1 colspan=1>523.6± 55.768</td><td rowspan=1 colspan=1>732.0± 44.98</td><td rowspan=1 colspan=1>817.4 ± 37.91</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>KL(model||data)</td><td rowspan=1 colspan=1>2.58 ±0.751</td><td rowspan=1 colspan=1>2.44 ±0.26</td><td rowspan=1 colspan=1>1.66 ± 0.090</td><td rowspan=1 colspan=1>1.43 ± 0.12</td></tr></table>
+
+Table 1: Unrolled GANs cover more discrete modes when modeling a dataset with 1,000 data modes, corresponding to all combinations of three MNIST digits $[ 1 0 ^ { 3 }$ digit combinations). The number of modes covered is given for different numbers of unrolling steps, and for two different architectures. The reverse KL divergence between model and data is also given. Standard error is provided for both measures.
+
+# 3.3 MODE AND MANIFOLD COLLAPSE USING AUGMENTED MNIST
+
+GANs suffer from two different types of model collapse – collapse to a subset of data modes, and collapse to a sub-manifold within the data distribution. In these experiments we isolate both effects using artificially constructed datasets, and demonstrate that unrolling can largely rescue both types of collapse.
+
+# 3.3.1 DISCRETE MODE COLLAPSE
+
+To explore the degree to which GANs drop discrete modes in a dataset, we use a technique similar to one from (Che et al., 2016). We construct a dataset by stacking three randomly chosen MNIST digits, so as to construct an RGB image with a different MNIST digit in each color channel. This new dataset has 1,000 distinct modes, corresponding to each combination of the ten MNIST classes in the three channels.
+
+We train a GAN on this dataset, and generate samples from the trained model (25,600 samples for all experiments). We then compute the predicted class label of each color channel using a pre-trained MNIST classifier. To evaluate performance, we use two metrics: the number of modes for which the generator produced at least one sample, and the KL divergence between the model and the expected data distribution. Within this discrete label space, a KL divergence can be estimated tractably between the generated samples and the data distribution over classes, where the data distribution is a uniform distribution over all 1,000 classes.
+
+As presented in Table 1, as the number of unrolling steps is increased, both mode coverage and reverse KL divergence improve. Contrary to (Che et al., 2016), we found that reasonably sized models (such as the one used in Section 3.4) covered all 1,000 modes even without unrolling. As such we use smaller convolutional GAN models. Details on the models used are provided in Appendix E.
+
+We observe an additional interesting effect in this experiment. The benefits of unrolling increase as the discriminator size is reduced. We believe unrolling effectively increases the capacity of the discriminator. The unrolled discriminator can better react to any specific way in which the generator is producing non-data-like samples. When the discriminator is weak, the positive impact of unrolling is thus larger.
+
+# 3.3.2 MANIFOLD COLLAPSE
+
+In addition to discrete modes, we examine the effect of unrolling when modeling continuous manifolds. To get at this quantity, we constructed a dataset consisting of colored MNIST digits. Unlike in the previous experiment, a single MNIST digit was chosen, and then assigned a single monochromatic color. With a perfect generator, one should be able to recover the distribution of colors used to generate the digits. We use colored MNIST digits so that the generator also has to model the digits, which makes the task sufficiently complex that the generator is unable to perfectly solve it. The color of each digit is sampled from a 3D normal distribution. Details of this dataset are provided in Appendix F. We will examine the distribution of colors in the samples generated by the trained GAN. As will also be true in the CIFAR10 example in Section 3.4, the lack of diversity in generated colors is almost invisible using only visual inspection of the samples. Samples can be found in Appendix F.
+
+Table 2: Unrolled GANs better model a continuous distribution. GANs are trained to model randomly colored MNIST digits, where the color is drawn from a Gaussian distribution. The JS divergence between the data and model distributions over digit colors is then reported, along with standard error in the JS divergence. More unrolling steps, and larger models, lead to better JS divergence.   
+
+<table><tr><td rowspan=1 colspan=1>Unrolling steps</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/4 layer size</td><td rowspan=1 colspan=1>0.073 ± 0.0058</td><td rowspan=1 colspan=1>0.142 ± 0.028</td><td rowspan=1 colspan=1>0.049 ± 0.0021</td><td rowspan=1 colspan=1>0.075 ± 0.012</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/2 layer size</td><td rowspan=1 colspan=1>0.095 ± 0.011</td><td rowspan=1 colspan=1>0.119 ± 0.010</td><td rowspan=1 colspan=1>0.055 ± 0.0049</td><td rowspan=1 colspan=1>0.074± 0.016</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/1 layer size</td><td rowspan=1 colspan=1>0.034 ± 0.0034</td><td rowspan=1 colspan=1>0.050± 0.0026</td><td rowspan=1 colspan=1>0.027 ± 0.0028</td><td rowspan=1 colspan=1>0.025 ± 0.00076</td></tr></table>
+
+![](images/3c9a4f15816f9475e20e64fe6d880999dbc71b9039d3a86102ace92afbbde1f2.jpg)  
+Figure 4: Visual perception of sample quality and diversity is very similar for models trained with different numbers of unrolling steps. Actual sample diversity is higher with more unrolling steps. Each pane shows samples generated after training a model on CIFAR10 with 0, 1, 5, and 10 steps of unrolling.
+
+In order to recover the color the GAN assigned to the digit, we used k-means with 2 clusters, to pick out the foreground color from the background. We then performed this transformation for both the training data and the generated images. Next we fit a Gaussian kernel density estimator to both distributions over digit colors. Finally, we computed the JS divergence between the model and data distributions over colors. Results can be found in Table 2 for several model sizes. Details of the models are provided in Appendix F.
+
+In general, the best performing models are unrolled for 5-10 steps, and larger models perform better than smaller models. Counter-intuitively, taking 1 unrolling step seems to hurt this measure of diversity. We suspect that this is due to it introducing oscillatory dynamics into training. Taking more unrolling steps however leads to improved performance with unrolling.
+
+# 3.4 IMAGE MODELING OF CIFAR10
+
+Here we test our technique on a more traditional convolutional GAN architecture and task, similar to those used in (Radford et al., 2015; Salimans et al., 2016). In the previous experiments we tested models where the standard GAN training algorithm would not converge. In this section we improve a standard model by reducing its tendency to engage in mode collapse. We ran 4 configurations of this model, varying the number of unrolling steps to be 0, 1, 5, or 10. Each configuration was run 5 times with different random seeds. For full training details see Appendix D. Samples from each of the 4 configurations can be found in Figure 4. There is no obvious difference in visual quality across these model configurations. Visual inspection however provides only a poor measure of sample diversity.
+
+By training with an unrolled discriminator, we expect to generate more diverse samples which more closely resemble the underlying data distribution. We introduce two techniques to examine sample diversity: inference via optimization, and pairwise distance distributions.
+
+<table><tr><td rowspan=1 colspan=1>Unrolling Steps</td><td rowspan=1 colspan=1>0 steps</td><td rowspan=1 colspan=1>1 step</td><td rowspan=1 colspan=1>5 steps</td><td rowspan=1 colspan=1>10 steps</td></tr><tr><td rowspan=1 colspan=1>Average MSE</td><td rowspan=1 colspan=1>0.0231± 0.0024</td><td rowspan=1 colspan=1>0.0195 ± 0.0021</td><td rowspan=1 colspan=1>0.0200± 0.0023</td><td rowspan=1 colspan=1>0.0181± 0.0018</td></tr><tr><td rowspan=1 colspan=1>PercentBestRank</td><td rowspan=1 colspan=1>0.63%</td><td rowspan=1 colspan=1>22.97%</td><td rowspan=1 colspan=1>15.31%</td><td rowspan=1 colspan=1>61.09 %</td></tr></table>
+
+Table 3: GANs trained with unrolling are better able to match images in the training set than standard GANs, likely due to mode dropping by the standard GAN. Results show the MSE between training images and the best reconstruction for a model with the given number of unrolling steps. The fraction of training images best reconstructed by a given model is given in the final column. The best reconstructions is found by optimizing the latent representation $z$ to produce the closest matching pixel output $G \left( z ; \theta _ { G } \right)$ . Results are averaged over all 5 runs of each model with different random seeds.
+
+# 3.4.1 INFERENCE VIA OPTIMIZATION
+
+Since likelihood cannot be tractably computed, over-fitting of GANs is typically tested by taking samples and computing the nearest-neighbor images in pixel space from the training data (Goodfellow et al., 2014). We will do the reverse, and measure the ability of the generative model to generate images that look like specific samples from the training data. If we did this by generating random samples from the model, we would need an exponentially large number of samples. We instead treat finding the nearest neighbor $x _ { \mathrm { n e a r e s t } }$ to a target image $x _ { \mathrm { { t a r g e t } } }$ as an optimization task,
+
+$$
+\begin{array} { r l } & { z _ { \mathrm { n e a r e s t } } = \underset { z } { \mathrm { a r g m i n } } | | G ( z ; \theta _ { G } ) - x _ { \mathrm { t a r g e t } } | \rvert _ { 2 } ^ { 2 } } \\ & { x _ { \mathrm { n e a r e s t } } = G ( z _ { \mathrm { n e a r e s t } } ; \theta _ { G } ) . } \end{array}
+$$
+
+This concept of backpropagating to generate images has been widely used in visualizing features from discriminative networks (Simonyan et al., 2013; Yosinski et al., 2015; Nguyen et al., 2016) and has been applied to explore the visual manifold of GANs in (Zhu et al., 2016).
+
+We apply this technique to each of the models trained. We optimize with 3 random starts using LBFGS, which is the optimizer typically used in similar settings such as style transfer (Johnson et al., 2016; Champandard, 2016). Results comparing average mean squared errors between xnearest and $x _ { \mathrm { { t a r g e t } } }$ in pixel space can be found in Table 3. In addition we compute the percent of images for which a certain configuration achieves the lowest loss when compared to the other configurations.
+
+In the zero step case, there is poor reconstruction and less than $1 \%$ of the time does it obtain the lowest error of the 4 configurations. Taking 1 unrolling step results in a significant improvement in MSE. Taking 10 unrolling steps results in more modest improvement, but continues to reduce the reconstruction MSE.
+
+To visually see this, we compare the result of the optimization process for 0, 1, 5, and 10 step configurations in Figure 5. To select for images where differences in behavior is most apparent, we sort the data by the absolute value of a fractional difference in MSE between the 0 and 10 step models, $\left| \frac { l _ { 0 s t e p } - l _ { 1 0 s t e p } } { \frac { 1 } { 2 } ( l _ { 0 s t e p } + l _ { 1 0 s t e p } ) } \right|$ This highlights examples where either the 0 or 10 step model cannot accurately fit the data example but the other can. In Appendix G we show the same comparison for models initialized using different random seeds. Many of the zero step images are fuzzy and illdefined suggesting that these images cannot be generated by the standard GAN generative model, and come from a dropped mode. As more unrolling steps are added, the outlines become more clear and well defined – the model covers more of the distribution and thus can recreate these samples.
+
+# 3.4.2 PAIRWISE DISTANCES
+
+A second complementary approach is to compare statistics of data samples to the corresponding statistics for samples generated by the various models. One particularly simple and relevant statistic is the distribution over pairwise distances between random pairs of samples. In the case of mode collapse, greater probability mass will be concentrated in smaller volumes, and the distribution over inter-sample distances should be skewed towards smaller distances. We sample random pairs of images from each model, as well as from the training data, and compute histograms of the $\ell _ { 2 }$ distances between those sample pairs. As illustrated in Figure 6, the standard GAN, with zero unrolling steps, has its probability mass skewed towards smaller $\ell _ { 2 }$ intersample distances, compared to real data. As the number of unrolling steps is increased, the histograms over intersample distances increasingly come to resemble that for the data distribution. This is further evidence in support of unrolling decreasing the mode collapse behavior of GANs.
+
+![](images/3f34af532391d651dea1658c3e0b24559b5a1f84c026a63aecddf941edff8580.jpg)  
+Figure 5: Training set images are more accurately reconstructed using GANs trained with unrolling than by a standard (0 step) GAN, likely due to mode dropping by the standard GAN. Raw data is on the left, and the optimized images to reach this target follow for 0, 1, 5, and 10 unrolling steps. The reconstruction MSE is listed below each sample. A random 1280 images where selected from the training set, and corresponding best reconstructions for each model were found via optimization. Shown here are the eight images with the largest absolute fractional difference between GANs trained with 0 and 10 unrolling steps.
+
+# 4 DISCUSSION
+
+In this work we developed a method to stabilize GAN training and reduce mode collapse by defining the generator objective with respect to unrolled optimization of the discriminator. We then demonstrated the application of this method to several tasks, where it either rescued unstable training, or reduced the tendency of the model to drop regions of the data distribution.
+
+The main drawback to this method is computational cost of each training step, which increases linearly with the number of unrolling steps. There is a tradeoff between better approximating the true generator loss and the computation required to make this estimate. Depending on the architecture, one unrolling step can be enough. In other more unstable models, such as the RNN case, more are needed to stabilize training. We have some initial positive results suggesting it may be sufficient to further perturb the training gradient in the same direction that a single unrolling step perturbs it. While this is more computationally efficient, further investigation is required.
+
+The method presented here bridges some of the gap between theoretical and practical results for training of GANs. We believe developing better update rules for the generator and discriminator is an important line of work for GAN training. In this work we have only considered a small fraction of the design space. For instance, the approach could be extended to unroll $G$ when updating $D$ as well – letting the discriminator react to how the generator would move. It is also possible to unroll sequences of $G$ and $D$ updates. This would make updates that are recursive: $G$ could react to maximize performance as if $G$ and $D$ had already updated.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Laurent Dinh, David Dohan, Vincent Dumoulin, Liam Fedus, Ishaan Gulrajani, Julian Ibarz, Eric Jang, Matthew Johnson, Marc Lanctot, Augustus Odena, Gabriel Pereyra,
+
+![](images/34368461772c640ab500044ba570ada5248c204871939191a4efbed64c4bf4ee.jpg)  
+Figure 6: As the number of unrolling steps in GAN training is increased, the distribution of pairwise distances between model samples more closely resembles the same distribution for the data. Here we plot histograms of pairwise distances between randomly selected samples. The red line gives pairwise distances in the data, while each of the five blue lines in each plot represents a model trained with a different random seed. The vertical lines are the medians of each distribution.
+
+Colin Raffel, Sam Schoenholz, Ayush Sekhari, Jon Shlens, and Dale Schuurmans for insightful conversation, as well as the rest of the Google Brain Team.
+
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+
+# Appendix
+
+# A 2D GAUSSIAN TRAINING DETAILS
+
+Network architecture and experimental details for the experiment in Section 3.1 are as follows:
+
+The dataset is sampled from a mixture of 8 Gaussians of standard deviation 0.02. The means are equally spaced around a circle of radius 2.
+
+The generator network consists of a fully connected network with 2 hidden layers of size 128 with relu activations followed by a linear projection to 2 dimensions. All weights are initialized to be orthogonal with scaling of 0.8.
+
+The discriminator network first scales its input down by a factor of 4 (to roughly scale to (-1,1)), followed by 1 layer fully connected network with relu activations to a linear layer to of size 1 to act as the logit.
+
+The generator minimizes $\mathcal { L } _ { G } = \log ( D ( x ) ) + \log ( 1 - D ( G ( z ) ) )$ and the discriminator minimizes $\mathcal { L } _ { D } \stackrel { = } { = } - \log ( D ( x ) ) - \log ( 1 - D ( \stackrel { . } { G } ( z ) ) )$ where $\mathbf { X }$ is sampled from the data distribution and $z \sim$ $\mathcal { N } ( 0 , I _ { 2 5 6 } )$ . Both networks are optimized using Adam (Kingma & Ba, 2014) with a learning rate of 1e-4 and $\beta _ { 1 } { = } 0 . 5$ .
+
+The network is trained by alternating updates of the generator and the discriminator. One step consists of either $\mathbf { G }$ or $\mathbf { D }$ updating.
+
+# B MORE MIXTURE OF GAUSSIAN EXPERIMENTS
+
+# B.1 EFFECTS OF TIME DELAY / HISTORICAL AVERAGING
+
+Another comparison we looked at was with regard to historical averaging based approaches. Recently similarly inspired approaches have been used in (Salimans et al., 2016) to stabilize training. For our study, we looked at taking an ensemble of discriminators over time.
+
+First, we looked at taking an ensemble of the last N steps, as shown in Figure App.1.
+
+![](images/ca00dc71a894afed5412e7d9c9476916ab9ab1f2648dedea7d529b310373bd67.jpg)  
+Figure App.1: Historical averaging does not visibly increase stability on the mixture of Gaussians task. Each row corresponds to an ensemble of discriminators which consists of the indicated number of immediately preceding discriminators. The columns correspond to different numbers of training steps.
+
+To further explore this idea, we ran experiments with an ensemble of 5 discriminators, but with different periods between replacing discriminators in the ensemble. For example, if I sample at a rate of 100, it would take 500 steps to replace all 5 discriminators. Results can be seen in Figure App.2.
+
+We observe that given longer and longer time delays, the model becomes less and less stable. We hypothesize that this is due to the initial shape of the discriminator loss surface. When training, the discriminator’s estimates of probability densities are only accurate on regions where it was trained. When fixing this discriminator, we are removing the feedback between the generator exploitation and the discriminators ability to move. As a result, the generator is able to exploit these fixed areas of poor performance for older discriminators in the ensemble. New discriminators (over)compensate for this, leading the system to diverge.
+
+![](images/04adbd77c06eb06d8623be8ab50fe44136a6a0398a13d8fdd2a583c617dbcff0.jpg)  
+Figure App.2: Introducing longer time delays between the discriminator ensemble results in instability and probability distributions that are not in the window being visualized. The $\mathbf { X }$ axis is the number of weight updates and the y axis is how many steps to skip between discriminator updates when selecting the ensemble of 5 discriminators.
+
+# B.2 EFFECTS OF THE SECOND GRADIENT
+
+A second factor we analyzed is the effect of backpropagating the learning signal through the unrolling in Equation 12. We can turn on or off this backpropagation through the unrolling by introducing stop gradient calls into our computation graph between each unrolling step. With the stop gradient in place, the update signal corresponds only to the first term in Equation 12. We looked at 3 configurations: without stop gradients; vanilla unrolled GAN, with stop gradients; and with stop gradients but taking the average over the $k$ unrolling steps instead of taking the final value. Results can be see in Figure App.3.
+
+We initially observed no difference between unrolling with and without the second gradient, as both required 3 unrolling steps to become stable. When the discriminator is unrolled to convergence, the second gradient term becomes zero. Due to the simplicity of the problem, we suspect that the discriminator nearly converged for every generator step, and the second gradient term was thus irrelevant.
+
+To test this, we modified the dynamics to perform five generator steps for each discriminator update. Results are shown in Figure App.4. With the discriminator now kept out of equilibrium, successful training can be achieved with half as many unrolling steps when using both terms in the gradient than when only including the first term.
+
+# C RNN MNIST TRAINING DETAILS
+
+The network architecture for the experiment in Section 3.2 is as follows:
+
+The MNIST dataset is scaled to [-1, 1).
+
+The generator first scales the 256D noise vector through a 256 unit fully connected layer with relu activation. This is then fed into the initial state of a 256D LSTM(Hochreiter & Schmidhuber, 1997) that runs 28 steps corresponding to the number of columns in MNIST. The resulting sequence of activations is projected through a fully connected layer with 28 outputs with a tanh activation function. All weights are initialized via the ”Xavier” initialization (Glorot & Bengio, 2010). The forget bias on the LSTM is initialized to 1.
+
+The discriminator network feeds the input into a Convolution(16, stride $^ { = 2 }$ ) followed by a Convolution(32, stride $^ { = 2 }$ ) followed by Convolution(32, stride ${ \boldsymbol { \mathbf { \mathit { \varepsilon } } } } = 2 { \boldsymbol { \mathbf { \mathit { \varepsilon } } } }$ ). All convolutions have stride 2. As in (Radford et al., 2015) leaky rectifiers are used with a 0.3 leak. Batch normalization is applied after each layer (Ioffe & Szegedy, 2015). The resulting 4D tensor is then flattened and a linear projection is performed to a single scalar.
+
+![](images/e082e1fa15fe3c8c65693a9ca8c06d3ede7c2ddcf2d0287c257abe4e8c2ff9b6.jpg)  
+Figure App.3: If the discriminator remains nearly at its optimum during learning, then performance is nearly identical with and without the second gradient term in Equation 12. As shown in Figure App.4, when the discriminator lags behind the generator, backpropagating through unrolling aids convergence.
+
+The generator network minimises $\mathcal { L } _ { G } = \log ( D ( G ( z ) ) )$ and the discriminator minimizes $\mathcal { L } _ { D } =$ $\log ( D ( x ) ) + \log ( 1 - D ( G ( z ) ) )$ . Both networks are trained with Adam(Kingma & Ba, 2014) with learning rates of 1e-4 and $\beta _ { 1 } { = } 0 . 5$ . The network is trained alternating updating the generator and the discriminator for 150k steps. One step consists of just 1 network update.
+
+# D CIFAR10/MNIST TRAINING DETAILS
+
+The network architectures for the discriminator, generator, and encoder as as follows. All convolutions have a kernel size of 3x3 with batch normalization and leaky ReLU’s with a 0.3 leak.
+
+The generator network is defined as:
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z ~ N(0,I256)Fully connectedReshape to image 4,4,512ConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4*4*512256128641or3</td><td rowspan=1 colspan=1>２２２１</td></tr></table>
+
+![](images/ddba8731c308e072b5532fd2f2dfbcf72fd2fc3c5d92dcb601c2c03608beee85.jpg)  
+Unrolled GAN with 5 G Steps per D without second gradient
+
+Unrolled GAN with 5 G Steps per D   
+
+<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x~ Pdata or G Transposed Convolution Transposed Convolution Transposed Convolution</td><td>64 128 256</td><td>２２2</td></tr><tr><td>Flatten</td><td></td><td></td></tr><tr><td>Fully Connected</td><td>1</td><td></td></tr></table>
+
+![](images/c5b3fa4bd5bb790af426480883819ba983aa6618ec11502817b7cc7fa39e22d5.jpg)  
+Figure App.4: Backpropagating through the unrolling process aids convergence when the discriminator does not fully converge between generator updates. When taking 5 generator steps per discriminator step unrolling greatly increases stability, requiring only 5 unrolling steps to converge. Without the second gradient it requires 10 unrolling steps. Also see Figure App.3.
+
+The discriminator network is defined as:
+
+The generator network minimises $\mathcal { L } _ { G } = \log ( D ( G ( z ) ) )$ and the discriminator minimizes $\mathcal { L } _ { D } =$ $\log ( D ( x ) ) + \log ( 1 - D ( G ( z ) ) )$ . The networks are trained with Adam with a generator learning rate of 1e-4, and a discriminator learning rate of 2e-4. The network is trained alternating updating the generator and the discriminator for $1 0 0 \mathrm { k }$ steps. One step consists of just 1 network update.
+
+# E 1000 CLASS MNIST
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z ~ N(0,I256)Fully connectedReshape to image 4,4,64ConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4 *4*64321683</td><td rowspan=1 colspan=1>２２２１</td></tr></table>
+
+The discriminator network is parametrized by a size $\mathrm { X }$ and is defined as follows. In our tests, we used X of 1/4 and 1/2.
+
+<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x ~ Pdata or G Transposed Convolution Transposed Convolution</td><td>8*X 16*X</td><td>２２2</td></tr><tr><td>Transposed Convolution Flatten</td><td>32*X</td><td></td></tr><tr><td>Fully Connected</td><td>1</td><td></td></tr></table>
+
+# F COLORED MNIST DATASET
+
+# F.1 DATASET
+
+To generate this dataset we first took the mnist digit, $I$ , scaled between 0 and 1. For each image we sample a color, $C$ , normally distributed with mean $\scriptstyle = 0$ and std $\scriptstyle \mathtt { = 0 . 5 }$ . To generate a colored digit between (-1, 1) we do $I * C + \mathsf { \bar { ( } } I - 1 )$ . Finally, we add a small amount of pixel independent noise sampled from a normal distribution with std $= 0 . 2$ , and the resulting values are cliped between (-1, 1). When visualized, this generates images and samples that can be seen in figure App.5. Once again it is very hard to visually see differences in sample diversity when comparing the 128 and the 512 sized models.
+
+![](images/44d318f018c78dbdaec0c5211309ab172baeb975108ebe77f750982a4193cef6.jpg)  
+Figure App.5: Right: samples from the data distribution. Middle: Samples from 1/4 size model with 0 look ahead steps (worst diversity). Left: Samples from 1/1 size model with 10 look ahead steps (most diversity).
+
+# F.2 MODELS
+
+The models used in this section are parametrized by a variable $\mathrm { X }$ to control capacity. A value of ${ \bf X } { = } 1$ is same architecture used in the cifar10 experiments. We used 1/4, 1/2 and 1 as these values.
+
+The generator network is defined as:
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z~ N(0,I256)Fully connectedReshape to image 4,4,512*XConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4*4*512*X256*X128*X64*X3</td><td rowspan=1 colspan=1>2221</td></tr></table>
+
+The discriminator network is defined as:
+
+<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x ~ Pdata or G Transposed Convolution Transposed Convolution Transposed Convolution Flatten Fully Connected</td><td>64*X 128*X 256*X</td><td>２２2</td></tr></table>
+
+# G OPTIMIZATION BASED VISUALIZATIONS
+
+More examples of model based optimization. We performed 5 runs with different seeds of each of of the unrolling steps configuration. Bellow are comparisons for each run index. Ideally this would be a many to many comparison, but for space efficiency we grouped the runs by the index in which they were run.
+
+![](images/8d38e597124cc936be0c8530d6fb7e8b1ecc41f5e50351124daf12a2d78e11fa.jpg)  
+Figure App.6: Samples from 1/5 with different random seeds.
+
+![](images/be94e2bb107541e788983b4ad2bdc31bebdff50fbbb8529c915238f8cca809e8.jpg)  
+Figure App.7: Samples from 2/5 with different random seeds.
+
+![](images/e17957e3a6c6fffe5b489567eb4830a1720a2ae85eadaf27b1b6017a82527e8b.jpg)  
+Figure App.8: Samples from 3/5 with different random seeds.
+
+![](images/fbca68637cf3798a1a739daff1548046fba7251170be090a9d1d669d0aa5603e.jpg)  
+Figure App.9: Samples from 4/5 with different random seeds.
+
+![](images/c67273dbe4f3a1a6aee81a0d0854850226af640cdd34215ec9e0663f82c11a7d.jpg)  
+Figure App.10: Samples from 5/5 with different random seeds.
\ No newline at end of file
diff --git a/parse/train/BydrOIcle/BydrOIcle_content_list.json b/parse/train/BydrOIcle/BydrOIcle_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..946e84320358b54b71d53206c206f42fbeabf0ef
--- /dev/null
+++ b/parse/train/BydrOIcle/BydrOIcle_content_list.json
@@ -0,0 +1,2528 @@
+[
+    {
+        "type": "text",
+        "text": "UNROLLED GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            99,
+            807,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Luke Metz∗   \nGoogle Brain   \nlmetz@google.com   \nBen Poole†   \nStanford University   \npoole@cs.stanford.edu ",
+        "bbox": [
+            184,
+            145,
+            343,
+            188
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+        "page_idx": 0
+    },
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+        "text": "",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Pfau Google DeepMind pfau@google.com ",
+        "bbox": [
+            663,
+            145,
+            813,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jascha Sohl-Dickstein Google Brain jaschasd@google.com ",
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+            183,
+            208,
+            372,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            287,
+            544,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a method to stabilize Generative Adversarial Networks (GANs) by defining the generator objective with respect to an unrolled optimization of the discriminator. This allows training to be adjusted between using the optimal discriminator in the generator’s objective, which is ideal but infeasible in practice, and using the current value of the discriminator, which is often unstable and leads to poor solutions. We show how this technique solves the common problem of mode collapse, stabilizes training of GANs with complex recurrent generators, and increases diversity and coverage of the data distribution by the generator. ",
+        "bbox": [
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+            318,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+            457,
+            336,
+            473
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The use of deep neural networks as generative models for complex data has made great advances in recent years. This success has been achieved through a surprising diversity of training losses and model architectures, including denoising autoencoders (Vincent et al., 2010), variational autoencoders (Kingma & Welling, 2013; Rezende et al., 2014; Gregor et al., 2015; Kulkarni et al., 2015; Burda et al., 2015; Kingma et al., 2016), generative stochastic networks (Alain et al., 2015), diffusion probabilistic models (Sohl-Dickstein et al., 2015), autoregressive models (Theis & Bethge, 2015; van den Oord et al., 2016a;b), real non-volume preserving transformations (Dinh et al., 2014; 2016), Helmholtz machines (Dayan et al., 1995; Bornschein et al., 2015), and Generative Adversarial Networks (GANs) (Goodfellow et al., 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1.1 GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While most deep generative models are trained by maximizing log likelihood or a lower bound on log likelihood, GANs take a radically different approach that does not require inference or explicit calculation of the data likelihood. Instead, two models are used to solve a minimax game: a generator which samples data, and a discriminator which classifies the data as real or generated. In theory these models are capable of modeling an arbitrarily complex probability distribution. When using the optimal discriminator for a given class of generators, the original GAN proposed by Goodfellow et al. minimizes the Jensen-Shannon divergence between the data distribution and the generator, and extensions generalize this to a wider class of divergences (Nowozin et al., 2016; Sonderby et al., 2016; Poole et al., 2016). ",
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+        "text": "The ability to train extremely flexible generating functions, without explicitly computing likelihoods or performing inference, and while targeting more mode-seeking divergences as made GANs extremely successful in image generation (Odena et al., 2016; Salimans et al., 2016; Radford et al., 2015), and image super resolution (Ledig et al., 2016). The flexibility of the GAN framework has also enabled a number of successful extensions of the technique, for instance for structured prediction (Reed et al., 2016a;b; Odena et al., 2016), training energy based models (Zhao et al., 2016), and combining the GAN loss with a mutual information loss (Chen et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "In practice, however, GANs suffer from many issues, particularly during training. One common failure mode involves the generator collapsing to produce only a single sample or a small family of very similar samples. Another involves the generator and discriminator oscillating during training, rather than converging to a fixed point. In addition, if one agent becomes much more powerful than the other, the learning signal to the other agent becomes useless, and the system does not learn. To train GANs many tricks must be employed, such as careful selection of architectures (Radford et al., 2015), minibatch discrimination (Salimans et al., 2016), and noise injection (Salimans et al., 2016; Sonderby et al., 2016). Even with these tricks the set of hyperparameters for which training is successful is generally very small in practice. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Once converged, the generative models produced by the GAN training procedure normally do not cover the whole distribution (Dumoulin et al., 2016; Che et al., 2016), even when targeting a modecovering divergence such as KL. Additionally, because it is intractable to compute the GAN training loss, and because approximate measures of performance such as Parzen window estimates suffer from major flaws (Theis et al., 2016), evaluation of GAN performance is challenging. Currently, human judgement of sample quality is one of the leading metrics for evaluating GANs. In practice this metric does not take into account mode dropping if the number of modes is greater than the number of samples one is visualizing. In fact, the mode dropping problem generally helps visual sample quality as the model can choose to focus on only the most common modes. These common modes correspond, by definition, to more typical samples. Additionally, the generative model is able to allocate more expressive power to the modes it does cover than it would if it attempted to cover all modes. ",
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+    {
+        "type": "text",
+        "text": "1.2 DIFFERENTIATING THROUGH OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Many optimization schemes, including SGD, RMSProp (Tieleman & Hinton, 2012), and Adam (Kingma & Ba, 2014), consist of a sequence of differentiable updates to parameters. Gradients can be backpropagated through unrolled optimization updates in a similar fashion to backpropagation through a recurrent neural network. The parameters output by the optimizer can thus be included, in a differentiable way, in another objective (Maclaurin et al., 2015). This idea was first suggested for minimax problems in (Pearlmutter & Siskind, 2008), while (Zhang & Lesser, 2010) provided a theoretical analysis and experimental results on differentiating through a single step of gradient ascent for simple matrix games. Differentiating through unrolled optimization was first scaled to deep networks in (Maclaurin et al., 2015), where it was used for hyperparameter optimization. More recently, (Belanger & McCallum, 2015; Han et al., 2016; Andrychowicz et al., 2016) backpropagate through optimization procedures in contexts unrelated to GANs or minimax games. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work we address the challenges of unstable optimization and mode collapse in GANs by unrolling optimization of the discriminator objective during training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The GAN learning problem is to find the optimal parameters $\\theta _ { G } ^ { * }$ for a generator function $G \\left( z ; \\theta _ { G } \\right)$ in a minimax objective, ",
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+    {
+        "type": "equation",
+        "img_path": "images/b2008d380b8f084085a49f2a632840f8a4144bbae5800b215f3679c79653344b.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\theta _ { G } ^ { * } = \\underset { \\theta _ { G } } { \\mathrm { a r g m i n } } \\underset { \\theta _ { D } } { \\mathrm { m a x } } f \\left( \\theta _ { G } , \\theta _ { D } \\right) } } \\\\ { { \\ \\mathrm { ~ } } } \\\\ { { \\displaystyle \\qquad = \\underset { \\theta _ { G } } { \\mathrm { a r g m i n } } f \\left( \\theta _ { G } , \\theta _ { D } ^ { * } \\left( \\theta _ { G } \\right) \\right) } } \\\\ { { \\theta _ { D } ^ { * } \\left( \\theta _ { G } \\right) = \\underset { \\theta _ { D } } { \\mathrm { a r g m a x } } f \\left( \\theta _ { G } , \\theta _ { D } \\right) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $f$ is commonly chosen to be ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ee6df4b3e18c286fdee617b2aa953c06b25b1f3424af4b30db2853ba740dd9e.jpg",
+        "text": "$$\nf \\left( \\theta _ { G } , \\theta _ { D } \\right) = \\mathbb { E } _ { x \\sim p _ { d a t a } } \\left[ \\log \\left( D \\left( x ; \\theta _ { D } \\right) \\right) \\right] + \\mathbb { E } _ { z \\sim N ( 0 , I ) } \\left[ \\log \\left( 1 - D \\left( G \\left( z ; \\theta _ { G } \\right) ; \\theta _ { D } \\right) \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here $x \\in \\mathcal { X }$ is the data variable, $z \\in { \\mathcal { Z } }$ is the latent variable, $p _ { d a t a }$ is the data distribution, the discriminator $D ( \\cdot ; \\theta _ { D } ) : \\mathcal { X }  [ 0 , 1 ]$ outputs the estimated probability that a sample $x$ comes from the data distribution, $\\theta _ { D }$ and $\\theta _ { G }$ are the discriminator and generator parameters, and the generator function $G ( \\cdot ; \\theta _ { G } ) : \\mathcal { Z }  \\mathcal { X }$ transforms a sample in the latent space into a sample in the data space. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the minimax loss in Eq. 4, the optimal discriminator $D ^ { \\ast } \\left( x \\right)$ is a known smooth function of the generator probability $p _ { G } \\left( x \\right)$ (Goodfellow et al., 2014), ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e2376c104ce7eecf400d8575cb56227acb0e45882302fbde8854a5f2d2b2c6b.jpg",
+        "text": "$$\nD ^ { * } \\left( x \\right) = \\frac { p _ { d a t a } \\left( x \\right) } { p _ { d a t a } \\left( x \\right) + p _ { G } \\left( x \\right) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When the generator loss in Eq. 2 is rewritten directly in terms of $p _ { G } \\left( x \\right)$ and Eq. 5 rather than $\\theta _ { G }$ and $\\theta _ { D } ^ { * } \\left( \\theta _ { G } ^ { - } \\right)$ , then it is similarly a smooth function of $p _ { G } \\left( x \\right)$ . These smoothness guarantees are typically lost when $D \\left( x ; \\theta _ { D } \\right)$ and $G \\left( z ; \\theta _ { G } \\right)$ are drawn from parametric families. They nonetheless suggest that the true generator objective in Eq. 2 will often be well behaved, and is a desirable target for direct optimization. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Explicitly solving for the optimal discriminator parameters $\\theta _ { D } ^ { * } \\left( \\theta _ { G } \\right) $ for every update step of the generator $G$ is computationally infeasible for discriminators based on neural networks. Therefore this minimax optimization problem is typically solved by alternating gradient descent on $\\theta _ { G }$ and ascent on $\\theta _ { D }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The optimal solution $\\theta ^ { * } = \\{ \\theta _ { G } ^ { * } , \\theta _ { D } ^ { * } \\}$ is a fixed point of these iterative learning dynamics. Additionally, if $f \\left( { \\theta } _ { G } , { \\theta } _ { D } \\right)$ is convex in $\\theta _ { G }$ and concave in $\\theta _ { D }$ , then alternating gradient descent (ascent) trust region updates are guaranteed to converge to the fixed point, under certain additional weak assumptions (Juditsky et al., 2011). However in practice $f \\left( { \\theta } _ { G } , { \\theta } _ { D } \\right)$ is typically very far from convex in $\\theta _ { G }$ and concave in $\\theta _ { D }$ , and updates are not constrained in an appropriate way. As a result GAN training suffers from mode collapse, undamped oscillations, and other problems detailed in Section 1.1. In order to address these difficulties, we will introduce a surrogate objective function $f _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right)$ for training the generator which more closely resembles the true generator objective $f \\left( \\theta _ { G } , \\theta _ { D } ^ { * } \\left( \\theta _ { G } \\right) \\right)$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 UNROLLING GANS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A local optimum of the discriminator parameters $\\theta _ { D } ^ { * }$ can be expressed as the fixed point of an iterative optimization procedure, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f45244ba45a91984f7cab7a13e843b9f8428ca6f8c95d11c332b9e18669248b2.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\theta _ { D } ^ { 0 } = \\theta _ { D } } } \\\\ { { \\displaystyle } } \\\\ { { \\theta _ { D } ^ { k + 1 } = \\theta _ { D } ^ { k } + \\eta ^ { k } \\displaystyle \\frac { \\mathrm { d } f ( \\theta _ { G } , \\theta _ { D } ^ { k } ) } { \\mathrm { d } \\theta _ { D } ^ { k } } } } \\\\ { { \\displaystyle } } \\\\ { { \\theta _ { D } ^ { * } ( \\theta _ { G } ) = \\displaystyle \\operatorname* { l i m } _ { k  \\infty } \\theta _ { D } ^ { k } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta ^ { k }$ is the learning rate schedule. For clarity, we have expressed Eq. 7 as a full batch steepest gradient ascent equation. More sophisticated optimizers can be similarly unrolled. In our experiments we unroll Adam (Kingma & Ba, 2014). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By unrolling for $K$ steps, we create a surrogate objective for the update of the generator, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/722de6d2f0bca528a8df9e9213bb3bcba721cff99ce6b981b4c7ab2eda98fd02.jpg",
+        "text": "$$\nf _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) = f \\left( \\theta _ { G } , \\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When $K = 0$ this objective corresponds exactly to the standard GAN objective, while as $K  \\infty$ it corresponds to the true generator objective function $f \\left( \\theta _ { G } , \\theta _ { D } ^ { * } \\left( G \\right) \\right)$ . By adjusting the number of unrolling steps $K$ , we are thus able to interpolate between standard GAN training dynamics with their associated pathologies, and more costly gradient descent on the true generator loss. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 PARAMETER UPDATES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generator and discriminator parameter updates using this surrogate loss are ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35895e40c05db7fa3394ccde440178bbdbe1633ad2e976da444caa1f30f35303.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\theta _ { G }  \\theta _ { G } - \\eta \\frac { \\mathrm { d } f _ { K } ( \\theta _ { G } , \\theta _ { D } ) } { \\mathrm { d } \\theta _ { G } } } \\\\ & { \\theta _ { D }  \\theta _ { D } + \\eta \\frac { \\mathrm { d } f ( \\theta _ { G } , \\theta _ { D } ) } { \\mathrm { d } \\theta _ { D } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For clarity we use full batch steepest gradient descent (ascent) with stepsize $\\eta$ above, while in experiments we instead use minibatch Adam for both updates. The gradient in Eq. 10 requires backpropagating through the optimization process in Eq. 7. A clear description of differentiation through gradient descent is given as Algorithm 2 in (Maclaurin et al., 2015), though in practice the use of an automatic differentiation package means this step does not need to be programmed explicitly. A pictorial representation of these updates is provided in Figure 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8d41d44f95393092b08110ba25482b02bb1bcb0a2930f383731e4e370791ca3.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of the computation graph for an unrolled GAN with 3 unrolling steps. The generator update in Equation 10 involves backpropagating the generator gradient (blue arrows) through the unrolled optimization. Each step $k$ in the unrolled optimization uses the gradients of $f _ { k }$ with respect to $\\theta _ { D } ^ { k }$ , as described in Equation 7 and indicated by the green arrows. The discriminator update in Equation 11 does not depend on the unrolled optimization (red arrow). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is important to distinguish this from an approach suggested in (Goodfellow et al., 2014), that several update steps of the discriminator parameters should be run before each single update step for the generator. In that approach, the update steps for both models are still gradient descent (ascent) with respect to fixed values of the other model parameters, rather than the surrogate loss we describe in Eq. 9. Performing $K$ steps of discriminator update between each single step of generator update corresponds to updating the generator parameters $\\theta _ { G }$ using only the first term in Eq. 12 below. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 THE MISSING GRADIENT TERM ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To better understand the behavior of the surrogate loss $f _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right)$ , we examine its gradient with respect to the generator parameters $\\theta _ { G }$ , ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/791d5164930dd3fb042c3335cdcf143313e5660a7175699577db372f61561f22.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } f _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) } { \\mathrm { d } \\theta _ { G } } = \\frac { \\partial f \\left( \\theta _ { G } , \\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) \\right) } { \\partial \\theta _ { G } } + \\frac { \\partial f \\left( \\theta _ { G } , \\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) \\right) } { \\partial \\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) } \\frac { \\mathrm { d } \\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) } { \\mathrm { d } \\theta _ { G } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Standard GAN training corresponds exactly to updating the generator parameters using only the first term in this gradient, with $\\theta _ { D } ^ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right) $ being the parameters resulting from the discriminator update step. An optimal generator for any fixed discriminator is a delta function at the $x$ to which the discriminator assigns highest data probability. Therefore, in standard GAN training, each generator update step is a partial collapse towards a delta function. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The second term captures how the discriminator would react to a change in the generator. It reduces the tendency of the generator to engage in mode collapse. For instance, the second term reflects that as the generator collapses towards a delta function, the discriminator reacts and assigns lower probability to that state, increasing the generator loss. It therefore discourages the generator from collapsing, and may improve stability. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As $K  \\infty$ , $\\theta _ { D } ^ { K }$ goes to a local optimum of $f$ , where $\\frac { \\partial f } { \\partial \\theta _ { D } ^ { K } } = 0$ , and therefore the second term in Eq. 12 goes to 0 (Danskin, 1967). The gradient of the unrolled surrogate loss $f _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right)$ with respect to $\\theta _ { G }$ is thus identical to the gradient of the standard GAN loss $f \\bar { ( \\theta _ { G } , \\theta _ { D } ) }$ both when $K = 0$ and when $K  \\infty$ , where we take $K  \\infty$ to imply that in the standard GAN the discriminator is also fully optimized between each generator update. Between these two extremes, $f _ { K } \\left( \\theta _ { G } , \\theta _ { D } \\right)$ captures additional information about the response of the discriminator to changes in the generator. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 CONSEQUENCES OF THE SURROGATE LOSS",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "GANs can be thought of as a game between the discriminator $( D )$ and the generator $( G )$ . The agents take turns taking actions and updating their parameters until a Nash equilibrium is reached. The optimal action for $D$ is to evaluate the probability ratio $\\frac { p _ { d a t a } ( x ) } { p _ { G } ( x ) + p _ { d a t a } ( x ) }$ for the generator’s move $x$ (Eq. 5). The optimal generator action is to move its mass to maximize this ratio. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The initial move for $G$ will be to move as much mass as its parametric family and update step permits to the single point that maximizes the ratio of probability densities. The action $D$ will then take is quite simple. It will track that point, and to the extent allowed by its own parametric family and update step assign low data probability to it, and uniform probability everywhere else. This cycle of $G$ moving and $D$ following will repeat forever or converge depending on the rate of change of the two agents. This is similar to the situation in simple matrix games like rock-paper-scissors and matching pennies, where alternating gradient descent (ascent) with a fixed learning rate is known not to converge (Singh et al., 2000; Bowling & Veloso, 2002). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the unrolled case, however, this undesirable behavior no longer occurs. Now $G$ ’s actions take into account how $D$ will respond. In particular, $G$ will try to make steps that $D$ will have a hard time responding to. This extra information helps the generator spread its mass to make the next $D$ step less effective instead of collapsing to a point. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In principle, a surrogate loss function could be used for both $D$ and $G$ . In the case of 1-step unrolled optimization this is known to lead to convergence for games in which gradient descent (ascent) fails (Zhang & Lesser, 2010). However, the motivation for using the surrogate generator loss in Section 2.2, of unrolling the inner of two nested min and max functions, does not apply to using a surrogate discriminator loss. Additionally, it is more common for the discriminator to overpower the generator than vice-versa when training a GAN. Giving more information to $G$ by allowing it to ‘see into the future’ may thus help the two models be more balanced. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we demonstrate improved mode coverage and stability by applying this technique to five datasets of increasing complexity. Evaluation of generative models is a notoriously hard problem (Theis et al., 2016). As such the de facto standard in GAN literature has become sample quality as evaluated by a human and/or evaluated by a heuristic (Inception score for example, (Salimans et al., 2016)). While these evaluation metrics do a reasonable job capturing sample quality, they fail to capture sample diversity. In our first 2 experiments diversity is easily evaluated via visual inspection. In our later experiments this is not the case, and we will use a variety of methods to quantify coverage of samples. Our measures are individually strongly suggestive of unrolling reducing mode-collapse and improving stability, but none of them alone are conclusive. We believe that taken together however, they provide extremely compelling evidence for the advantages of unrolling. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When doing stochastic optimization, we must choose which minibatches to use in the unrolling updates in Eq. 7. We experimented with both a fixed minibatch and re-sampled minibatches for each unrolling step, and found it did not significantly impact the result. We use fixed minibatches for all experiments in this section. ",
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+    },
+    {
+        "type": "text",
+        "text": "We provide a reference implementation of this technique at github.com/poolio/unrolled gan. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 MIXTURE OF GAUSSIANS DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To illustrate the impact of discriminator unrolling, we train a simple GAN architecture on a 2D mixture of 8 Gaussians arranged in a circle. For a detailed list of architecture and hyperparameters see Appendix A. Figure 2 shows the dynamics of this model through time. Without unrolling the generator rotates around the valid modes of the data distribution but is never able to spread out mass. When adding in unrolling steps $\\mathbf { G }$ quickly learns to spread probability mass and the system converges to the data distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Appendix B we perform further experiments on this toy dataset. We explore how unrolling compares to historical averaging, and compares to using the unrolled discriminator to update the generator, but without backpropagating through the generator. In both cases we find that the unrolled objective performs better. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a4459b06fa7e9fcdf667d911bf408b5c5ccfe73af1d14e29428d98326008c54a.jpg",
+        "image_caption": [
+            "Figure 2: Unrolling the discriminator stabilizes GAN training on a toy 2D mixture of Gaussians dataset. Columns show a heatmap of the generator distribution after increasing numbers of training steps. The final column shows the data distribution. The top row shows training for a GAN with 10 unrolling steps. Its generator quickly spreads out and converges to the target distribution. The bottom row shows standard GAN training. The generator rotates through the modes of the data distribution. It never converges to a fixed distribution, and only ever assigns significant probability mass to a single data mode at once. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff67f1c079a9b83431ee462581bfb81893ea4eadc9da332a9fff1968becf8c08.jpg",
+        "image_caption": [
+            "Figure 3: Unrolled GAN training increases stability for an RNN generator and convolutional discriminator trained on MNIST. The top row was run with 20 unrolling steps. The bottom row is a standard GAN, with 0 unrolling steps. Images are samples from the generator after the indicated number of training steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 PATHOLOGICAL MODEL WITH MISMATCHED GENERATOR AND DISCRIMINATOR ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate the ability of this approach to improve trainability, we look to a traditionally challenging family of models to train – recurrent neural networks (RNNs). In this experiment we try to generate MNIST samples using an LSTM (Hochreiter & Schmidhuber, 1997). MNIST digits are $2 8 \\mathbf { x } 2 8$ pixel images. At each timestep of the generator LSTM, it outputs one column of this image, so that after 28 timesteps it has output the entire sample. We use a convolutional neural network as the discriminator. See Appendix C for the full model and training details. Unlike in all previously successful GAN models, there is no symmetry between the generator and the discriminator in this task, resulting in a more complex power balance. Results can be seen in Figure 3. Once again, without unrolling the model quickly collapses, and rotates through a sequence of single modes. Instead of rotating spatially, it cycles through proto-digit like blobs. When running with unrolling steps the generator disperses and appears to cover the whole data distribution, as in the 2D example. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/e4cb5f019901d91438706efd6a3956d78e353190e6df31aa58c09fecfa15f59b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Discriminator Size</td><td rowspan=1 colspan=1>Unrolling steps</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>1/4 size of D compared to G</td><td rowspan=1 colspan=1>Modes generated</td><td rowspan=1 colspan=1>30.6± 20.73</td><td rowspan=1 colspan=1>65.4 ± 34.75</td><td rowspan=1 colspan=1>236.4 ± 63.30</td><td rowspan=1 colspan=1>327.2 ± 74.67</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>KL(model||data)</td><td rowspan=1 colspan=1>5.99± 0.42</td><td rowspan=1 colspan=1>5.911 ± 0.14</td><td rowspan=1 colspan=1>4.67 ± 0.43</td><td rowspan=1 colspan=1>4.66 ± 0.46</td></tr><tr><td rowspan=1 colspan=1>1/2 size of D compared to G</td><td rowspan=1 colspan=1>Modes generated</td><td rowspan=1 colspan=1>628.0± 140.9</td><td rowspan=1 colspan=1>523.6± 55.768</td><td rowspan=1 colspan=1>732.0± 44.98</td><td rowspan=1 colspan=1>817.4 ± 37.91</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>KL(model||data)</td><td rowspan=1 colspan=1>2.58 ±0.751</td><td rowspan=1 colspan=1>2.44 ±0.26</td><td rowspan=1 colspan=1>1.66 ± 0.090</td><td rowspan=1 colspan=1>1.43 ± 0.12</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Unrolled GANs cover more discrete modes when modeling a dataset with 1,000 data modes, corresponding to all combinations of three MNIST digits $[ 1 0 ^ { 3 }$ digit combinations). The number of modes covered is given for different numbers of unrolling steps, and for two different architectures. The reverse KL divergence between model and data is also given. Standard error is provided for both measures. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 MODE AND MANIFOLD COLLAPSE USING AUGMENTED MNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GANs suffer from two different types of model collapse – collapse to a subset of data modes, and collapse to a sub-manifold within the data distribution. In these experiments we isolate both effects using artificially constructed datasets, and demonstrate that unrolling can largely rescue both types of collapse. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 DISCRETE MODE COLLAPSE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To explore the degree to which GANs drop discrete modes in a dataset, we use a technique similar to one from (Che et al., 2016). We construct a dataset by stacking three randomly chosen MNIST digits, so as to construct an RGB image with a different MNIST digit in each color channel. This new dataset has 1,000 distinct modes, corresponding to each combination of the ten MNIST classes in the three channels. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We train a GAN on this dataset, and generate samples from the trained model (25,600 samples for all experiments). We then compute the predicted class label of each color channel using a pre-trained MNIST classifier. To evaluate performance, we use two metrics: the number of modes for which the generator produced at least one sample, and the KL divergence between the model and the expected data distribution. Within this discrete label space, a KL divergence can be estimated tractably between the generated samples and the data distribution over classes, where the data distribution is a uniform distribution over all 1,000 classes. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As presented in Table 1, as the number of unrolling steps is increased, both mode coverage and reverse KL divergence improve. Contrary to (Che et al., 2016), we found that reasonably sized models (such as the one used in Section 3.4) covered all 1,000 modes even without unrolling. As such we use smaller convolutional GAN models. Details on the models used are provided in Appendix E. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We observe an additional interesting effect in this experiment. The benefits of unrolling increase as the discriminator size is reduced. We believe unrolling effectively increases the capacity of the discriminator. The unrolled discriminator can better react to any specific way in which the generator is producing non-data-like samples. When the discriminator is weak, the positive impact of unrolling is thus larger. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 MANIFOLD COLLAPSE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In addition to discrete modes, we examine the effect of unrolling when modeling continuous manifolds. To get at this quantity, we constructed a dataset consisting of colored MNIST digits. Unlike in the previous experiment, a single MNIST digit was chosen, and then assigned a single monochromatic color. With a perfect generator, one should be able to recover the distribution of colors used to generate the digits. We use colored MNIST digits so that the generator also has to model the digits, which makes the task sufficiently complex that the generator is unable to perfectly solve it. The color of each digit is sampled from a 3D normal distribution. Details of this dataset are provided in Appendix F. We will examine the distribution of colors in the samples generated by the trained GAN. As will also be true in the CIFAR10 example in Section 3.4, the lack of diversity in generated colors is almost invisible using only visual inspection of the samples. Samples can be found in Appendix F. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/bdf448f52fbc7d7ff1a8589d6a25dd7b7fe2e14cdf8e504055b7f3a333d00a6e.jpg",
+        "table_caption": [
+            "Table 2: Unrolled GANs better model a continuous distribution. GANs are trained to model randomly colored MNIST digits, where the color is drawn from a Gaussian distribution. The JS divergence between the data and model distributions over digit colors is then reported, along with standard error in the JS divergence. More unrolling steps, and larger models, lead to better JS divergence. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Unrolling steps</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/4 layer size</td><td rowspan=1 colspan=1>0.073 ± 0.0058</td><td rowspan=1 colspan=1>0.142 ± 0.028</td><td rowspan=1 colspan=1>0.049 ± 0.0021</td><td rowspan=1 colspan=1>0.075 ± 0.012</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/2 layer size</td><td rowspan=1 colspan=1>0.095 ± 0.011</td><td rowspan=1 colspan=1>0.119 ± 0.010</td><td rowspan=1 colspan=1>0.055 ± 0.0049</td><td rowspan=1 colspan=1>0.074± 0.016</td></tr><tr><td rowspan=1 colspan=1>JS divergence with 1/1 layer size</td><td rowspan=1 colspan=1>0.034 ± 0.0034</td><td rowspan=1 colspan=1>0.050± 0.0026</td><td rowspan=1 colspan=1>0.027 ± 0.0028</td><td rowspan=1 colspan=1>0.025 ± 0.00076</td></tr></table>",
+        "bbox": [
+            133,
+            101,
+            864,
+            161
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3c9a4f15816f9475e20e64fe6d880999dbc71b9039d3a86102ace92afbbde1f2.jpg",
+        "image_caption": [
+            "Figure 4: Visual perception of sample quality and diversity is very similar for models trained with different numbers of unrolling steps. Actual sample diversity is higher with more unrolling steps. Each pane shows samples generated after training a model on CIFAR10 with 0, 1, 5, and 10 steps of unrolling. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In order to recover the color the GAN assigned to the digit, we used k-means with 2 clusters, to pick out the foreground color from the background. We then performed this transformation for both the training data and the generated images. Next we fit a Gaussian kernel density estimator to both distributions over digit colors. Finally, we computed the JS divergence between the model and data distributions over colors. Results can be found in Table 2 for several model sizes. Details of the models are provided in Appendix F. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In general, the best performing models are unrolled for 5-10 steps, and larger models perform better than smaller models. Counter-intuitively, taking 1 unrolling step seems to hurt this measure of diversity. We suspect that this is due to it introducing oscillatory dynamics into training. Taking more unrolling steps however leads to improved performance with unrolling. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.4 IMAGE MODELING OF CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            722,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here we test our technique on a more traditional convolutional GAN architecture and task, similar to those used in (Radford et al., 2015; Salimans et al., 2016). In the previous experiments we tested models where the standard GAN training algorithm would not converge. In this section we improve a standard model by reducing its tendency to engage in mode collapse. We ran 4 configurations of this model, varying the number of unrolling steps to be 0, 1, 5, or 10. Each configuration was run 5 times with different random seeds. For full training details see Appendix D. Samples from each of the 4 configurations can be found in Figure 4. There is no obvious difference in visual quality across these model configurations. Visual inspection however provides only a poor measure of sample diversity. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "By training with an unrolled discriminator, we expect to generate more diverse samples which more closely resemble the underlying data distribution. We introduce two techniques to examine sample diversity: inference via optimization, and pairwise distance distributions. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b50d528ebc1d7e4f4ff566671309621016069f0eaa4be9ec232cad39b3046c6b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Unrolling Steps</td><td rowspan=1 colspan=1>0 steps</td><td rowspan=1 colspan=1>1 step</td><td rowspan=1 colspan=1>5 steps</td><td rowspan=1 colspan=1>10 steps</td></tr><tr><td rowspan=1 colspan=1>Average MSE</td><td rowspan=1 colspan=1>0.0231± 0.0024</td><td rowspan=1 colspan=1>0.0195 ± 0.0021</td><td rowspan=1 colspan=1>0.0200± 0.0023</td><td rowspan=1 colspan=1>0.0181± 0.0018</td></tr><tr><td rowspan=1 colspan=1>PercentBestRank</td><td rowspan=1 colspan=1>0.63%</td><td rowspan=1 colspan=1>22.97%</td><td rowspan=1 colspan=1>15.31%</td><td rowspan=1 colspan=1>61.09 %</td></tr></table>",
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 3: GANs trained with unrolling are better able to match images in the training set than standard GANs, likely due to mode dropping by the standard GAN. Results show the MSE between training images and the best reconstruction for a model with the given number of unrolling steps. The fraction of training images best reconstructed by a given model is given in the final column. The best reconstructions is found by optimizing the latent representation $z$ to produce the closest matching pixel output $G \\left( z ; \\theta _ { G } \\right)$ . Results are averaged over all 5 runs of each model with different random seeds. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.4.1 INFERENCE VIA OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            277,
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+            291
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Since likelihood cannot be tractably computed, over-fitting of GANs is typically tested by taking samples and computing the nearest-neighbor images in pixel space from the training data (Goodfellow et al., 2014). We will do the reverse, and measure the ability of the generative model to generate images that look like specific samples from the training data. If we did this by generating random samples from the model, we would need an exponentially large number of samples. We instead treat finding the nearest neighbor $x _ { \\mathrm { n e a r e s t } }$ to a target image $x _ { \\mathrm { { t a r g e t } } }$ as an optimization task, ",
+        "bbox": [
+            174,
+            300,
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ebe230db022f5349155b2c7b8f508d347322a1d01e98e07a32852a61e22205d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z _ { \\mathrm { n e a r e s t } } = \\underset { z } { \\mathrm { a r g m i n } } | | G ( z ; \\theta _ { G } ) - x _ { \\mathrm { t a r g e t } } | \\rvert _ { 2 } ^ { 2 } } \\\\ & { x _ { \\mathrm { n e a r e s t } } = G ( z _ { \\mathrm { n e a r e s t } } ; \\theta _ { G } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            391,
+            632,
+            440
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This concept of backpropagating to generate images has been widely used in visualizing features from discriminative networks (Simonyan et al., 2013; Yosinski et al., 2015; Nguyen et al., 2016) and has been applied to explore the visual manifold of GANs in (Zhu et al., 2016). ",
+        "bbox": [
+            174,
+            450,
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+            492
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We apply this technique to each of the models trained. We optimize with 3 random starts using LBFGS, which is the optimizer typically used in similar settings such as style transfer (Johnson et al., 2016; Champandard, 2016). Results comparing average mean squared errors between xnearest and $x _ { \\mathrm { { t a r g e t } } }$ in pixel space can be found in Table 3. In addition we compute the percent of images for which a certain configuration achieves the lowest loss when compared to the other configurations. ",
+        "bbox": [
+            174,
+            500,
+            825,
+            569
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the zero step case, there is poor reconstruction and less than $1 \\%$ of the time does it obtain the lowest error of the 4 configurations. Taking 1 unrolling step results in a significant improvement in MSE. Taking 10 unrolling steps results in more modest improvement, but continues to reduce the reconstruction MSE. ",
+        "bbox": [
+            174,
+            575,
+            825,
+            632
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To visually see this, we compare the result of the optimization process for 0, 1, 5, and 10 step configurations in Figure 5. To select for images where differences in behavior is most apparent, we sort the data by the absolute value of a fractional difference in MSE between the 0 and 10 step models, $\\left| \\frac { l _ { 0 s t e p } - l _ { 1 0 s t e p } } { \\frac { 1 } { 2 } ( l _ { 0 s t e p } + l _ { 1 0 s t e p } ) } \\right|$ This highlights examples where either the 0 or 10 step model cannot accurately fit the data example but the other can. In Appendix G we show the same comparison for models initialized using different random seeds. Many of the zero step images are fuzzy and illdefined suggesting that these images cannot be generated by the standard GAN generative model, and come from a dropped mode. As more unrolling steps are added, the outlines become more clear and well defined – the model covers more of the distribution and thus can recreate these samples. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.4.2 PAIRWISE DISTANCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            787,
+            380,
+            803
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A second complementary approach is to compare statistics of data samples to the corresponding statistics for samples generated by the various models. One particularly simple and relevant statistic is the distribution over pairwise distances between random pairs of samples. In the case of mode collapse, greater probability mass will be concentrated in smaller volumes, and the distribution over inter-sample distances should be skewed towards smaller distances. We sample random pairs of images from each model, as well as from the training data, and compute histograms of the $\\ell _ { 2 }$ distances between those sample pairs. As illustrated in Figure 6, the standard GAN, with zero unrolling steps, has its probability mass skewed towards smaller $\\ell _ { 2 }$ intersample distances, compared to real data. As the number of unrolling steps is increased, the histograms over intersample distances increasingly come to resemble that for the data distribution. This is further evidence in support of unrolling decreasing the mode collapse behavior of GANs. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/3f34af532391d651dea1658c3e0b24559b5a1f84c026a63aecddf941edff8580.jpg",
+        "image_caption": [
+            "Figure 5: Training set images are more accurately reconstructed using GANs trained with unrolling than by a standard (0 step) GAN, likely due to mode dropping by the standard GAN. Raw data is on the left, and the optimized images to reach this target follow for 0, 1, 5, and 10 unrolling steps. The reconstruction MSE is listed below each sample. A random 1280 images where selected from the training set, and corresponding best reconstructions for each model were found via optimization. Shown here are the eight images with the largest absolute fractional difference between GANs trained with 0 and 10 unrolling steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work we developed a method to stabilize GAN training and reduce mode collapse by defining the generator objective with respect to unrolled optimization of the discriminator. We then demonstrated the application of this method to several tasks, where it either rescued unstable training, or reduced the tendency of the model to drop regions of the data distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The main drawback to this method is computational cost of each training step, which increases linearly with the number of unrolling steps. There is a tradeoff between better approximating the true generator loss and the computation required to make this estimate. Depending on the architecture, one unrolling step can be enough. In other more unstable models, such as the RNN case, more are needed to stabilize training. We have some initial positive results suggesting it may be sufficient to further perturb the training gradient in the same direction that a single unrolling step perturbs it. While this is more computationally efficient, further investigation is required. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The method presented here bridges some of the gap between theoretical and practical results for training of GANs. We believe developing better update rules for the generator and discriminator is an important line of work for GAN training. In this work we have only considered a small fraction of the design space. For instance, the approach could be extended to unroll $G$ when updating $D$ as well – letting the discriminator react to how the generator would move. It is also possible to unroll sequences of $G$ and $D$ updates. This would make updates that are recursive: $G$ could react to maximize performance as if $G$ and $D$ had already updated. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Laurent Dinh, David Dohan, Vincent Dumoulin, Liam Fedus, Ishaan Gulrajani, Julian Ibarz, Eric Jang, Matthew Johnson, Marc Lanctot, Augustus Odena, Gabriel Pereyra, ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/34368461772c640ab500044ba570ada5248c204871939191a4efbed64c4bf4ee.jpg",
+        "image_caption": [
+            "Figure 6: As the number of unrolling steps in GAN training is increased, the distribution of pairwise distances between model samples more closely resembles the same distribution for the data. Here we plot histograms of pairwise distances between randomly selected samples. The red line gives pairwise distances in the data, while each of the five blue lines in each plot represents a model trained with a different random seed. The vertical lines are the medians of each distribution. "
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+    {
+        "type": "text",
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+        "page_idx": 10
+    },
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+        "bbox": [
+            173,
+            746,
+            823,
+            776
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. ",
+        "bbox": [
+            173,
+            784,
+            821,
+            814
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Philipp Krahenb ¨ uhl, Eli Shechtman, and Alexei A. Efros. Generative visual manipula- ¨ tion on the natural image manifold. In Proceedings of European Conference on Computer Vision (ECCV), 2016. ",
+        "bbox": [
+            173,
+            821,
+            825,
+            864
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Appendix ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            316,
+            127
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A 2D GAUSSIAN TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            148,
+            500,
+            166
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Network architecture and experimental details for the experiment in Section 3.1 are as follows: ",
+        "bbox": [
+            173,
+            180,
+            794,
+            195
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The dataset is sampled from a mixture of 8 Gaussians of standard deviation 0.02. The means are equally spaced around a circle of radius 2. ",
+        "bbox": [
+            173,
+            202,
+            823,
+            229
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The generator network consists of a fully connected network with 2 hidden layers of size 128 with relu activations followed by a linear projection to 2 dimensions. All weights are initialized to be orthogonal with scaling of 0.8. ",
+        "bbox": [
+            174,
+            237,
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+            279
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network first scales its input down by a factor of 4 (to roughly scale to (-1,1)), followed by 1 layer fully connected network with relu activations to a linear layer to of size 1 to act as the logit. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The generator minimizes $\\mathcal { L } _ { G } = \\log ( D ( x ) ) + \\log ( 1 - D ( G ( z ) ) )$ and the discriminator minimizes $\\mathcal { L } _ { D } \\stackrel { = } { = } - \\log ( D ( x ) ) - \\log ( 1 - D ( \\stackrel { . } { G } ( z ) ) )$ where $\\mathbf { X }$ is sampled from the data distribution and $z \\sim$ $\\mathcal { N } ( 0 , I _ { 2 5 6 } )$ . Both networks are optimized using Adam (Kingma & Ba, 2014) with a learning rate of 1e-4 and $\\beta _ { 1 } { = } 0 . 5$ . ",
+        "bbox": [
+            174,
+            334,
+            825,
+            391
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The network is trained by alternating updates of the generator and the discriminator. One step consists of either $\\mathbf { G }$ or $\\mathbf { D }$ updating. ",
+        "bbox": [
+            174,
+            398,
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+            426
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B MORE MIXTURE OF GAUSSIAN EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            445,
+            594,
+            462
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.1 EFFECTS OF TIME DELAY / HISTORICAL AVERAGING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            477,
+            576,
+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Another comparison we looked at was with regard to historical averaging based approaches. Recently similarly inspired approaches have been used in (Salimans et al., 2016) to stabilize training. For our study, we looked at taking an ensemble of discriminators over time. ",
+        "bbox": [
+            174,
+            502,
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+            545
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "First, we looked at taking an ensemble of the last N steps, as shown in Figure App.1. ",
+        "bbox": [
+            174,
+            551,
+            725,
+            568
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca00dc71a894afed5412e7d9c9476916ab9ab1f2648dedea7d529b310373bd67.jpg",
+        "image_caption": [
+            "Figure App.1: Historical averaging does not visibly increase stability on the mixture of Gaussians task. Each row corresponds to an ensemble of discriminators which consists of the indicated number of immediately preceding discriminators. The columns correspond to different numbers of training steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            582,
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+            720
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To further explore this idea, we ran experiments with an ensemble of 5 discriminators, but with different periods between replacing discriminators in the ensemble. For example, if I sample at a rate of 100, it would take 500 steps to replace all 5 discriminators. Results can be seen in Figure App.2. ",
+        "bbox": [
+            174,
+            805,
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+            861
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We observe that given longer and longer time delays, the model becomes less and less stable. We hypothesize that this is due to the initial shape of the discriminator loss surface. When training, the discriminator’s estimates of probability densities are only accurate on regions where it was trained. When fixing this discriminator, we are removing the feedback between the generator exploitation and the discriminators ability to move. As a result, the generator is able to exploit these fixed areas of poor performance for older discriminators in the ensemble. New discriminators (over)compensate for this, leading the system to diverge. ",
+        "bbox": [
+            174,
+            867,
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+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/04adbd77c06eb06d8623be8ab50fe44136a6a0398a13d8fdd2a583c617dbcff0.jpg",
+        "image_caption": [
+            "Figure App.2: Introducing longer time delays between the discriminator ensemble results in instability and probability distributions that are not in the window being visualized. The $\\mathbf { X }$ axis is the number of weight updates and the y axis is how many steps to skip between discriminator updates when selecting the ensemble of 5 discriminators. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            102,
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+            242
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            342,
+            823,
+            383
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.2 EFFECTS OF THE SECOND GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            402,
+            470,
+            416
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A second factor we analyzed is the effect of backpropagating the learning signal through the unrolling in Equation 12. We can turn on or off this backpropagation through the unrolling by introducing stop gradient calls into our computation graph between each unrolling step. With the stop gradient in place, the update signal corresponds only to the first term in Equation 12. We looked at 3 configurations: without stop gradients; vanilla unrolled GAN, with stop gradients; and with stop gradients but taking the average over the $k$ unrolling steps instead of taking the final value. Results can be see in Figure App.3. ",
+        "bbox": [
+            174,
+            429,
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+            526
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We initially observed no difference between unrolling with and without the second gradient, as both required 3 unrolling steps to become stable. When the discriminator is unrolled to convergence, the second gradient term becomes zero. Due to the simplicity of the problem, we suspect that the discriminator nearly converged for every generator step, and the second gradient term was thus irrelevant. ",
+        "bbox": [
+            174,
+            532,
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+            603
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To test this, we modified the dynamics to perform five generator steps for each discriminator update. Results are shown in Figure App.4. With the discriminator now kept out of equilibrium, successful training can be achieved with half as many unrolling steps when using both terms in the gradient than when only including the first term. ",
+        "bbox": [
+            174,
+            609,
+            823,
+            665
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C RNN MNIST TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            688,
+            493,
+            704
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The network architecture for the experiment in Section 3.2 is as follows: ",
+        "bbox": [
+            176,
+            720,
+            647,
+            734
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The MNIST dataset is scaled to [-1, 1). ",
+        "bbox": [
+            176,
+            742,
+            429,
+            756
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The generator first scales the 256D noise vector through a 256 unit fully connected layer with relu activation. This is then fed into the initial state of a 256D LSTM(Hochreiter & Schmidhuber, 1997) that runs 28 steps corresponding to the number of columns in MNIST. The resulting sequence of activations is projected through a fully connected layer with 28 outputs with a tanh activation function. All weights are initialized via the ”Xavier” initialization (Glorot & Bengio, 2010). The forget bias on the LSTM is initialized to 1. ",
+        "bbox": [
+            174,
+            762,
+            825,
+            847
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network feeds the input into a Convolution(16, stride $^ { = 2 }$ ) followed by a Convolution(32, stride $^ { = 2 }$ ) followed by Convolution(32, stride ${ \\boldsymbol { \\mathbf { \\mathit { \\varepsilon } } } } = 2 { \\boldsymbol { \\mathbf { \\mathit { \\varepsilon } } } }$ ). All convolutions have stride 2. As in (Radford et al., 2015) leaky rectifiers are used with a 0.3 leak. Batch normalization is applied after each layer (Ioffe & Szegedy, 2015). The resulting 4D tensor is then flattened and a linear projection is performed to a single scalar. ",
+        "bbox": [
+            176,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e082e1fa15fe3c8c65693a9ca8c06d3ede7c2ddcf2d0287c257abe4e8c2ff9b6.jpg",
+        "image_caption": [
+            "Figure App.3: If the discriminator remains nearly at its optimum during learning, then performance is nearly identical with and without the second gradient term in Equation 12. As shown in Figure App.4, when the discriminator lags behind the generator, backpropagating through unrolling aids convergence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            97,
+            825,
+            535
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The generator network minimises $\\mathcal { L } _ { G } = \\log ( D ( G ( z ) ) )$ and the discriminator minimizes $\\mathcal { L } _ { D } =$ $\\log ( D ( x ) ) + \\log ( 1 - D ( G ( z ) ) )$ . Both networks are trained with Adam(Kingma & Ba, 2014) with learning rates of 1e-4 and $\\beta _ { 1 } { = } 0 . 5$ . The network is trained alternating updating the generator and the discriminator for 150k steps. One step consists of just 1 network update. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            694
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D CIFAR10/MNIST TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            720,
+            534,
+            738
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The network architectures for the discriminator, generator, and encoder as as follows. All convolutions have a kernel size of 3x3 with batch normalization and leaky ReLU’s with a 0.3 leak. ",
+        "bbox": [
+            176,
+            757,
+            823,
+            786
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The generator network is defined as: ",
+        "bbox": [
+            176,
+            792,
+            413,
+            808
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/6be1d7b077edb834c2bd6ee68d568dbb574dbb6746300a22e384363f25abe428.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z ~ N(0,I256)Fully connectedReshape to image 4,4,512ConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4*4*512256128641or3</td><td rowspan=1 colspan=1>２２２１</td></tr></table>",
+        "bbox": [
+            174,
+            814,
+            542,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ddba8731c308e072b5532fd2f2dfbcf72fd2fc3c5d92dcb601c2c03608beee85.jpg",
+        "image_caption": [
+            "Unrolled GAN with 5 G Steps per D without second gradient "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
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+            316
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/21b4b254a8e79c21939f50e66bbaf743c06f0c33a79d176b714a375db90c6857.jpg",
+        "table_caption": [
+            "Unrolled GAN with 5 G Steps per D "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x~ Pdata or G Transposed Convolution Transposed Convolution Transposed Convolution</td><td>64 128 256</td><td>２２2</td></tr><tr><td>Flatten</td><td></td><td></td></tr><tr><td>Fully Connected</td><td>1</td><td></td></tr></table>",
+        "bbox": [
+            174,
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+            862
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5b3fa4bd5bb790af426480883819ba983aa6618ec11502817b7cc7fa39e22d5.jpg",
+        "image_caption": [
+            "Figure App.4: Backpropagating through the unrolling process aids convergence when the discriminator does not fully converge between generator updates. When taking 5 generator steps per discriminator step unrolling greatly increases stability, requiring only 5 unrolling steps to converge. Without the second gradient it requires 10 unrolling steps. Also see Figure App.3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            338,
+            821,
+            536
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network is defined as: ",
+        "bbox": [
+            174,
+            743,
+            439,
+            757
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The generator network minimises $\\mathcal { L } _ { G } = \\log ( D ( G ( z ) ) )$ and the discriminator minimizes $\\mathcal { L } _ { D } =$ $\\log ( D ( x ) ) + \\log ( 1 - D ( G ( z ) ) )$ . The networks are trained with Adam with a generator learning rate of 1e-4, and a discriminator learning rate of 2e-4. The network is trained alternating updating the generator and the discriminator for $1 0 0 \\mathrm { k }$ steps. One step consists of just 1 network update. ",
+        "bbox": [
+            173,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "E 1000 CLASS MNIST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            380,
+            118
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/6d161339c120af4e827f0777703e338fce4dd94b3a55a7a2c9761991b51c1f31.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z ~ N(0,I256)Fully connectedReshape to image 4,4,64ConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4 *4*64321683</td><td rowspan=1 colspan=1>２２２１</td></tr></table>",
+        "bbox": [
+            173,
+            136,
+            532,
+            250
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network is parametrized by a size $\\mathrm { X }$ and is defined as follows. In our tests, we used X of 1/4 and 1/2. ",
+        "bbox": [
+            171,
+            255,
+            826,
+            284
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/d861a766290ac6cab3787aa8042ed8fa20fa08a94a861ed7ebddde0d0efa7d41.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x ~ Pdata or G Transposed Convolution Transposed Convolution</td><td>8*X 16*X</td><td>２２2</td></tr><tr><td>Transposed Convolution Flatten</td><td>32*X</td><td></td></tr><tr><td>Fully Connected</td><td>1</td><td></td></tr></table>",
+        "bbox": [
+            174,
+            287,
+            531,
+            388
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "F COLORED MNIST DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            411,
+            439,
+            429
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "F.1 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            449,
+            279,
+            463
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "To generate this dataset we first took the mnist digit, $I$ , scaled between 0 and 1. For each image we sample a color, $C$ , normally distributed with mean $\\scriptstyle = 0$ and std $\\scriptstyle \\mathtt { = 0 . 5 }$ . To generate a colored digit between (-1, 1) we do $I * C + \\mathsf { \\bar { ( } } I - 1 )$ . Finally, we add a small amount of pixel independent noise sampled from a normal distribution with std $= 0 . 2$ , and the resulting values are cliped between (-1, 1). When visualized, this generates images and samples that can be seen in figure App.5. Once again it is very hard to visually see differences in sample diversity when comparing the 128 and the 512 sized models. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/44d318f018c78dbdaec0c5211309ab172baeb975108ebe77f750982a4193cef6.jpg",
+        "image_caption": [
+            "Figure App.5: Right: samples from the data distribution. Middle: Samples from 1/4 size model with 0 look ahead steps (worst diversity). Left: Samples from 1/1 size model with 10 look ahead steps (most diversity). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            593,
+            795,
+            746
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "F.2 MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            844,
+            276,
+            859
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The models used in this section are parametrized by a variable $\\mathrm { X }$ to control capacity. A value of ${ \\bf X } { = } 1$ is same architecture used in the cifar10 experiments. We used 1/4, 1/2 and 1 as these values. ",
+        "bbox": [
+            176,
+            873,
+            825,
+            902
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The generator network is defined as: ",
+        "bbox": [
+            176,
+            909,
+            413,
+            924
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/de1a5ae153f5aca675e1d81f0274ce2bb52042972834a41845eb58fa8a6854f0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>number outputs</td><td rowspan=1 colspan=1>stride</td></tr><tr><td rowspan=1 colspan=1>Input: z~ N(0,I256)Fully connectedReshape to image 4,4,512*XConvolutionConvolutionConvolutionConvolution</td><td rowspan=1 colspan=1>4*4*512*X256*X128*X64*X3</td><td rowspan=1 colspan=1>2221</td></tr></table>",
+        "bbox": [
+            174,
+            101,
+            562,
+            215
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network is defined as: ",
+        "bbox": [
+            176,
+            220,
+            439,
+            234
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/8185cc38813b0aac909bf7ec1d0d173e742b2ff60e4a2f788d6a7b905fcd642a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>number outputs</td><td>stride</td></tr><tr><td>Input: x ~ Pdata or G Transposed Convolution Transposed Convolution Transposed Convolution Flatten Fully Connected</td><td>64*X 128*X 256*X</td><td>２２2</td></tr></table>",
+        "bbox": [
+            174,
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+    {
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+        "text": "G OPTIMIZATION BASED VISUALIZATIONS ",
+        "text_level": 1,
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+            547,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "More examples of model based optimization. We performed 5 runs with different seeds of each of of the unrolling steps configuration. Bellow are comparisons for each run index. Ideally this would be a many to many comparison, but for space efficiency we grouped the runs by the index in which they were run. ",
+        "bbox": [
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+        "img_path": "images/8d38e597124cc936be0c8530d6fb7e8b1ecc41f5e50351124daf12a2d78e11fa.jpg",
+        "image_caption": [
+            "Figure App.6: Samples from 1/5 with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            821,
+            883
+        ],
+        "page_idx": 20
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+        "type": "image",
+        "img_path": "images/be94e2bb107541e788983b4ad2bdc31bebdff50fbbb8529c915238f8cca809e8.jpg",
+        "image_caption": [
+            "Figure App.7: Samples from 2/5 with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            113,
+            820,
+            881
+        ],
+        "page_idx": 21
+    },
+    {
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+        "img_path": "images/e17957e3a6c6fffe5b489567eb4830a1720a2ae85eadaf27b1b6017a82527e8b.jpg",
+        "image_caption": [
+            "Figure App.8: Samples from 3/5 with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            112,
+            820,
+            880
+        ],
+        "page_idx": 22
+    },
+    {
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+        "img_path": "images/fbca68637cf3798a1a739daff1548046fba7251170be090a9d1d669d0aa5603e.jpg",
+        "image_caption": [
+            "Figure App.9: Samples from 4/5 with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            117,
+            820,
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+        ],
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+    {
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+        "img_path": "images/c67273dbe4f3a1a6aee81a0d0854850226af640cdd34215ec9e0663f82c11a7d.jpg",
+        "image_caption": [
+            "Figure App.10: Samples from 5/5 with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            112,
+            820,
+            881
+        ],
+        "page_idx": 24
+    }
+]
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diff --git a/parse/train/BydrOIcle/BydrOIcle_middle.json b/parse/train/BydrOIcle/BydrOIcle_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..07177144a4c5bec38051aaf0e358bcfb9b9bcf93
--- /dev/null
+++ b/parse/train/BydrOIcle/BydrOIcle_middle.json
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+                                    "score": 1.0,
+                                    "content": "responding to. This extra information helps the generator spread its mass to make the next",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "optimization this is known to lead to convergence for games in which gradient descent (ascent) fails",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "(Zhang & Lesser, 2010). However, the motivation for using the surrogate generator loss in Section",
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+                                    "content": "2.2, of unrolling the inner of two nested min and max functions, does not apply to using a surrogate",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "discriminator loss. Additionally, it is more common for the discriminator to overpower the generator",
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+                                    "score": 1.0,
+                                    "content": "than vice-versa when training a GAN. Giving more information to",
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+                                    "score": 1.0,
+                                    "content": "by allowing it to ‘see into the",
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+                            "index": 22
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+                        {
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+                                    "score": 1.0,
+                                    "content": "future’ may thus help the two models be more balanced.",
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+                            "index": 23
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+                    ],
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+                                    "content": "3 EXPERIMENTS",
+                                    "type": "text"
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+                            ],
+                            "index": 24
+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "five datasets of increasing complexity. Evaluation of generative models is a notoriously hard problem",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "(Theis et al., 2016). As such the de facto standard in GAN literature has become sample quality as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "evaluated by a human and/or evaluated by a heuristic (Inception score for example, (Salimans et al.,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "2016)). While these evaluation metrics do a reasonable job capturing sample quality, they fail to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "capture sample diversity. In our first 2 experiments diversity is easily evaluated via visual inspection.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In our later experiments this is not the case, and we will use a variety of methods to quantify coverage",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "of samples. Our measures are individually strongly suggestive of unrolling reducing mode-collapse",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and improving stability, but none of them alone are conclusive. We believe that taken together",
+                                    "type": "text"
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+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "however, they provide extremely compelling evidence for the advantages of unrolling.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
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+                    ],
+                    "index": 29.5
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+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "When doing stochastic optimization, we must choose which minibatches to use in the unrolling",
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+                            ],
+                            "index": 35
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "updates in Eq. 7. We experimented with both a fixed minibatch and re-sampled minibatches for",
+                                    "type": "text"
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+                            "index": 36
+                        },
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+                                    "score": 1.0,
+                                    "content": "each unrolling step, and found it did not significantly impact the result. We use fixed minibatches",
+                                    "type": "text"
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+                            ],
+                            "index": 37
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for all experiments in this section.",
+                                    "type": "text"
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+                            ],
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+                        }
+                    ],
+                    "index": 36.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We provide a reference implementation of this technique at github.com/poolio/unrolled gan.",
+                                    "type": "text"
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+                            ],
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+                    ],
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+                                    "content": "3.1 MIXTURE OF GAUSSIANS DATASET",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        }
+                    ],
+                    "index": 40
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+                                    "score": 1.0,
+                                    "content": "To illustrate the impact of discriminator unrolling, we train a simple GAN architecture on a 2D",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "mixture of 8 Gaussians arranged in a circle. For a detailed list of architecture and hyperparameters",
+                                    "type": "text"
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+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "see Appendix A. Figure 2 shows the dynamics of this model through time. Without unrolling the",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "generator rotates around the valid modes of the data distribution but is never able to spread out",
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+                                    "score": 0.37,
+                                    "content": "\\mathbf { G }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "quickly learns to spread probability mass and the system",
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+                                    "score": 1.0,
+                                    "content": "converges to the data distribution.",
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+                                    "score": 1.0,
+                                    "content": ". In the case of 1-step unrolled",
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+                                    "score": 1.0,
+                                    "content": "optimization this is known to lead to convergence for games in which gradient descent (ascent) fails",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "(Zhang & Lesser, 2010). However, the motivation for using the surrogate generator loss in Section",
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+                                    "score": 1.0,
+                                    "content": "than vice-versa when training a GAN. Giving more information to",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "future’ may thus help the two models be more balanced.",
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+                                    "score": 1.0,
+                                    "content": "(Theis et al., 2016). As such the de facto standard in GAN literature has become sample quality as",
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+                                    "score": 1.0,
+                                    "content": "evaluated by a human and/or evaluated by a heuristic (Inception score for example, (Salimans et al.,",
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+                            "index": 28
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "2016)). While these evaluation metrics do a reasonable job capturing sample quality, they fail to",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "capture sample diversity. In our first 2 experiments diversity is easily evaluated via visual inspection.",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In our later experiments this is not the case, and we will use a variety of methods to quantify coverage",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "of samples. Our measures are individually strongly suggestive of unrolling reducing mode-collapse",
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+                                    "score": 1.0,
+                                    "content": "and improving stability, but none of them alone are conclusive. We believe that taken together",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "however, they provide extremely compelling evidence for the advantages of unrolling.",
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+                    ],
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+                                    "content": "updates in Eq. 7. We experimented with both a fixed minibatch and re-sampled minibatches for",
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+                                    "score": 1.0,
+                                    "content": "each unrolling step, and found it did not significantly impact the result. We use fixed minibatches",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for all experiments in this section.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "We provide a reference implementation of this technique at github.com/poolio/unrolled gan.",
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+                    ],
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+                                    "content": "3.1 MIXTURE OF GAUSSIANS DATASET",
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+                                    "content": "To illustrate the impact of discriminator unrolling, we train a simple GAN architecture on a 2D",
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+                                    "score": 1.0,
+                                    "content": "mixture of 8 Gaussians arranged in a circle. For a detailed list of architecture and hyperparameters",
+                                    "type": "text"
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+                            "index": 42
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+                        {
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+                                    "score": 1.0,
+                                    "content": "see Appendix A. Figure 2 shows the dynamics of this model through time. Without unrolling the",
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+                                    "score": 0.37,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "To evaluate the ability of this approach to improve trainability, we look to a traditionally challenging",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "task, resulting in a more complex power balance. Results can be seen in Figure 3. Once again,",
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+                                    "content": "to one from (Che et al., 2016). We construct a dataset by stacking three randomly chosen MNIST",
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+                                    "content": "folds. To get at this quantity, we constructed a dataset consisting of colored MNIST digits. Unlike",
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+                                    "content": "in the previous experiment, a single MNIST digit was chosen, and then assigned a single monochro-",
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+                                    "content": "generate the digits. We use colored MNIST digits so that the generator also has to model the digits,",
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+                                    "score": 1.0,
+                                    "content": "which makes the task sufficiently complex that the generator is unable to perfectly solve it. The",
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+                                    "content": "color of each digit is sampled from a 3D normal distribution. Details of this dataset are provided",
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+                                    "score": 1.0,
+                                    "content": "in Appendix F. We will examine the distribution of colors in the samples generated by the trained",
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+                                    "content": "ated colors is almost invisible using only visual inspection of the samples. Samples can be found in",
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+                                    "score": 1.0,
+                                    "content": "The reverse KL divergence between model and data is also given. Standard error is provided for",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "both measures.",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "using artificially constructed datasets, and demonstrate that unrolling can largely rescue both types",
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+                                    "score": 1.0,
+                                    "content": "of collapse.",
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "content": "to one from (Che et al., 2016). We construct a dataset by stacking three randomly chosen MNIST",
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+                                    "content": "digits, so as to construct an RGB image with a different MNIST digit in each color channel. This",
+                                    "type": "text"
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+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "new dataset has 1,000 distinct modes, corresponding to each combination of the ten MNIST classes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "in the three channels.",
+                                    "type": "text"
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+                            "index": 18
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "We train a GAN on this dataset, and generate samples from the trained model (25,600 samples for all",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
+                                    "content": "experiments). We then compute the predicted class label of each color channel using a pre-trained",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "MNIST classifier. To evaluate performance, we use two metrics: the number of modes for which the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "generator produced at least one sample, and the KL divergence between the model and the expected",
+                                    "type": "text"
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+                            ],
+                            "index": 22
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+                        {
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+                                    "score": 1.0,
+                                    "content": "data distribution. Within this discrete label space, a KL divergence can be estimated tractably be-",
+                                    "type": "text"
+                                }
+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "tween the generated samples and the data distribution over classes, where the data distribution is a",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "uniform distribution over all 1,000 classes.",
+                                    "type": "text"
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+                            "index": 25
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "As presented in Table 1, as the number of unrolling steps is increased, both mode coverage and re-",
+                                    "type": "text"
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+                            ],
+                            "index": 26
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+                                    "score": 1.0,
+                                    "content": "verse KL divergence improve. Contrary to (Che et al., 2016), we found that reasonably sized models",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "(such as the one used in Section 3.4) covered all 1,000 modes even without unrolling. As such we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "use smaller convolutional GAN models. Details on the models used are provided in Appendix E.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        }
+                    ],
+                    "index": 27.5,
+                    "bbox_fs": [
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+                                    "score": 1.0,
+                                    "content": "We observe an additional interesting effect in this experiment. The benefits of unrolling increase",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "as the discriminator size is reduced. We believe unrolling effectively increases the capacity of the",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "discriminator. The unrolled discriminator can better react to any specific way in which the generator",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "is producing non-data-like samples. When the discriminator is weak, the positive impact of unrolling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
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+                        {
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+                                    "score": 1.0,
+                                    "content": "is thus larger.",
+                                    "type": "text"
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+                            "index": 34
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+                                    "content": "3.3.2 MANIFOLD COLLAPSE",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "folds. To get at this quantity, we constructed a dataset consisting of colored MNIST digits. Unlike",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "in the previous experiment, a single MNIST digit was chosen, and then assigned a single monochro-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "matic color. With a perfect generator, one should be able to recover the distribution of colors used to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "generate the digits. We use colored MNIST digits so that the generator also has to model the digits,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "which makes the task sufficiently complex that the generator is unable to perfectly solve it. The",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "color of each digit is sampled from a 3D normal distribution. Details of this dataset are provided",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "in Appendix F. We will examine the distribution of colors in the samples generated by the trained",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
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+                                    "content": "In order to recover the color the GAN assigned to the digit, we used k-means with 2 clusters, to",
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+                                    "content": "pick out the foreground color from the background. We then performed this transformation for both",
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+                                    "content": "distributions over digit colors. Finally, we computed the JS divergence between the model and data",
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+                                    "content": "distributions over colors. Results can be found in Table 2 for several model sizes. Details of the",
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+                                    "content": "models are provided in Appendix F.",
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+                                    "score": 1.0,
+                                    "content": "Here we test our technique on a more traditional convolutional GAN architecture and task, similar",
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+                                    "score": 1.0,
+                                    "content": "to those used in (Radford et al., 2015; Salimans et al., 2016). In the previous experiments we tested",
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+                                    "content": "models where the standard GAN training algorithm would not converge. In this section we improve",
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+                                    "score": 1.0,
+                                    "content": "a standard model by reducing its tendency to engage in mode collapse. We ran 4 configurations of",
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+                                    "score": 1.0,
+                                    "content": "this model, varying the number of unrolling steps to be 0, 1, 5, or 10. Each configuration was run 5",
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+                            "index": 29
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+                                    "score": 1.0,
+                                    "content": "times with different random seeds. For full training details see Appendix D. Samples from each of",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the 4 configurations can be found in Figure 4. There is no obvious difference in visual quality across",
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+                                    "score": 1.0,
+                                    "content": "these model configurations. Visual inspection however provides only a poor measure of sample",
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+                                    "score": 1.0,
+                                    "content": "Here we test our technique on a more traditional convolutional GAN architecture and task, similar",
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+                                    "content": "to those used in (Radford et al., 2015; Salimans et al., 2016). In the previous experiments we tested",
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+                                    "content": "models where the standard GAN training algorithm would not converge. In this section we improve",
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+                                    "content": "a standard model by reducing its tendency to engage in mode collapse. We ran 4 configurations of",
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+                                    "score": 1.0,
+                                    "content": "this model, varying the number of unrolling steps to be 0, 1, 5, or 10. Each configuration was run 5",
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+                                    "score": 1.0,
+                                    "content": "times with different random seeds. For full training details see Appendix D. Samples from each of",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the 4 configurations can be found in Figure 4. There is no obvious difference in visual quality across",
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+                                    "score": 1.0,
+                                    "content": "these model configurations. Visual inspection however provides only a poor measure of sample",
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+                                    "score": 1.0,
+                                    "content": "low et al., 2014). We will do the reverse, and measure the ability of the generative model to generate",
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+                                    "content": "et al., 2016; Champandard, 2016). Results comparing average mean squared errors between xnearest",
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+                                    "score": 1.0,
+                                    "content": "in pixel space can be found in Table 3. In addition we compute the percent of images for",
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+                                    "score": 1.0,
+                                    "content": "of the time does it obtain the",
+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        },
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+                                    "score": 1.0,
+                                    "content": "lowest error of the 4 configurations. Taking 1 unrolling step results in a significant improvement in",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "MSE. Taking 10 unrolling steps results in more modest improvement, but continues to reduce the",
+                                    "type": "text"
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+                            "index": 30
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+                        {
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+                                    "score": 1.0,
+                                    "content": "reconstruction MSE.",
+                                    "type": "text"
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+                            ],
+                            "index": 31
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+                                    "score": 1.0,
+                                    "content": "To visually see this, we compare the result of the optimization process for 0, 1, 5, and 10 step",
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+                                    "content": "configurations in Figure 5. To select for images where differences in behavior is most apparent,",
+                                    "type": "text"
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+                                    "content": "models,",
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+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "\\left| \\frac { l _ { 0 s t e p } - l _ { 1 0 s t e p } } { \\frac { 1 } { 2 } ( l _ { 0 s t e p } + l _ { 1 0 s t e p } ) } \\right|",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "This highlights examples where either the 0 or 10 step model cannot",
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+                                    "score": 1.0,
+                                    "content": "accurately fit the data example but the other can. In Appendix G we show the same comparison for",
+                                    "type": "text"
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+                            ],
+                            "index": 36
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "models initialized using different random seeds. Many of the zero step images are fuzzy and ill-",
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+                                    "score": 1.0,
+                                    "content": "defined suggesting that these images cannot be generated by the standard GAN generative model,",
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+                            "index": 38
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+                                    "score": 1.0,
+                                    "content": "and come from a dropped mode. As more unrolling steps are added, the outlines become more clear",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and well defined – the model covers more of the distribution and thus can recreate these samples.",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "over inter-sample distances should be skewed towards smaller distances. We sample random pairs",
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+                                    "score": 1.0,
+                                    "content": "of images from each model, as well as from the training data, and compute histograms of the",
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+                                    "score": 0.85,
+                                    "content": "\\ell _ { 2 }",
+                                    "type": "inline_equation"
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+                            "index": 47
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+                                    "score": 1.0,
+                                    "content": "distances between those sample pairs. As illustrated in Figure 6, the standard GAN, with zero",
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+                                    "content": "Table 3: GANs trained with unrolling are better able to match images in the training set than",
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+                                    "content": "The fraction of training images best reconstructed by a given model is given in the final column.",
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+                                    "score": 1.0,
+                                    "content": "The best reconstructions is found by optimizing the latent representation",
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+                                    "content": "z",
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+                                    "content": "to produce the closest",
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+                                    "score": 1.0,
+                                    "content": "matching pixel output",
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+                                    "bbox": [
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+                                    "score": 0.94,
+                                    "content": "G \\left( z ; \\theta _ { G } \\right)",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Results are averaged over all 5 runs of each model with different",
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+                                    "content": "3.4.1 INFERENCE VIA OPTIMIZATION",
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+                                    "score": 1.0,
+                                    "content": "low et al., 2014). We will do the reverse, and measure the ability of the generative model to generate",
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+                                    "content": "images that look like specific samples from the training data. If we did this by generating random",
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+                                    "score": 1.0,
+                                    "content": "samples from the model, we would need an exponentially large number of samples. We instead treat",
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+                                    "content": "This concept of backpropagating to generate images has been widely used in visualizing features",
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+                            ],
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "We apply this technique to each of the models trained. We optimize with 3 random starts using",
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+                                    "content": "LBFGS, which is the optimizer typically used in similar settings such as style transfer (Johnson",
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+                                    "score": 1.0,
+                                    "content": "et al., 2016; Champandard, 2016). Results comparing average mean squared errors between xnearest",
+                                    "type": "text"
+                                }
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+                                    "content": "and",
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+                                    "score": 0.83,
+                                    "content": "x _ { \\mathrm { { t a r g e t } } }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "in pixel space can be found in Table 3. In addition we compute the percent of images for",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "which a certain configuration achieves the lowest loss when compared to the other configurations.",
+                                    "type": "text"
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+                        }
+                    ],
+                    "index": 25,
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+                                    "score": 1.0,
+                                    "content": "In the zero step case, there is poor reconstruction and less than",
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+                                    "score": 1.0,
+                                    "content": "lowest error of the 4 configurations. Taking 1 unrolling step results in a significant improvement in",
+                                    "type": "text"
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+                                    "content": "MSE. Taking 10 unrolling steps results in more modest improvement, but continues to reduce the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "reconstruction MSE.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "To visually see this, we compare the result of the optimization process for 0, 1, 5, and 10 step",
+                                    "type": "text"
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+                            ],
+                            "index": 32
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "configurations in Figure 5. To select for images where differences in behavior is most apparent,",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "content": "we sort the data by the absolute value of a fractional difference in MSE between the 0 and 10 step",
+                                    "type": "text"
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+                                    "content": "models,",
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+                                {
+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "\\left| \\frac { l _ { 0 s t e p } - l _ { 1 0 s t e p } } { \\frac { 1 } { 2 } ( l _ { 0 s t e p } + l _ { 1 0 s t e p } ) } \\right|",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "This highlights examples where either the 0 or 10 step model cannot",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "accurately fit the data example but the other can. In Appendix G we show the same comparison for",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "models initialized using different random seeds. Many of the zero step images are fuzzy and ill-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "defined suggesting that these images cannot be generated by the standard GAN generative model,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and come from a dropped mode. As more unrolling steps are added, the outlines become more clear",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "and well defined – the model covers more of the distribution and thus can recreate these samples.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "3.4.2 PAIRWISE DISTANCES",
+                                    "type": "text"
+                                }
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+                            "index": 41
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+                    ],
+                    "index": 41
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "A second complementary approach is to compare statistics of data samples to the corresponding",
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+                                    "content": "statistics for samples generated by the various models. One particularly simple and relevant statistic",
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+                                    "score": 1.0,
+                                    "content": "is the distribution over pairwise distances between random pairs of samples. In the case of mode",
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+                                            "content": "the training set, and corresponding best reconstructions for each model were found via optimiza-",
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+                                    "content": "4 DISCUSSION",
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+                                    "content": "In this work we developed a method to stabilize GAN training and reduce mode collapse by defining",
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+                                    "content": "the generator objective with respect to unrolled optimization of the discriminator. We then demon-",
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+                                    "score": 1.0,
+                                    "content": "strated the application of this method to several tasks, where it either rescued unstable training, or",
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+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "reduced the tendency of the model to drop regions of the data distribution.",
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+                    ],
+                    "index": 15.5
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+                                    "content": "The main drawback to this method is computational cost of each training step, which increases",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "linearly with the number of unrolling steps. There is a tradeoff between better approximating the true",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "generator loss and the computation required to make this estimate. Depending on the architecture,",
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+                                    "score": 1.0,
+                                    "content": "to further perturb the training gradient in the same direction that a single unrolling step perturbs it.",
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+                                    "content": "as well – letting the discriminator react to how the generator would move. It is also possible to",
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+                                    "score": 1.0,
+                                    "content": "rate of 100, it would take 500 steps to replace all 5 discriminators. Results can be seen in Figure",
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+                                    "content": "Results are shown in Figure App.4. With the discriminator now kept out of equilibrium, successful",
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+                                    "content": "( G _ { m } ^ { ( i ) } , G _ { M } ^ { ( i ) } )",
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+                        },
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "entire joint space of events (the lattice). However, neither is factorization of a conditional probability",
+                                    "type": "text"
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+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "table using a logistic sigmoid link function, which is commonly used for the similar tasks. Our",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "approach retains the inductive bias of the original box model, is equivalent in the limit, and satisfies",
+                                    "type": "text"
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+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the necessary condition that",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.93,
+                                    "content": "p ( \\mathbf { x } , \\mathbf { x } ) = p ( \\mathbf { x } )",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ". A comparison of the 3 different functions is given in",
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+                        },
+                        {
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+                                    "content": "Figure 3, with the softplus overlap showing much better behavior for highly disjoint boxes than the",
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+                                    "score": 1.0,
+                                    "content": "Gaussian model, while also preserving the meet property.",
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+                                    "score": 1.0,
+                                    "content": "parison with existing results we use the exact same dataset from Vilnis et al. (2018). In this case, we",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "ratings to make sure that counting statistics are significant enough. This leads to 8545 movies in our",
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+                                    "score": 1.0,
+                                    "content": "We thank Travis Wolfe, Colin Evans, Rob Zinkov, Ben Poole, and Laurent Dinh for helpful dis-",
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diff --git a/parse/train/J28lNO4p3ki/J28lNO4p3ki.md b/parse/train/J28lNO4p3ki/J28lNO4p3ki.md
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+# STEM: A Stochastic Two-Sided Momentum Algorithm Achieving Near-Optimal Sample and Communication Complexities for Federated Learning
+
+Prashant Khanduri University of Minnesota khand095@umn.edu
+
+Pranay Sharma Carnegie Mellon University pranaysh@andrew.cmu.edu
+
+Haibo Yang The Ohio State University yang.5952@buckeyemail.osu.edu
+
+Mingyi Hong⇤ University of Minnesota mhong@umn.edu
+
+Jia Liu The Ohio State University liu@ece.osu.edu
+
+Ketan Rajawat Indian Institute of Technology Kanpur ketan@iitk.ac.in
+
+Pramod K. Varshney Syracuse University varshney@syr.edu
+
+# Abstract
+
+Federated Learning (FL) refers to the paradigm where multiple worker nodes (WNs) build a joint model by using local data. Despite extensive research, for a generic non-convex FL problem, it is not clear, how to choose the WNs’ and the server’s update directions, the minibatch sizes, and the number of local updates, so that the WNs use the minimum number of samples and communication rounds to achieve the desired solution. This work addresses the above question and considers a class of stochastic algorithms where the WNs perform a few local updates before communication. We show that when both the WN’s and the server’s directions are chosen based on certain stochastic momentum estimator, the algorithm requires $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ samples and $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ communication rounds to compute an $\epsilon$ -stationary solution. To the best of our knowledge, this is the first FL algorithm that achieves such near-optimal sample and communication complexities simultaneously. Further, we show that there is a trade-off curve between the number of local updates and the minibatch sizes, on which the above sample and communication complexities can be maintained. Finally, we show that for the classical FedAvg (a.k.a. Local SGD, which is a momentum-less special case of the STEM), a similar trade-off curve exists, albeit with worse sample and communication complexities. Our insights on this trade-off provides guidelines for choosing the four important design elements for FL algorithms, the number of local updates, WNs’ and server’s update directions, and minibatch sizes to achieve the best performance.
+
+# 1 Introduction
+
+In Federated Learning (FL), multiple worker nodes (WNs) collaborate with the goal of learning a joint model, by only using local data. Therefore it has become popular for machine learning problems where datasets are massively distributed [1]. In FL, the data is often collected at or off-loaded to multiple WNs which in collaboration with a server node (SN) jointly aim to learn a centralized model [2, 3]. The local WNs share the computational load and since the data is local to each WN, FL also provides some level of data privacy $\mathbb { \lVert \rVert }$ . A classical distributed optimization problem that $K$ WNs aim to solve:
+
+![](images/8b109c70673d05d4fb4863253d56729a5c2b37b6a73924b30a5e660d7b6d1078.jpg)  
+Figure 1: The 3D surface in (a) plots the communication complexity of the proposed STEM for different minibatch sizes and number of local updates. The surface is generated such that each point represents STEM with a particular choice of $( b , I )$ , so that it requires $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ samples to achieve $\epsilon$ -stationarity. Plot (b) shows the optimal trade off between the minibatch sizes and the number of local updates at each WN (i.e., achieving the lowest communication and sample complexities). Both plots are generated for an accuracy of $\epsilon = 1 0 ^ { - 3 }$ and all the constants dependent on system parameters (variance of stochastic gradients, heterogeneity parameter, optimality gap, Lipschitz constants, etc.) are assumed to be 1. Fed STEM is a special case of STEM where $\mathcal { O } ( 1 )$ minibatch is used; Minibatch STEM is a special case of STEM where $\mathcal { O } ( 1 )$ local updates are used.
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } \bigg \{ f ( x ) : = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } f ^ { ( k ) } ( x ) : = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { E } _ { \xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) } } \big [ f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) \big ] \bigg \} .
+$$
+
+where $f ^ { ( k ) } : \mathbb { R } ^ { d }  \mathbb { R }$ denotes the smooth (possibly non-convex) objective function and $\xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) }$ represents the sample/s drawn from distribution $\mathcal { D } ^ { ( k ) }$ at the $k ^ { \mathrm { { t h } } }$ WN with $k \in [ K ]$ . When the distributions $\mathcal { D } ^ { ( k ) }$ are different across the WNs, it is referred to as the heterogeneous data setting.
+
+The optimization performance of non-convex FL algorithms is typically measured by the total number of samples accessed (cf. Definition $2 . 2 )$ and the total rounds of communication (cf. Definition $2 . 3 )$ required by each WN to achieve an $\epsilon$ -stationary solution (cf. Definition $2 . 1 )$ . To minimize the sample and the communication complexities, FL algorithms rely on the following four key design elements: (i) the WNs’ local model update directions, (ii) Minibatch size to compute each local direction, (iii) the number of local updates before WNs share their parameters, and (iv) the SN’s update direction. How to find effective FL algorithms by (optimally) designing these parameters has received significant research interest recently.
+
+Contributions. The main contributions of this work are listed below:
+
+1) We propose the Stochastic Two-Sided Momentum (STEM) algorithm, that utilizes certain momentum-assisted stochastic gradient directions for both the WNs and SN updates. We show that there exists an optimal trade off between the minibatch sizes and number of local updates, such that on the trade-off curve STEM requires $\underline { { \tilde { \mathcal { O } } } } ( \epsilon ^ { - 3 / 2 } ) \underline { { \left[ \frac { 2 } { } \right] } }$ samples and $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ communication rounds to reach an $\epsilon$ -stationary solution; see Figure $\bigstar$ for an illustration. These complexity results are the best achievable for first-order stochastic FL algorithms (under certain assumptions, cf. Assumption $\bigstar \bigstar$ ; see $\pm \boxed { 5 } \boxed { 8 } \parallel$ and $\mathbb { B } \mathbb { n o }$ , as well as Remark $\checkmark$ of this paper for discussions regarding optimality. To the best of our knowledge, STEM is the first algorithm which – (i) simultaneously achieves the optimal sample and communication complexities for FL and (ii) can optimally trade off the minibatch sizes and the number of local updates.
+
+2) A momentum-less special case of our STEM result further reveals some interesting insights of the classical FedAvg algorithm (a.k.a. the Local SGD) [11–13]. Specifically, we show that for FedAvg, there also exists a trade-off between the minibatch sizes and the number of local updates, such that it requires $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples and $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ communication rounds to achieve an $\epsilon$ -stationary solution.
+
+<table><tr><td>Algorithm</td><td>Work</td><td>Sample</td><td>Comm.</td><td>Minibatch (b)</td><td>Local Updates (I) /round</td></tr><tr><td>FedAvg</td><td>国园 国国</td><td>0(€-2)</td><td>0(c-3/2) 0(c-2)</td><td>0(1) 0(1) 2(1-v)</td><td>0(c-1/2) 0(1) 3v</td></tr><tr><td>SCAFFOLD*</td><td>this work 国</td><td>0(c-2）</td><td>O(e-3/2) 0（c-2）</td><td>O(c 4-v) 0(1)</td><td>O(c−2（4-D)） 0(1)</td></tr><tr><td>FedPD/FedProx*</td><td>四/□</td><td>O(c-2）</td><td>0(e-1)</td><td>0(1)</td><td>0(e-1）</td></tr><tr><td>MIME†/FedGLOMO</td><td>/8</td><td>0(c-3/2)</td><td>O(€-3/2)</td><td>0(1)</td><td>0(1)</td></tr><tr><td>STEM Fed STEM Minibatch STEM*</td><td> this work</td><td>O(€-3/2)</td><td>O(e-1)</td><td>( 0(1) O(e-1/2)</td><td>O(∈−(3)) O(∈-1/2) 0(1)</td></tr></table>
+
+Table 1: Comparison of FedAvg and STEM with different FL algorithms for various choices of the minibatch sizes $( b )$ and the number of per node local updates between two rounds of communication $( I )$ . $^ \circ \nu \in [ 0 , 1 ]$ trades off $^ { b }$ and $I$ ; $\nu = 1$ (resp. $\nu = 0$ ) uses multiple (resp. $\mathcal { O } ( 1 ) .$ ) local updates and $\mathcal { O } ( 1 )$ (resp. multiple) samples. Fed STEM and Minibatch STEM are two variants of the proposed STEM. ‡The data heterogeneity assumption is weaker than Assumption $2$ (please see $\bigstar \bigstar$ for details). †Requires bounded Hessian dissimilarity to model data heterogeneity across WNs. ⇤Guarantees for Minibatch STEM with $I = 1$ and SCAFFOLD are independent of the data heterogeneity.
+
+Collectively, our insights on the trade-offs provide practical guidelines for choosing different design elements for FL algorithms.
+
+Related Works. FL algorithms were first proposed in the form of FedAvg [11], where the local update directions at each WN were chosen to be the SGD updates. Earlier works analyzed these algorithms in the homogeneous data setting [19–25], while many recent studies have focused on designing new algorithms to deal with heterogeneous data settings, as well as problems where the local loss functions are non-convex [9, 10, 12–16, 18, 26–32]. In $\bar { \mathbb { E } 2 } \mathbb { I }$ , the authors showed that Parallel Restarted SGD (Local SGD or FedAvg [11]) achieves linear speed up while requiring $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples and $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ rounds of communication to reach an $\epsilon$ -stationary solution. In $[ \bar { \lVert { 4 } } ]$ , a Momentum SGD was proposed, which achieved the same sample and communication complexities as Parallel Restarted SGD $\mathbb { \lVert 1 2 \rVert }$ , without requiring that the second moments of the gradients be bounded. Further, it was shown that under the homogeneous data setting, the communication complexity can be improved to $\mathcal { O } ( \epsilon ^ { - 1 } )$ while maintaining the same sample complexity. The works in $\boxed { 1 5 } , \boxed { 1 6 }$ conducted tighter analysis for FedAvg with partial WN participation with $\mathcal { O } ( 1 )$ local updates and batch sizes. Their analysis showed that FedAvg’s sample and communication complexities are both $\mathcal { O } ( \epsilon ^ { - 2 } )$ . Additionally, SCAFFOLD was proposed in $\bar { \| 1 5 \| }$ , which utilized variance reduction based local update directions $\pmb { \mathbb { B 3 } }$ to achieve the same sample and communication complexities as FedAvg. Similarly, VRL-SGD proposed in $\left[ \left[ 2 9 \right] \right]$ also utilized variance reduction and showed improved communication complexity of $\mathcal { O } ( \epsilon ^ { - 1 } )$ , while requiring the same computations as FedAvg. Importantly, both SCAFFOLD and VRL-SGD’s guarantees were independent of the data heterogeneity. The FedProx proposed in $\mathbb { \ m }$ used a penalty based method to improve the communication complexity of FedAvg (i.e., the Parallel Restarted and Momentum SGD [14, $\mathbb { L } 2 \mathbb { I }$ ) to $\mathcal { O } ( \epsilon ^ { - 1 } )$ . FedProx used a gradient similarity assumption to model data heterogeneity which can be stringent for many practical applications. This assumption was relaxed by FedPD proposed in $\pmb { \mathbb { B } }$ .
+
+Recently, the works [17, 18] proposed to utilize hybrid momentum gradient estimators [7, 8]. The MIME algorithm $\mathbb { \lVert 1 7 \rVert }$ matched the optimal sample complexity (under certain smoothness assumptions) of $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ of the centralized non-convex stochastic optimization algorithms [5–8]. Similarly, Fed-GLOMO $[ \overline { { 1 8 } } ]$ achieved the same sample complexity while employing compression to further reduce communication. Both MIME and Fed-GLOMO required $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ communication rounds to achieve an $\epsilon$ -stationary solution. Please see Table $^ 1$ for a summary of the above discussion.
+
+The comparison of Local SGD (FedAvg) to Minibatch SGD for convex and strongly convex problems with homogeneous data setting was first conducted in $\mathbb { \lVert 1 9 \rVert }$ and later extended to heterogeneous setting in $\mathbb { \lVert \rVert 3 \rVert }$ . It was shown that Minibatch SGD almost always dominates the Local SGD. In contrast, it was shown in $\pmb { \Vert 2 4 \Vert }$ that Local SGD dominates Minibatch SGD in terms of generalization performance. Although existing FL results are rich, but they are somehow ad hoc and there is a lack of principled understanding of the algorithms. We note that the proposed STEM algorithmic framework provides a theoretical framework that unifies all existing $\mathrm { F L }$ results on sample and communication complexities.
+
+Notations. The expected value of a random variable $X$ is denoted by $\mathbb { E } [ X ]$ and its expectation conditioned on an Event $A$ is denoted as $\mathbb { E } [ X | \mathrm { E v e n t ~ } A ]$ . We denote by $\mathbb { R }$ (and $\mathbb { R } ^ { d }$ ) the real line (and the $d$ -dimensional Euclidean space). The set of natural numbers is denoted by $\mathbb { N }$ . Given a positive integer $K \in \mathbb N$ , we denote $[ K ] \triangleq \{ 1 , 2 , \dots , K \}$ . Notation $\| \cdot \|$ denotes the $\ell _ { 2 }$ -norm and $\langle \cdot , \cdot \rangle$ the Euclidean inner product. For a discrete set $\boldsymbol { B }$ , $| B |$ denotes the cardinality of the set. Uniform distribution over a discrete set $\{ 1 , \ldots , T \}$ is denoted as ${ \dot { \mathcal { U } } } \{ 1 , \dots , T \}$ .
+
+# 2 Preliminaries
+
+Before we proceed to the algorithms, we make the following assumptions about problem $( 1 )$ .
+
+Assumption 1 (Sample Gradient Lipschitz Smoothness). The stochastic functions $f ^ { ( k ) } ( \cdot , \xi ^ { ( k ) } )$ with $\xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) }$ for all $k \in [ K ]$ , satisfy the mean squared smoothness property, i.e, we have
+
+$$
+\begin{array} { r } { \mathbb { E } \| \nabla f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) - \nabla f ^ { ( k ) } ( y ; \xi ^ { ( k ) } ) \| ^ { 2 } \leq L ^ { 2 } \| x - y \| ^ { 2 } \mathrm { ~ f o r ~ a l l ~ } x , y \in \mathbb { R } ^ { d } . } \end{array}
+$$
+
+Assumption 2 (Unbiased gradient and Variance Bounds). (i) Unbiased Gradient. The stochastic gradients computed at each WN are unbiased
+
+$$
+\mathbb { E } [ \nabla f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) ] = \nabla f ^ { ( k ) } ( x ) , \forall \xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) } , \forall k \in [ K ] .
+$$
+
+(ii) Intra- and inter- node Variance Bound. The following bounds hold:
+
+$$
+\begin{array} { r } { \mathbb { \tilde { z } } \| \nabla f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) - \nabla f ^ { ( k ) } ( x ) \| ^ { 2 } \leq \sigma ^ { 2 } , \| \nabla f ^ { ( k ) } ( x ) - \nabla f ^ { ( \ell ) } ( x ) \| ^ { 2 } \leq \zeta ^ { 2 } , \forall \xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) } , \forall k , \ell \in [ K ] . } \end{array}
+$$
+
+Note that Assumption $\boxed { 1 }$ is stronger than directly assuming $f ^ { ( k ) }$ ’s are Lipschitz smooth (which we will refer to as the averaged gradient Lipschitz smooth condition), but it is still a rather standard assumption in SGD analysis. For example it has been used in analyzing centralized SGD algorithms such as SPIDER $ { \mathbb { I } }$ , SNVRG $\pmb { \Vert 6 \Vert }$ , STORM $\mathbb { [ [ \big ] ] }$ (and many others) as well as in FL algorithms such as MIME $ { \mathbb { I } } ^ { [ 1 2 ] }$ and Fed-GLOMO $ { \mathbb { I } }  { \mathrm { 1 8 } }  { \Vert }$ . The second relation in Assumption $2 \cdot$ (ii) quantifies the data heterogeneity, and we call $\zeta > 0$ as the heterogeneity parameter. This is a typical assumption required to evaluate the performance of FL algorithms. If data distributions across individual WNs are identical, i.e., $\mathcal { D } ^ { ( k ) } \stackrel { = } { = } \mathcal { D } ^ { ( \ell ) }$ for all $k , \ell \in [ K ]$ then we have $\zeta = 0$ .
+
+Next, we define the $\epsilon$ -stationary solution for non-convex optimization problems, as well as quantify the computation and communication complexities to achieve an $\epsilon$ -stationary point.
+
+Definition 2.1 $\epsilon$ -Stationary Point). A point $x$ is called $\epsilon$ -stationary if $\| \nabla f ( x ) \| ^ { 2 } \leq \epsilon$ . Moreover, a stochastic algorithm is said to achieve an $\epsilon$ -stationary point in $t$ iterations if $\begin{array} { r } { \ddot { \mathbb { E } } [ \| \nabla f ( x _ { t } ) \| ^ { 2 } ] \le \epsilon . } \end{array}$ where the expectation is over the stochasticity of the algorithm until time instant $t$ .
+
+Definition 2.2 (Sample complexity). We assume an Incremental First-order Oracle (IFO) framework $\textcircled { \lVert { 3 4 } \rVert }$ where, given a sample $\xi ^ { ( k ) } \sim \mathcal { D } ^ { ( k ) }$ at the $k ^ { \mathrm { { t h } } }$ node and iterate $x$ , the oracle returns $( f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) , \nabla f ^ { ( k ) } ( x ; \xi ^ { ( k ) } ) )$ . Each access to the oracle is counted as a single IFO operation. We measure the sample (and computational) complexity in terms of the total number of calls to the IFO by all WNs to achieve an $\epsilon$ -stationary point given in Definition 2.1.
+
+Definition 2.3 (Communication complexity). We define a communication round as a one back-andforth sharing of parameters between the WNs and the SN. Then the communication complexity is defined to be the total number of communication rounds between any WN and the SN required to achieve an $\epsilon$ -stationary point given in Definition 2.1.
+
+# 3 The STEM algorithm and the trade-off analysis
+
+In this section, we discuss the proposed algorithm and present the main results. The key in the algorithm design is to carefully balance all the four design elements mentioned in Sec. $^ { 1 , }$ so that sufficient and useful progress can be made between two rounds of communication.
+
+Let us discuss the key steps of STEM, listed in Algorithm $^ { 1 . }$ In Step 10, each node locally updates its model parameters using the local direction $d _ { t } ^ { k }$ , computed by using $b$ stochastic gradients at two
+
+1: Input: Parameters: $c > 0$ , the number of local updates $I$ , batch size $b$ , stepsizes $\{ \eta _ { t } \}$ .   
+2: Initialize: Iterate $\begin{array} { r } { x _ { 1 } ^ { ( k ) } = \bar { x } _ { 1 } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } x _ { 1 } ^ { ( k ) } } \end{array}$ , descent direction $\begin{array} { r } { d _ { 1 } ^ { ( k ) } = \bar { d } _ { 1 } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } d _ { 1 } ^ { ( k ) } } \end{array}$   
+with $\begin{array} { r } { d _ { 1 } ^ { ( k ) } = \frac { 1 } { B } \sum _ { \xi _ { 1 } ^ { ( k ) } \in \mathcal { B } _ { 1 } ^ { ( k ) } } \nabla f ^ { ( k ) } ( x _ { 1 } ^ { ( k ) } ; \xi _ { 1 } ^ { ( k ) } ) } \end{array}$ and $| B _ { 1 } ^ { ( k ) } | = B$ for $k \in [ K ]$ .   
+3: Perform: $x _ { 2 } ^ { ( k ) } = x _ { 1 } ^ { k } - \eta _ { 1 } d _ { 1 } ^ { ( k ) }$ , $\forall k \in [ K ]$   
+4: for $t = 1$ to $T$ do   
+5: for $k = 1$ to $K$ do #at the WN   
+6: $\mathcal { d } _ { t + 1 } ^ { ( k ) } = \frac { 1 } { b } \sum _ { \xi _ { t + 1 } ^ { ( k ) } \in \mathcal { B } _ { t + 1 } ^ { ( k ) } } ^ { \pi \Delta } \nabla f ^ { ( k ) } ( x _ { t + 1 } ^ { ( k ) } ; \xi _ { t + 1 } ^ { ( k ) } ) + \left( 1 - a _ { t + 1 } \right) \bigg ( d _ { t } ^ { ( k ) } - \frac { 1 } { b } \sum _ { \xi _ { t + 1 } ^ { ( k ) } \in \mathcal { B } _ { t + 1 } ^ { ( k ) } } \nabla f ^ { ( k ) } ( x _ { t } ^ { ( k ) } ; \xi _ { t + 1 } ^ { ( k ) } ) \bigg )$   
+where we choose $| B _ { t + 1 } ^ { ( k ) } | = b$ , and $a _ { t + 1 } = c \cdot \eta _ { t } ^ { 2 }$ ;   
+7: 8: if $t$ $I = 0$ #at the SN   
+$\begin{array} { r } { d _ { t + 1 } ^ { ( k ) } = \bar { d } _ { t + 1 } : = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } d _ { t + 1 } ^ { ( k ) } } \end{array}$   
+9: 10: e $\begin{array} { r l } & { x _ { t + 2 } ^ { ( k ) ^ { \bot } } : = \bar { x } _ { t + 1 } - \eta _ { t + 1 } \bar { d } _ { t + 1 } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } x _ { t + 1 } ^ { ( k ) } - \eta _ { t + 1 } \bar { d } _ { t + 1 } } \end{array}$ $x _ { t + 2 } ^ { ( k ) } = x _ { t + 1 } ^ { ( k ) } - \eta _ { t + 1 } d _ { t + 1 } ^ { ( k ) }$ #server-side momentum   
+11: end if   
+12: end for   
+13: end for   
+14: Return: $\scriptstyle { \bar { x } } _ { a }$ where $a \sim \mathcal { U } \{ 1 , . . . , T \}$ .
+
+consecutive iterates $x _ { t + 1 } ^ { ( k ) }$ and $x _ { t } ^ { ( k ) }$ . After every $I$ local steps, the WNs share their current local models $\{ x _ { t + 1 } ^ { ( k ) } \} _ { k = 1 } ^ { K }$ and directions $\{ d _ { t + 1 } ^ { ( k ) } \} _ { k = 1 } ^ { K }$ with the SN. The SN aggregates these quantities, and performs a server-side momentum step, before returning $\bar { x } _ { t + 1 }$ and $\bar { d } _ { t + 1 }$ to all the WNs. Because both the WNs and the SN perform momentum based updates, we call the algorithm a stochastic two-sided momentum algorithm. The key parameters are: $b$ the minibatch size, $I$ the local update steps between two communication rounds, $\eta _ { t }$ the stepsizes, and $a _ { t }$ the momentum parameters.
+
+One key technical innovation of our algorithm design is to identify the most suitable way to incorporate momentum based directions in FL algorithms. Although the momentum-based gradient estimator itself is not new and has been used in the literature before (see e.g., in $\mathbb { \left[ \bigstar \bigstar \right] }$ and $\overline { { \mathbb { D } \mathbb { Z } \mathbb { G } \mathbb { S } } }$ to improve the sample complexities of centralized and decentralized stochastic optimization problems, respectively), it is by no means clear if and how it can contribute to improve the communication complexity of FL algorithms. We show that in the FL setting, the local directions together with the local models have to be aggregated by the SN so to avoid being influenced too much by the local data. More importantly, besides the WNs, the SN also needs to perform updates using the (aggregated) momentum directions. Finally, such two-sided momentum updates have to be done carefully with the correct choice of minibatch size $b$ , and the number of local updates $I$ . Overall, it is the judicious choice of all these design elements that results in the optimal sample and communication complexities.
+
+Next, we present the convergence guarantees of the STEM algorithm.
+
+# 3.1 Main results: convergence guarantees for STEM
+
+In this section, we analyze the performance of STEM. We first present our main result, and then provide discussions about a few parameter choices. In the next subsection, we discuss a special case of STEM related to the classical FedAvg and minibatch SGD algorithms.
+
+Theorem 3.1. Under the Assumptions 1 and 2, suppose the stepsize sequence is chosen as:
+
+$$
+\eta _ { t } = \frac { \bar { \kappa } } { ( w _ { t } + \sigma ^ { 2 } t ) ^ { 1 / 3 } } ,
+$$
+
+where we define :
+
+$$
+\bar { \kappa } = \frac { ( b K ) ^ { 2 / 3 } \sigma ^ { 2 / 3 } } { L } , \quad w _ { t } = \operatorname * { m a x } \bigg \{ 2 \sigma ^ { 2 } , 4 0 9 6 L ^ { 3 } I ^ { 3 } \bar { \kappa } ^ { 3 } - \sigma ^ { 2 } t , \frac { c ^ { 3 } \bar { \kappa } ^ { 3 } } { 4 0 9 6 L ^ { 3 } I ^ { 3 } } \bigg \} .
+$$
+
+Further, let us set $\begin{array} { r } { c = \frac { 6 4 L ^ { 2 } } { b K } + \frac { \sigma ^ { 2 } } { 2 4 \bar { \kappa } ^ { 3 } L I } = L ^ { 2 } \bigg ( \frac { 6 4 } { b K } + \frac { 1 } { 2 4 ( b K ) ^ { 2 } I } \bigg ) } \end{array}$ and set the initial batch size as $B = b I$ ; set the local updates $I$ and minibatch size b as follows:
+
+$$
+I = \mathcal { O } \big ( ( T / K ^ { 2 } ) ^ { \nu / 3 } \big ) , \quad b = \mathcal { O } \big ( ( T / K ^ { 2 } ) ^ { 1 / 2 - \nu / 2 } \big )
+$$
+
+where $\nu$ satisfies $\nu \in [ 0 , 1 ]$ . Then for STEM the following holds:
+
+(i) For $\scriptstyle { \bar { x } } _ { a }$ chosen according to Algorithm $\boldsymbol { l } ,$ we have:
+
+$$
+\mathbb { E } \Vert \nabla f ( \bar { x } _ { a } ) \Vert ^ { 2 } = \mathcal { O } \Bigg ( \frac { f ( \bar { x } _ { 1 } ) - f ^ { * } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \Bigg ) + \tilde { \mathcal { O } } \Bigg ( \frac { \sigma ^ { 2 } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \Bigg ) + \tilde { \mathcal { O } } \Bigg ( \frac { \zeta ^ { 2 } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \Bigg ) .
+$$
+
+(ii) For any $\nu \in [ 0 , 1 ]$ , we have
+
+Sample Complexity: The sample complexity of STEM is $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ . This implies that each WN requires at most $\tilde { \mathcal { O } } ( K ^ { - 1 } \epsilon ^ { - 3 / 2 } )$ gradient computations, thereby achieving linear speedup with the number of WNs present in the network.
+
+Communication Complexity: The communication complexity of STEM is $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ .
+
+The proof of this result is relegated to the Supplemental Material. A few remarks are in order.
+
+Remark 1 (Near-Optimal sample and communication complexities). Theorem $3 . 1$ suggests that when $I$ and $b$ are selected appropriately, then STEM achieves $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ and $\tilde { \mathcal { O } } ( \overline { { \epsilon ^ { - 1 } } } )$ sample and communication complexities. Taking them separately, these complexity bounds are the best achievable by the existing FL algorithms (upto logarithmic factors regardless of sample or batch Lipschitz smooth assumption) $[ [ 3 5 ] ]$ ; see Table $1 .$ We note that the $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ complexity is the best possible that can be achieved by centralized SGD with the sample Lipschitz gradient assumption; see $ { \mathbb { I } } ^ {  { \left[ 5 \right] } }$ . On the other hand, the $\bar { \mathcal { O } } ( \epsilon ^ { - 1 } )$ complexity bound is also likely to be the optimal, since in $\mathbb { P }$ the authors showed that even when the local steps use a class of (deterministic) first-order algorithms, $\mathcal { O } ( \epsilon ^ { - 1 } )$ is the best achievable communication complexity. The only difference is that $\bigstar \bigstar$ does not explicitly assume the inter-node variance bound (i.e., the second relation in Assumption $2 \cdot$ -(ii)). We leave the precise characterization of the communication lower bound with inter-node variance as future work. □
+
+Remark 2 (Large Batch Sizes and/or Local Updates). At first glance, it may seem that the requirement of STEM to compute large mini-batches and/or local updates (cf. Table $^ { 1 ) }$ to achieve this (near) optimal performance is a drawback, however, we note that it is in fact an advantage of STEM that it allows the WNs to perform larger number of local updates (or compute large minibatches) without communicating often. This follows from the fact that irrespective of the number of local updates (or batch sizes) STEM achieves near-optimal communication complexity while attaining optimal overall sample complexity. Moreover, note that even with $b = I = \mathcal { O } ( 1 )$ (i.e., $b$ and $I$ are chosen as constants), STEM achieves the same (optimal) sample and communication complexities as achieved by FedGLOMO [18] and MIME [17]. We further note that to the best of our knowledge the algorithms that achieve the communication complexity of $\mathcal { O } ( \epsilon ^ { - 1 } )$ either require the number of local updates or the batch-sizes that depend on the solution accuracy $\epsilon$ . For example, FedProx $\mathbb { I O } ]$ , FedPD $\bigstar \bigstar$ , and FedDyn $\pmb { \Vert 3 6 \Vert }$ rely on solving the “local problems" to achieve an $\epsilon$ -accuracy, which implies that the number of local updates (or the batch sizes) implicitly depends on the desired solution accuracy $\epsilon$ , as is the case for STEM. Similarly, as shown in $\bar { \mathbb { E } 2 } \mathbb { I }$ and $\bar { \textregistered 4 } \bar { 1 }$ the communication complexity of FedAvg and its momentum version can be improved from $\mathcal { O } ( \epsilon ^ { - 2 } )$ to $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ when the number of local updates (or batch size) is chosen as $\mathcal { O } ( \epsilon ^ { - 1 / 2 } )$ (cf. Section $3 . 2$ for a more detailed discussion).
+
+Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter $\nu \in [ 0 , 1 ]$ is used to balance the local minibatch sizes $b$ , and the number of local updates $I$ . Eqs. in $\textcircled { 3 }$ suggest that when $\nu$ increases from 0 to 1, $b$ decreases and $I$ increases. Specifically, if $\nu = 1$ , then $b$ is a constant but $I = \mathcal { O } ( T ^ { 1 / 3 } / K ^ { 2 / 3 } )$ . In this case, each WN chooses a small minibatch while executing multiple local updates, and STEM resembles a FedAvg (a.k.a. Local SGD) algorithm but with double-sided momentum update directions, and is referred to as Fed STEM. In contrast, if $\nu = 0$ , then $b = \mathcal { O } ( T ^ { 1 / 2 } / K )$ but $I$ is a constant. In this case, each WN chooses a large batch size while executing only a few, or even one, local updates, and STEM resembles the Minibatch SGD, but again with different update directions, and is referred to as Minibatch STEM. Such a trade-off can be seen in Fig. 1b. Due to space limitation, these two special cases will be precisely stated in the supplementary materials as corollaries of Theorem 3.1. □
+
+# Algorithm 2 The FedAvg Algorithm
+
+1: Input: $\{ \eta _ { t } \} _ { t = 0 } ^ { T } ; I$ , the # of local updates per communication round; $b$ , the minibatch sizes.   
+2: for $t = 1$ to $T$ do   
+3: 4: 5: for $\begin{array} { r l } & { \mathcal { \kappa } _ { t } ^ { = } \stackrel { \mathrm { ~ L ~ U ~ O ~ } \Lambda } { = } \mathbf { 0 } } \\ & { d _ { t } ^ { ( k ) } = \frac { 1 } { b } \sum _ { \xi _ { t } ^ { ( k ) } \in \mathcal { B } _ { t } ^ { ( k ) } } \nabla f ^ { ( k ) } ( x _ { t } ^ { ( k ) } ; \xi _ { t } ^ { ( k ) } ) \mathrm { ~ w i t h ~ } | \mathcal { B } _ { t } ^ { ( k ) } | = b } \\ & { x _ { t + 1 } ^ { ( k ) } = x _ { t } ^ { ( k ) } - \eta _ { t } d _ { t } ^ { ( k ) } } \\ & { \mathbf { i f } t \operatorname* { m o d } I = 0 \mathbf { \Lambda } \mathbf { t h e n } } \\ & { ~ x _ { t + 1 } ^ { ( k ) } = \bar { x } _ { t + 1 } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } x _ { t + 1 } ^ { ( k ) } } \\ & { \mathbf { e n d } \mathbf { \Phi } \mathbf { i f } } \end{array}$ $k = 1$ $K$   
+6:   
+7:   
+8:   
+9: end for   
+10: end for   
+11: Return: $\scriptstyle { \bar { x } } _ { a }$ where $a \sim \mathcal { U } \{ 1 , . . . , T \}$ .
+
+Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem $\left| \overline { { \mathbf { C . 1 0 } } } \right|$ included in the supplemental material), we can see that STEM requires $\bar { \tilde { O } } ( \operatorname* { m a x } \big \{ ( b \cdot$ $I ) \epsilon ^ { - 1 } , K ^ { - 1 } \epsilon ^ { - 3 / 2 } \rbrace )$ samples and $\tilde { \mathcal { O } } \big ( \operatorname* { m a x } \big \{ \epsilon ^ { - 1 } , ( b \cdot I ) ^ { - 1 } K ^ { - 1 } \epsilon ^ { - 3 / 2 } \big \} \big )$  and communication rounds. According to the above expressions, if $b \cdot I$ increases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \cdot I$ decreases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . For instance, if we choose $b = \mathcal { O } ( 1 )$ and $I = { \mathcal { O } } ( 1 )$ the communication complexity becomes $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\boxed { 1 \mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □
+
+Remark 5 (Data Heterogeneity). The term $\begin{array} { r } { \tilde { \mathcal { O } } \biggl ( \frac { \zeta ^ { 2 } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \biggr ) } \end{array}$ in the gradient bound $\textcircled{4}$ captures the effect of the heterogeneity of data across WNs, where $\zeta$ is the parameter characterizing the intra-node variance and has been defined in Assumption $\bigstar$ (ii). Highly heterogeneous data with large $\zeta ^ { 2 }$ can adversely impact the performance of STEM. Note that such a dependency on $\zeta$ also appears in other existing FL algorithms, such as $[ \bigcirc , \bigcirc , \bigcirc , \bigcirc , \bigcirc ]$ . However, there is one special case of STEM that does not depend on the parameter $\zeta$ . This is the case where $I = 1$ , i.e., the minibatch SGD counterpart of STEM where only a single local iteration is performed between two communication rounds. We have the following corollary. □
+
+Corollary 1 (Minibatch STEM). Under Assumptions $\bigstar \bigstar \bigstar | \bigstar |$ , and choose the algorithm parameters as in Theorem $3 . I .$ At each WN, choose $I = 1$ , $b = ( T / K ^ { 2 } ) ^ { 1 / 2 }$ , and the initial batch size $B = \boldsymbol { b } \cdot \boldsymbol { I }$ . Then STEM satisfies:
+
+(i) For $\bar { x } _ { a }$ chosen according to Algorithm $\boldsymbol { l } ,$ we have
+
+$$
+\mathbb { E } \| \nabla f ( \bar { x } _ { a } ) \| ^ { 2 } = \mathcal { O } \Big ( \frac { f ( \bar { x } _ { 1 } ) - f ^ { * } } { T } \Big ) + \tilde { \mathcal { O } } \Big ( \frac { \sigma ^ { 2 } } { T } \Big ) .
+$$
+
+(ii) Minibatch STEM achieves $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ sample and $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ communication complexity.
+
+Next, we show that FedAvg also exhibits a trade-off similar to that of STEM but with worse sample and communication complexities.
+
+# 3.2 Special cases: The FedAvg algorithm
+
+We briefly discuss another interesting special case of STEM, where the local momentum update is replaced by the conventional SGD (i.e., $a _ { t } = 1 , ~ \forall ~ t )$ , while the server does not perform the momentum update (i.e., $\bar { d } _ { t } = 0 , \forall t )$ . This is essentially the classical FedAvg algorithm, just that it balances the number of local updates $I$ and the minibatch size $b$ . We show that this algorithm also exhibits a trade-off between $b$ and $I$ and on the trade-off curve it achieves $\mathcal { O } ( \epsilon ^ { - 2 } )$ sample complexity and $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ communication complexity.
+
+<table><tr><td>Algorithm</td><td> Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>78.2</td><td>74.1</td></tr><tr><td>FedProx</td><td>79.2</td><td>74.8</td></tr><tr><td>FedDyn</td><td>68.9</td><td>66.0</td></tr><tr><td>SCAFFOLD</td><td>71.9</td><td>74.0</td></tr><tr><td>MIME</td><td>82.6</td><td>76.8</td></tr><tr><td>FedGLOMO</td><td>76.1</td><td>72.8</td></tr><tr><td> STEM</td><td>80.1</td><td>78.8</td></tr></table>
+
+(a) Mild heterogeneity, $b = 6 4$ , and $I = 7$ .
+
+<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>73.6</td><td>75.4</td></tr><tr><td>FedProx</td><td>80.0</td><td>75.2</td></tr><tr><td>FedDyn</td><td>76.1</td><td>71.3</td></tr><tr><td>SCAFFOLD</td><td>72.5</td><td>73.7</td></tr><tr><td>MIME</td><td>61.5</td><td>58.6</td></tr><tr><td>FedGLOMO</td><td>10.0</td><td>10.0</td></tr><tr><td>STEM</td><td>81.1</td><td>78.5</td></tr></table>
+
+(b) Moderate heterogeneity, $b = 8$ , and $I = 6 1$
+
+Table 2: Training and testing accuracy of different algorithms on CIFAR-10 dataset for different batch-sizes, number of local updates, and heteregeneity settings.
+
+Theorem 3.2 (The FedAvg Algorithm). Under Assumptions 1 and 2, suppose the stepsize is chosen as: $\begin{array} { r } { \eta = \sqrt { \frac { b K } { T } } } \end{array}$ ; Let us set:
+
+$$
+I = \mathcal { O } \big ( ( T / K ^ { 3 } ) ^ { \nu / 4 } \big ) , \quad b = \mathcal { O } \big ( ( T / K ^ { 3 } ) ^ { 1 / 3 - \nu / 3 } \big )
+$$
+
+where $\nu \in [ 0 , 1 ]$ is a constant. Then for FedAvg with $T \geq 8 1 L ^ { 2 } I ^ { 2 } b K$ , the following holds
+
+(i) For $\scriptstyle { \bar { x } } _ { a }$ chosen according to Algorithm $\perp$ we have
+
+$$
+\mathbb { E } \Vert \nabla f ( \bar { x } _ { a } ) \Vert ^ { 2 } = \mathcal { O } \Bigg ( \frac { f ( \bar { x } _ { 1 } ) - f ^ { * } } { K ^ { \nu / 2 } T ^ { 2 / 3 - \nu / 6 } } \Bigg ) + \mathcal { O } \Bigg ( \frac { \sigma ^ { 2 } } { K ^ { \nu / 2 } T ^ { 2 / 3 - \nu / 6 } } \Bigg ) + \mathcal { O } \Bigg ( \frac { \zeta ^ { 2 } } { K ^ { \nu / 2 } T ^ { 2 / 3 - \nu / 6 } } \Bigg ) .
+$$
+
+(ii) For any choice of $\nu \in [ 0 , 1 ]$ we have:
+
+Sample Complexity: The sample complexity of FedAvg is $\mathcal { O } ( \epsilon ^ { - 2 } )$ . This implies that each WN requires at most $\tilde { \mathcal { O } } ( K ^ { - 1 } \epsilon ^ { - 2 } )$ gradient computations, thereby achieving linear speedup with the number of WNs in the network.
+
+Communication Complexity: The communication complexity of FedAvg is $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ .
+
+Note that the requirement on $T$ being lower bounded is only relevant for theoretical purposes, a similar requirement was also imposed in $[ \mathbb { 1 4 } ]$ to prove convergence. Again, the parameter $\nu \in [ 0 , 1 ]$ in the statement of Theorem $3 . 2$ balances $I$ and $b$ at each WN while maintaining state-of-the-art sample and communication complexities; please see Table $^ 1$ for a comparison of those bounds with existing FedAvg bounds. For $\nu = 1$ , FedAvg (cf. Theorem $\textcircled { 3 . 2 }$ reduces to FedAvg proposed in [12, 14] and for $\nu = 0$ , the algorithm can be viewed as a large batch FedAvg with constant local updates [15, 16]. Note that similar to STEM, it is known that for $I = 1$ , the Minibatch SGD’s performance is independent of the heterogeneity parameter, $\zeta \equiv \mathbb { I I } 3 \mathbb { I }$ . We also point out that if Algorithm $^ 1$ uses Nesterov’s or Polyak’s momentum $[ \textcircled { 1 4 } ]$ at local WNs instead of the recursive momentum estimator we get the same guarantees as in Theorem $3 . 2 .$
+
+In summary, this section established that once the WN’s and the SN’s update directions (SGD in FedAvg and momentum based directions in STEM) are fixed, there exists a sequence of optimal choices of the number of local updates $I$ , and the batch sizes $b$ , which guarantees the best possible sample and communication complexities for the particular algorithm. The trade-off analysis presented in this section provides some useful guidelines for how to best select $b$ and $I$ in practice. Our subsequent numerical results will also verify that if $b$ or $I$ are not chosen judiciously, then the practical performance of the algorithms can degrade significantly.
+
+# 4 Numerical results
+
+In this section, we validate the proposed STEM algorithm and compare its performance with the de facto standard FedAvg [11], and the algorithms stated in Table $^ { 1 . }$ Note that instead of FedPD we include the performance comparison with FedDyn $\pmb { \mathbb { B } } 6 \|$ since they are known to be very closely related. The goal of our experiments are three-fold: (1) To show that STEM performs on par, if not better, compared to other algorithms in different heterogeneity settings, (2) there are multiple ways to reach the desired solution accuracy, one can either choose a large batch size and perform only a few local updates or select a smaller batch size and perform multiple local updates, and finally, (3) if the local updates and the batch sizes are not chosen appropriately, the WNs might need to perform excessive computations to achieve the desired solution accuracy, thereby slowing down convergence.
+
+Table 3: Training and testing accuracy on CIFAR-10 dataset for high heterogeneity, $b =$ 128 and $I = 6$ .   
+
+<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>57.6</td><td>57.1</td></tr><tr><td>FedProx</td><td>59.1</td><td>58.5</td></tr><tr><td>FedDyn</td><td>51.2</td><td>51.3</td></tr><tr><td>SCAFFOLD</td><td>53.1</td><td>54.7</td></tr><tr><td>MIME</td><td>56.1</td><td>55.1</td></tr><tr><td>FedGLOMO</td><td>56.8</td><td>56.1</td></tr><tr><td> STEM</td><td>58.5</td><td>57.4</td></tr></table>
+
+Table 4: Training and testing accuracy on Shakespeare dataset.   
+
+<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>40.1</td><td>39.2</td></tr><tr><td>FedProx</td><td>43.5</td><td>43.2</td></tr><tr><td>FedDyn</td><td>43.7</td><td>43.2</td></tr><tr><td>SCAFFOLD</td><td>40.3</td><td>41.3</td></tr><tr><td>MIME</td><td>32.1</td><td>32.1</td></tr><tr><td>FedGLOMO</td><td>40.3</td><td>40.1</td></tr><tr><td> STEM</td><td>44.5</td><td>43.8</td></tr></table>
+
+![](images/1a3649642909a08bbdd852d92b8ec00817216ffd484e8ea8263d27aa9fb977f4.jpg)  
+Figure 2: Training loss and the testing accuracy for classification on MNIST data set against the number of samples accessed at each WN for moderate heterogeneity setting with $b = 8$ .
+
+Data and Parameter Settings: We compare the algorithms for image classification tasks on CIFAR10 and MNIST data sets with 100 WNs, and for next character prediction task on Shakespeare dataset $\pmb { \Vert 3 7 } \Vert$ with 143 WNs in the network. For both CIFAR-10 and MNIST, each WN implements a two-hidden-layer convolutional neural network (CNN) architecture followed by three linear layers for CIFAR-10 and two for MNIST. For CIFAR-10 (and MNIST) datset, we consider three settings with mild, moderate and high heterogeneity. For all the three settings, the data is partitioned into disjoint sets among the WNs. In the mild heterogeneity setting, the WNs have access to partitioned data from all the classes. In the moderate (resp. high) heterogeneity setting the data is partitioned such that each WN can access data from only 5 (resp. 2) out of 10 classes. For CIFAR-10 (resp. MNIST), each WN has access to 490 (resp. 540) samples for training and 90 (resp. 80) samples for testing purposes.
+
+We also compare the performance of algorithms on a popular FL benchmarking dataset, Shakespeare dataset $ { \mathbb { I } } ^ { \smash { \sum } }$ . For this task, we adopt the settings from $\dot { \left[ \left| 1 0 \right| \right] }$ and utilize a 2-Layer LSTM network with 100 hidden units and an 8-D embedding layer at each WN. Each WN has access to 3616 samples on average, and the samples are randomly split into an $80 \%$ training set and a $20 \%$ testing set. We randomly sample 10 nodes out of 143 for the training purpose. All the experiments are implemented on a single NVIDIA Quadro RTX 5000 GPU. More details are provided in appendix.
+
+For the proposed STEM algorithm, recall that the step-size is $\eta _ { t } = \bar { \kappa } / ( w _ { t } + \sigma ^ { 2 } t ) ^ { 1 / 3 }$ with momentum parameter defined as $a _ { t } = { c } \eta _ { t } ^ { 2 }$ . The step-size is used to update the iterates while the momentum parameter is used to construct the stochastic gradient estimate (cf. Algorithm 1 and Theorem $\boxed { 3 . 1 }$ . For the experiments, we set $w _ { t } = \sigma ^ { 2 } = 1$ and $c \doteq \bar { c } / \bar { \kappa } ^ { 2 }$ and tune for $\bar { \kappa } \in [ 1 0 ^ { - 1 } , \overline { { 1 0 } } ^ { - 2 } ]$ for the CIFAR-10 dataset and for ${ \bar { \kappa } } \in \{ 1 0 ^ { 1 } , 1 0 ^ { 0 } , 1 0 ^ { - 1 } , 1 0 ^ { - 2 } \}$ for the Shakespeare dataset. For both the datasets we tune for $\bar { c }$ in the range [1, 10]. For FedProx $[ \equiv ]$ and FedDyn $\lVert \bar { 3 6 } \rVert$ we choose the regularization constant to be 0.1. The momentum parameters for FedGLOMO $\pm \textcircled { 1 8 } \textcircled { 1 }$ and MIME $\mathbb { \lVert 1 7 \rVert }$ are set based on the choices given in the respective papers. Specifically, for FedGLOMO we choose the parameter $\beta _ { k } = 0 . 2$ and design the momentum gradient using a damping factor given in Appendix A.4 of FedGLOMO $\mathbb { \left[ \left[ 8 \right] \right] }$ . Moreover, for MIME we choose the momentum parameter as 0.9. For the rest of the algorithms (including FedAvg and SCAFFOLD), the step-size is tuned from the set $\{ 1 0 ^ { 1 } , 1 0 ^ { 0 } , 1 0 ^ { - 1 } , \hat { 1 0 } ^ { - 2 } \}$ .
+
+Discussion: We evaluate the training and testing performance of STEM against multiple algorithms for different heterogeneity settings, minibatch sizes, and number of local updates. In Tables $\bigstar$ $\boxed { 2 \mathbf { b } }$ and $\bigstar ,$ we compare the training and testing accuracy of STEM to that of other algorithms on the CIFAR-10 dataset. Specific, heterogeneity settings, the choices of minibatches, and number of local updates are stated along with the tables. Note that STEM performs uniformly well under all the conditions. Moreover, note from Table $\bigstar$ that FedGLOMO diverges once the number of local updates are high. Also, note from Table $\textcircled { 3 }$ that FedProx and STEM adapt well to high heterogeneity. Finally, with the next set of experiments we emphasize the importance of choosing $b$ and $I$ carefully. In Figure $\bigstar ,$ we compare the training and testing performance of STEM, FedAvg and SCAFFOLD, against the number of samples accessed at each WN for the classification task on MNIST dataset with moderate heterogeneity. We fix $b = 8$ and conduct experiments under two settings, one with $I = 6 7$ , and the other with $I = 5 3 6$ local updates at each WN. Note that although a large number of local updates might lead to fewer communication rounds but it can make the sample complexity extremely high as is demonstrated by Figure $2 .$ For example, Figure $\bigtriangledown$ shows that to reach testing accuracy of $9 6 - 9 7 \%$ with $I = 6 7$ , STEM requires approximately $5 0 0 0 - 6 0 0 0$ samples, in contrast with $I = 5 3 6$ it requires more than 25000 samples at each WN. Similar behavior can be observed if we fix $I > 1$ and increase the local batch sizes. This implies not choosing the local updates and the batch sizes judiciously might lead to increased sample complexity. Additional experiments are included in the supplementary material to further evaluate the performance of the proposed algorithms.
+
+# Conclusion
+
+In this work, we proposed a novel algorithm STEM, for distributed stochastic non-convex optimization with applications to FL. We showed that STEM reaches an $\epsilon$ -stationary point with $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ sample complexity while achieving linear speed-up with the number of WNs. Moreover, the algorithm achieves a communication complexity of $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . We established a (optimal) trade-off that allows interpolation between varying choices of local updates and the batch sizes at each WN while maintaining (near optimal) sample and communication complexities. We showed that FedAvg (a.k.a LocalSGD) also exhibits a similar trade-off while achieving worse complexities. Our results provide guidelines to carefully choose the number of local updates, update directions, and minibatch sizes to achieve the best performance. The future directions of this work include developing lower bounds on communication complexity that establishes the tightness of the analysis conducted in this work.
+
+# Acknowledgement
+
+We thank the anonymous reviewers for their valuable comments and suggestions. The work of Prashant Khanduri and Mingyi Hong was supported by NSF grant CMMI-1727757, AFOSR grant 19RT0424 and ARO grant W911NF-19-1-0247. The work of Mingyi Hong was also supported by an IBM Faculty Research award. The work of Jia Liu has been supported in part by NSF grants CAREER CNS-2110259, CNS-2112471, CNS-2102233, CCF-2110252, ECCS-2140277, and a Google Faculty Research Award.
+
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\ No newline at end of file
diff --git a/parse/train/J28lNO4p3ki/J28lNO4p3ki_content_list.json b/parse/train/J28lNO4p3ki/J28lNO4p3ki_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..09b005298f844ff63df32ee9321440dca1cb9d2a
--- /dev/null
+++ b/parse/train/J28lNO4p3ki/J28lNO4p3ki_content_list.json
@@ -0,0 +1,1234 @@
+[
+    {
+        "type": "text",
+        "text": "STEM: A Stochastic Two-Sided Momentum Algorithm Achieving Near-Optimal Sample and Communication Complexities for Federated Learning ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Prashant Khanduri University of Minnesota khand095@umn.edu ",
+        "bbox": [
+            269,
+            251,
+            429,
+            292
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pranay Sharma Carnegie Mellon University pranaysh@andrew.cmu.edu ",
+        "bbox": [
+            529,
+            251,
+            730,
+            292
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Haibo Yang The Ohio State University yang.5952@buckeyemail.osu.edu ",
+        "bbox": [
+            186,
+            314,
+            436,
+            356
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mingyi Hong⇤ University of Minnesota mhong@umn.edu ",
+        "bbox": [
+            455,
+            314,
+            617,
+            356
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jia Liu The Ohio State University liu@ece.osu.edu ",
+        "bbox": [
+            637,
+            314,
+            812,
+            356
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ketan Rajawat Indian Institute of Technology Kanpur ketan@iitk.ac.in ",
+        "bbox": [
+            253,
+            377,
+            508,
+            419
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pramod K. Varshney Syracuse University varshney@syr.edu ",
+        "bbox": [
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+            420
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            455,
+            535,
+            472
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning (FL) refers to the paradigm where multiple worker nodes (WNs) build a joint model by using local data. Despite extensive research, for a generic non-convex FL problem, it is not clear, how to choose the WNs’ and the server’s update directions, the minibatch sizes, and the number of local updates, so that the WNs use the minimum number of samples and communication rounds to achieve the desired solution. This work addresses the above question and considers a class of stochastic algorithms where the WNs perform a few local updates before communication. We show that when both the WN’s and the server’s directions are chosen based on certain stochastic momentum estimator, the algorithm requires $\\tilde { \\mathcal O } ( \\epsilon ^ { - 3 / 2 } )$ samples and $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ communication rounds to compute an $\\epsilon$ -stationary solution. To the best of our knowledge, this is the first FL algorithm that achieves such near-optimal sample and communication complexities simultaneously. Further, we show that there is a trade-off curve between the number of local updates and the minibatch sizes, on which the above sample and communication complexities can be maintained. Finally, we show that for the classical FedAvg (a.k.a. Local SGD, which is a momentum-less special case of the STEM), a similar trade-off curve exists, albeit with worse sample and communication complexities. Our insights on this trade-off provides guidelines for choosing the four important design elements for FL algorithms, the number of local updates, WNs’ and server’s update directions, and minibatch sizes to achieve the best performance. ",
+        "bbox": [
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+            486,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In Federated Learning (FL), multiple worker nodes (WNs) collaborate with the goal of learning a joint model, by only using local data. Therefore it has become popular for machine learning problems where datasets are massively distributed [1]. In FL, the data is often collected at or off-loaded to multiple WNs which in collaboration with a server node (SN) jointly aim to learn a centralized model [2, 3]. The local WNs share the computational load and since the data is local to each WN, FL also provides some level of data privacy $\\mathbb { \\lVert \\rVert }$ . A classical distributed optimization problem that $K$ WNs aim to solve: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b109c70673d05d4fb4863253d56729a5c2b37b6a73924b30a5e660d7b6d1078.jpg",
+        "image_caption": [
+            "Figure 1: The 3D surface in (a) plots the communication complexity of the proposed STEM for different minibatch sizes and number of local updates. The surface is generated such that each point represents STEM with a particular choice of $( b , I )$ , so that it requires $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ samples to achieve $\\epsilon$ -stationarity. Plot (b) shows the optimal trade off between the minibatch sizes and the number of local updates at each WN (i.e., achieving the lowest communication and sample complexities). Both plots are generated for an accuracy of $\\epsilon = 1 0 ^ { - 3 }$ and all the constants dependent on system parameters (variance of stochastic gradients, heterogeneity parameter, optimality gap, Lipschitz constants, etc.) are assumed to be 1. Fed STEM is a special case of STEM where $\\mathcal { O } ( 1 )$ minibatch is used; Minibatch STEM is a special case of STEM where $\\mathcal { O } ( 1 )$ local updates are used. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/2805a58b3a852bb789307ddca3e7e2aea04c24beb34df4fd0d38807ed0628944.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } \\bigg \\{ f ( x ) : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } f ^ { ( k ) } ( x ) : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbb { E } _ { \\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) } } \\big [ f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) \\big ] \\bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            444,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $f ^ { ( k ) } : \\mathbb { R } ^ { d }  \\mathbb { R }$ denotes the smooth (possibly non-convex) objective function and $\\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) }$ represents the sample/s drawn from distribution $\\mathcal { D } ^ { ( k ) }$ at the $k ^ { \\mathrm { { t h } } }$ WN with $k \\in [ K ]$ . When the distributions $\\mathcal { D } ^ { ( k ) }$ are different across the WNs, it is referred to as the heterogeneous data setting. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The optimization performance of non-convex FL algorithms is typically measured by the total number of samples accessed (cf. Definition $2 . 2 )$ and the total rounds of communication (cf. Definition $2 . 3 )$ required by each WN to achieve an $\\epsilon$ -stationary solution (cf. Definition $2 . 1 )$ . To minimize the sample and the communication complexities, FL algorithms rely on the following four key design elements: (i) the WNs’ local model update directions, (ii) Minibatch size to compute each local direction, (iii) the number of local updates before WNs share their parameters, and (iv) the SN’s update direction. How to find effective FL algorithms by (optimally) designing these parameters has received significant research interest recently. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions. The main contributions of this work are listed below: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1) We propose the Stochastic Two-Sided Momentum (STEM) algorithm, that utilizes certain momentum-assisted stochastic gradient directions for both the WNs and SN updates. We show that there exists an optimal trade off between the minibatch sizes and number of local updates, such that on the trade-off curve STEM requires $\\underline { { \\tilde { \\mathcal { O } } } } ( \\epsilon ^ { - 3 / 2 } ) \\underline { { \\left[ \\frac { 2 } { } \\right] } }$ samples and $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ communication rounds to reach an $\\epsilon$ -stationary solution; see Figure $\\bigstar$ for an illustration. These complexity results are the best achievable for first-order stochastic FL algorithms (under certain assumptions, cf. Assumption $\\bigstar \\bigstar$ ; see $\\pm \\boxed { 5 } \\boxed { 8 } \\parallel$ and $\\mathbb { B } \\mathbb { n o }$ , as well as Remark $\\checkmark$ of this paper for discussions regarding optimality. To the best of our knowledge, STEM is the first algorithm which – (i) simultaneously achieves the optimal sample and communication complexities for FL and (ii) can optimally trade off the minibatch sizes and the number of local updates. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2) A momentum-less special case of our STEM result further reveals some interesting insights of the classical FedAvg algorithm (a.k.a. the Local SGD) [11–13]. Specifically, we show that for FedAvg, there also exists a trade-off between the minibatch sizes and the number of local updates, such that it requires $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ samples and $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ communication rounds to achieve an $\\epsilon$ -stationary solution. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/c128a8f8c828914de2ebd0394fe1679cfb1eb64efae4ae9cdf73ecbb43acff9d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Comparison of FedAvg and STEM with different FL algorithms for various choices of the minibatch sizes $( b )$ and the number of per node local updates between two rounds of communication $( I )$ . $^ \\circ \\nu \\in [ 0 , 1 ]$ trades off $^ { b }$ and $I$ ; $\\nu = 1$ (resp. $\\nu = 0$ ) uses multiple (resp. $\\mathcal { O } ( 1 ) .$ ) local updates and $\\mathcal { O } ( 1 )$ (resp. multiple) samples. Fed STEM and Minibatch STEM are two variants of the proposed STEM. ‡The data heterogeneity assumption is weaker than Assumption $2$ (please see $\\bigstar \\bigstar$ for details). †Requires bounded Hessian dissimilarity to model data heterogeneity across WNs. ⇤Guarantees for Minibatch STEM with $I = 1$ and SCAFFOLD are independent of the data heterogeneity. "
+        ],
+        "table_body": "<table><tr><td>Algorithm</td><td>Work</td><td>Sample</td><td>Comm.</td><td>Minibatch (b)</td><td>Local Updates (I) /round</td></tr><tr><td>FedAvg</td><td>国园 国国</td><td>0(€-2)</td><td>0(c-3/2) 0(c-2)</td><td>0(1) 0(1) 2(1-v)</td><td>0(c-1/2) 0(1) 3v</td></tr><tr><td>SCAFFOLD*</td><td>this work 国</td><td>0(c-2）</td><td>O(e-3/2) 0（c-2）</td><td>O(c 4-v) 0(1)</td><td>O(c−2（4-D)） 0(1)</td></tr><tr><td>FedPD/FedProx*</td><td>四/□</td><td>O(c-2）</td><td>0(e-1)</td><td>0(1)</td><td>0(e-1）</td></tr><tr><td>MIME†/FedGLOMO</td><td>/8</td><td>0(c-3/2)</td><td>O(€-3/2)</td><td>0(1)</td><td>0(1)</td></tr><tr><td>STEM Fed STEM Minibatch STEM*</td><td> this work</td><td>O(€-3/2)</td><td>O(e-1)</td><td>( 0(1) O(e-1/2)</td><td>O(∈−(3)) O(∈-1/2) 0(1)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Collectively, our insights on the trade-offs provide practical guidelines for choosing different design elements for FL algorithms. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Related Works. FL algorithms were first proposed in the form of FedAvg [11], where the local update directions at each WN were chosen to be the SGD updates. Earlier works analyzed these algorithms in the homogeneous data setting [19–25], while many recent studies have focused on designing new algorithms to deal with heterogeneous data settings, as well as problems where the local loss functions are non-convex [9, 10, 12–16, 18, 26–32]. In $\\bar { \\mathbb { E } 2 } \\mathbb { I }$ , the authors showed that Parallel Restarted SGD (Local SGD or FedAvg [11]) achieves linear speed up while requiring $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ samples and $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ rounds of communication to reach an $\\epsilon$ -stationary solution. In $[ \\bar { \\lVert { 4 } } ]$ , a Momentum SGD was proposed, which achieved the same sample and communication complexities as Parallel Restarted SGD $\\mathbb { \\lVert 1 2 \\rVert }$ , without requiring that the second moments of the gradients be bounded. Further, it was shown that under the homogeneous data setting, the communication complexity can be improved to $\\mathcal { O } ( \\epsilon ^ { - 1 } )$ while maintaining the same sample complexity. The works in $\\boxed { 1 5 } , \\boxed { 1 6 }$ conducted tighter analysis for FedAvg with partial WN participation with $\\mathcal { O } ( 1 )$ local updates and batch sizes. Their analysis showed that FedAvg’s sample and communication complexities are both $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ . Additionally, SCAFFOLD was proposed in $\\bar { \\| 1 5 \\| }$ , which utilized variance reduction based local update directions $\\pmb { \\mathbb { B 3 } }$ to achieve the same sample and communication complexities as FedAvg. Similarly, VRL-SGD proposed in $\\left[ \\left[ 2 9 \\right] \\right]$ also utilized variance reduction and showed improved communication complexity of $\\mathcal { O } ( \\epsilon ^ { - 1 } )$ , while requiring the same computations as FedAvg. Importantly, both SCAFFOLD and VRL-SGD’s guarantees were independent of the data heterogeneity. The FedProx proposed in $\\mathbb { \\ m }$ used a penalty based method to improve the communication complexity of FedAvg (i.e., the Parallel Restarted and Momentum SGD [14, $\\mathbb { L } 2 \\mathbb { I }$ ) to $\\mathcal { O } ( \\epsilon ^ { - 1 } )$ . FedProx used a gradient similarity assumption to model data heterogeneity which can be stringent for many practical applications. This assumption was relaxed by FedPD proposed in $\\pmb { \\mathbb { B } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, the works [17, 18] proposed to utilize hybrid momentum gradient estimators [7, 8]. The MIME algorithm $\\mathbb { \\lVert 1 7 \\rVert }$ matched the optimal sample complexity (under certain smoothness assumptions) of $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ of the centralized non-convex stochastic optimization algorithms [5–8]. Similarly, Fed-GLOMO $[ \\overline { { 1 8 } } ]$ achieved the same sample complexity while employing compression to further reduce communication. Both MIME and Fed-GLOMO required $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ communication rounds to achieve an $\\epsilon$ -stationary solution. Please see Table $^ 1$ for a summary of the above discussion. ",
+        "bbox": [
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+            748,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The comparison of Local SGD (FedAvg) to Minibatch SGD for convex and strongly convex problems with homogeneous data setting was first conducted in $\\mathbb { \\lVert 1 9 \\rVert }$ and later extended to heterogeneous setting in $\\mathbb { \\lVert \\rVert 3 \\rVert }$ . It was shown that Minibatch SGD almost always dominates the Local SGD. In contrast, it was shown in $\\pmb { \\Vert 2 4 \\Vert }$ that Local SGD dominates Minibatch SGD in terms of generalization performance. Although existing FL results are rich, but they are somehow ad hoc and there is a lack of principled understanding of the algorithms. We note that the proposed STEM algorithmic framework provides a theoretical framework that unifies all existing $\\mathrm { F L }$ results on sample and communication complexities. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            825,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notations. The expected value of a random variable $X$ is denoted by $\\mathbb { E } [ X ]$ and its expectation conditioned on an Event $A$ is denoted as $\\mathbb { E } [ X | \\mathrm { E v e n t ~ } A ]$ . We denote by $\\mathbb { R }$ (and $\\mathbb { R } ^ { d }$ ) the real line (and the $d$ -dimensional Euclidean space). The set of natural numbers is denoted by $\\mathbb { N }$ . Given a positive integer $K \\in \\mathbb N$ , we denote $[ K ] \\triangleq \\{ 1 , 2 , \\dots , K \\}$ . Notation $\\| \\cdot \\|$ denotes the $\\ell _ { 2 }$ -norm and $\\langle \\cdot , \\cdot \\rangle$ the Euclidean inner product. For a discrete set $\\boldsymbol { B }$ , $| B |$ denotes the cardinality of the set. Uniform distribution over a discrete set $\\{ 1 , \\ldots , T \\}$ is denoted as ${ \\dot { \\mathcal { U } } } \\{ 1 , \\dots , T \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+            231,
+            318,
+            248
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Before we proceed to the algorithms, we make the following assumptions about problem $( 1 )$ . ",
+        "bbox": [
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+            277
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 (Sample Gradient Lipschitz Smoothness). The stochastic functions $f ^ { ( k ) } ( \\cdot , \\xi ^ { ( k ) } )$ with $\\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) }$ for all $k \\in [ K ]$ , satisfy the mean squared smoothness property, i.e, we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3cfe57005b8e58ae1bfc3b55d7cca26c0bbb487296b4468bd38bfdc42e09c0e.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\| \\nabla f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) - \\nabla f ^ { ( k ) } ( y ; \\xi ^ { ( k ) } ) \\| ^ { 2 } \\leq L ^ { 2 } \\| x - y \\| ^ { 2 } \\mathrm { ~ f o r ~ a l l ~ } x , y \\in \\mathbb { R } ^ { d } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 (Unbiased gradient and Variance Bounds). (i) Unbiased Gradient. The stochastic gradients computed at each WN are unbiased ",
+        "bbox": [
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+            372
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+        "page_idx": 3
+    },
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+        "type": "equation",
+        "img_path": "images/f06c119c8d26c099d56bae28e2fa443fdee4af7121660da108f532b9c8355105.jpg",
+        "text": "$$\n\\mathbb { E } [ \\nabla f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) ] = \\nabla f ^ { ( k ) } ( x ) , \\forall \\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) } , \\forall k \\in [ K ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(ii) Intra- and inter- node Variance Bound. The following bounds hold: ",
+        "bbox": [
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+            400,
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+            415
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b27d78c5e7a746fe9622599d0a5137acd7faea00d13d76004e33686cafb79bc5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { \\tilde { z } } \\| \\nabla f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) - \\nabla f ^ { ( k ) } ( x ) \\| ^ { 2 } \\leq \\sigma ^ { 2 } , \\| \\nabla f ^ { ( k ) } ( x ) - \\nabla f ^ { ( \\ell ) } ( x ) \\| ^ { 2 } \\leq \\zeta ^ { 2 } , \\forall \\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) } , \\forall k , \\ell \\in [ K ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            420,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that Assumption $\\boxed { 1 }$ is stronger than directly assuming $f ^ { ( k ) }$ ’s are Lipschitz smooth (which we will refer to as the averaged gradient Lipschitz smooth condition), but it is still a rather standard assumption in SGD analysis. For example it has been used in analyzing centralized SGD algorithms such as SPIDER $ { \\mathbb { I } }$ , SNVRG $\\pmb { \\Vert 6 \\Vert }$ , STORM $\\mathbb { [ [ \\big ] ] }$ (and many others) as well as in FL algorithms such as MIME $ { \\mathbb { I } } ^ { [ 1 2 ] }$ and Fed-GLOMO $ { \\mathbb { I } }  { \\mathrm { 1 8 } }  { \\Vert }$ . The second relation in Assumption $2 \\cdot$ (ii) quantifies the data heterogeneity, and we call $\\zeta > 0$ as the heterogeneity parameter. This is a typical assumption required to evaluate the performance of FL algorithms. If data distributions across individual WNs are identical, i.e., $\\mathcal { D } ^ { ( k ) } \\stackrel { = } { = } \\mathcal { D } ^ { ( \\ell ) }$ for all $k , \\ell \\in [ K ]$ then we have $\\zeta = 0$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we define the $\\epsilon$ -stationary solution for non-convex optimization problems, as well as quantify the computation and communication complexities to achieve an $\\epsilon$ -stationary point. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 $\\epsilon$ -Stationary Point). A point $x$ is called $\\epsilon$ -stationary if $\\| \\nabla f ( x ) \\| ^ { 2 } \\leq \\epsilon$ . Moreover, a stochastic algorithm is said to achieve an $\\epsilon$ -stationary point in $t$ iterations if $\\begin{array} { r } { \\ddot { \\mathbb { E } } [ \\| \\nabla f ( x _ { t } ) \\| ^ { 2 } ] \\le \\epsilon . } \\end{array}$ where the expectation is over the stochasticity of the algorithm until time instant $t$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Definition 2.2 (Sample complexity). We assume an Incremental First-order Oracle (IFO) framework $\\textcircled { \\lVert { 3 4 } \\rVert }$ where, given a sample $\\xi ^ { ( k ) } \\sim \\mathcal { D } ^ { ( k ) }$ at the $k ^ { \\mathrm { { t h } } }$ node and iterate $x$ , the oracle returns $( f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) , \\nabla f ^ { ( k ) } ( x ; \\xi ^ { ( k ) } ) )$ . Each access to the oracle is counted as a single IFO operation. We measure the sample (and computational) complexity in terms of the total number of calls to the IFO by all WNs to achieve an $\\epsilon$ -stationary point given in Definition 2.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 2.3 (Communication complexity). We define a communication round as a one back-andforth sharing of parameters between the WNs and the SN. Then the communication complexity is defined to be the total number of communication rounds between any WN and the SN required to achieve an $\\epsilon$ -stationary point given in Definition 2.1. ",
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+        "type": "text",
+        "text": "3 The STEM algorithm and the trade-off analysis ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we discuss the proposed algorithm and present the main results. The key in the algorithm design is to carefully balance all the four design elements mentioned in Sec. $^ { 1 , }$ so that sufficient and useful progress can be made between two rounds of communication. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let us discuss the key steps of STEM, listed in Algorithm $^ { 1 . }$ In Step 10, each node locally updates its model parameters using the local direction $d _ { t } ^ { k }$ , computed by using $b$ stochastic gradients at two ",
+        "bbox": [
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+        "type": "text",
+        "text": "1: Input: Parameters: $c > 0$ , the number of local updates $I$ , batch size $b$ , stepsizes $\\{ \\eta _ { t } \\}$ .   \n2: Initialize: Iterate $\\begin{array} { r } { x _ { 1 } ^ { ( k ) } = \\bar { x } _ { 1 } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } x _ { 1 } ^ { ( k ) } } \\end{array}$ , descent direction $\\begin{array} { r } { d _ { 1 } ^ { ( k ) } = \\bar { d } _ { 1 } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } d _ { 1 } ^ { ( k ) } } \\end{array}$   \nwith $\\begin{array} { r } { d _ { 1 } ^ { ( k ) } = \\frac { 1 } { B } \\sum _ { \\xi _ { 1 } ^ { ( k ) } \\in \\mathcal { B } _ { 1 } ^ { ( k ) } } \\nabla f ^ { ( k ) } ( x _ { 1 } ^ { ( k ) } ; \\xi _ { 1 } ^ { ( k ) } ) } \\end{array}$ and $| B _ { 1 } ^ { ( k ) } | = B$ for $k \\in [ K ]$ .   \n3: Perform: $x _ { 2 } ^ { ( k ) } = x _ { 1 } ^ { k } - \\eta _ { 1 } d _ { 1 } ^ { ( k ) }$ , $\\forall k \\in [ K ]$   \n4: for $t = 1$ to $T$ do   \n5: for $k = 1$ to $K$ do #at the WN   \n6: $\\mathcal { d } _ { t + 1 } ^ { ( k ) } = \\frac { 1 } { b } \\sum _ { \\xi _ { t + 1 } ^ { ( k ) } \\in \\mathcal { B } _ { t + 1 } ^ { ( k ) } } ^ { \\pi \\Delta } \\nabla f ^ { ( k ) } ( x _ { t + 1 } ^ { ( k ) } ; \\xi _ { t + 1 } ^ { ( k ) } ) + \\left( 1 - a _ { t + 1 } \\right) \\bigg ( d _ { t } ^ { ( k ) } - \\frac { 1 } { b } \\sum _ { \\xi _ { t + 1 } ^ { ( k ) } \\in \\mathcal { B } _ { t + 1 } ^ { ( k ) } } \\nabla f ^ { ( k ) } ( x _ { t } ^ { ( k ) } ; \\xi _ { t + 1 } ^ { ( k ) } ) \\bigg )$   \nwhere we choose $| B _ { t + 1 } ^ { ( k ) } | = b$ , and $a _ { t + 1 } = c \\cdot \\eta _ { t } ^ { 2 }$ ;   \n7: 8: if $t$ $I = 0$ #at the SN   \n$\\begin{array} { r } { d _ { t + 1 } ^ { ( k ) } = \\bar { d } _ { t + 1 } : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } d _ { t + 1 } ^ { ( k ) } } \\end{array}$   \n9: 10: e $\\begin{array} { r l } & { x _ { t + 2 } ^ { ( k ) ^ { \\bot } } : = \\bar { x } _ { t + 1 } - \\eta _ { t + 1 } \\bar { d } _ { t + 1 } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } x _ { t + 1 } ^ { ( k ) } - \\eta _ { t + 1 } \\bar { d } _ { t + 1 } } \\end{array}$ $x _ { t + 2 } ^ { ( k ) } = x _ { t + 1 } ^ { ( k ) } - \\eta _ { t + 1 } d _ { t + 1 } ^ { ( k ) }$ #server-side momentum   \n11: end if   \n12: end for   \n13: end for   \n14: Return: $\\scriptstyle { \\bar { x } } _ { a }$ where $a \\sim \\mathcal { U } \\{ 1 , . . . , T \\}$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "consecutive iterates $x _ { t + 1 } ^ { ( k ) }$ and $x _ { t } ^ { ( k ) }$ . After every $I$ local steps, the WNs share their current local models $\\{ x _ { t + 1 } ^ { ( k ) } \\} _ { k = 1 } ^ { K }$ and directions $\\{ d _ { t + 1 } ^ { ( k ) } \\} _ { k = 1 } ^ { K }$ with the SN. The SN aggregates these quantities, and performs a server-side momentum step, before returning $\\bar { x } _ { t + 1 }$ and $\\bar { d } _ { t + 1 }$ to all the WNs. Because both the WNs and the SN perform momentum based updates, we call the algorithm a stochastic two-sided momentum algorithm. The key parameters are: $b$ the minibatch size, $I$ the local update steps between two communication rounds, $\\eta _ { t }$ the stepsizes, and $a _ { t }$ the momentum parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One key technical innovation of our algorithm design is to identify the most suitable way to incorporate momentum based directions in FL algorithms. Although the momentum-based gradient estimator itself is not new and has been used in the literature before (see e.g., in $\\mathbb { \\left[ \\bigstar \\bigstar \\right] }$ and $\\overline { { \\mathbb { D } \\mathbb { Z } \\mathbb { G } \\mathbb { S } } }$ to improve the sample complexities of centralized and decentralized stochastic optimization problems, respectively), it is by no means clear if and how it can contribute to improve the communication complexity of FL algorithms. We show that in the FL setting, the local directions together with the local models have to be aggregated by the SN so to avoid being influenced too much by the local data. More importantly, besides the WNs, the SN also needs to perform updates using the (aggregated) momentum directions. Finally, such two-sided momentum updates have to be done carefully with the correct choice of minibatch size $b$ , and the number of local updates $I$ . Overall, it is the judicious choice of all these design elements that results in the optimal sample and communication complexities. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Next, we present the convergence guarantees of the STEM algorithm. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.1 Main results: convergence guarantees for STEM ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we analyze the performance of STEM. We first present our main result, and then provide discussions about a few parameter choices. In the next subsection, we discuss a special case of STEM related to the classical FedAvg and minibatch SGD algorithms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3.1. Under the Assumptions 1 and 2, suppose the stepsize sequence is chosen as: ",
+        "bbox": [
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+            785,
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/c7b9d663b7230014554f849f65c327948834a1aadf2fddbf4fd4727068678b71.jpg",
+        "text": "$$\n\\eta _ { t } = \\frac { \\bar { \\kappa } } { ( w _ { t } + \\sigma ^ { 2 } t ) ^ { 1 / 3 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where we define : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1ab3f1b7115347d40a7bc1f3343dd7707b65537827e4b8370ab97442863fcfd1.jpg",
+        "text": "$$\n\\bar { \\kappa } = \\frac { ( b K ) ^ { 2 / 3 } \\sigma ^ { 2 / 3 } } { L } , \\quad w _ { t } = \\operatorname * { m a x } \\bigg \\{ 2 \\sigma ^ { 2 } , 4 0 9 6 L ^ { 3 } I ^ { 3 } \\bar { \\kappa } ^ { 3 } - \\sigma ^ { 2 } t , \\frac { c ^ { 3 } \\bar { \\kappa } ^ { 3 } } { 4 0 9 6 L ^ { 3 } I ^ { 3 } } \\bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Further, let us set $\\begin{array} { r } { c = \\frac { 6 4 L ^ { 2 } } { b K } + \\frac { \\sigma ^ { 2 } } { 2 4 \\bar { \\kappa } ^ { 3 } L I } = L ^ { 2 } \\bigg ( \\frac { 6 4 } { b K } + \\frac { 1 } { 2 4 ( b K ) ^ { 2 } I } \\bigg ) } \\end{array}$ and set the initial batch size as $B = b I$ ; set the local updates $I$ and minibatch size b as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/571cd8b7f6de9fecec247a32a77a5fc7ff44b9b1a1446281dd4865af343e8ec8.jpg",
+        "text": "$$\nI = \\mathcal { O } \\big ( ( T / K ^ { 2 } ) ^ { \\nu / 3 } \\big ) , \\quad b = \\mathcal { O } \\big ( ( T / K ^ { 2 } ) ^ { 1 / 2 - \\nu / 2 } \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\nu$ satisfies $\\nu \\in [ 0 , 1 ]$ . Then for STEM the following holds: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(i) For $\\scriptstyle { \\bar { x } } _ { a }$ chosen according to Algorithm $\\boldsymbol { l } ,$ we have: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2c146df205c691e32ce24c2ee693fb97ea38edc895ba31bbeb511db58c5a452.jpg",
+        "text": "$$\n\\mathbb { E } \\Vert \\nabla f ( \\bar { x } _ { a } ) \\Vert ^ { 2 } = \\mathcal { O } \\Bigg ( \\frac { f ( \\bar { x } _ { 1 } ) - f ^ { * } } { K ^ { 2 \\nu / 3 } T ^ { 1 - \\nu / 3 } } \\Bigg ) + \\tilde { \\mathcal { O } } \\Bigg ( \\frac { \\sigma ^ { 2 } } { K ^ { 2 \\nu / 3 } T ^ { 1 - \\nu / 3 } } \\Bigg ) + \\tilde { \\mathcal { O } } \\Bigg ( \\frac { \\zeta ^ { 2 } } { K ^ { 2 \\nu / 3 } T ^ { 1 - \\nu / 3 } } \\Bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(ii) For any $\\nu \\in [ 0 , 1 ]$ , we have ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sample Complexity: The sample complexity of STEM is $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ . This implies that each WN requires at most $\\tilde { \\mathcal { O } } ( K ^ { - 1 } \\epsilon ^ { - 3 / 2 } )$ gradient computations, thereby achieving linear speedup with the number of WNs present in the network. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Communication Complexity: The communication complexity of STEM is $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proof of this result is relegated to the Supplemental Material. A few remarks are in order. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Remark 1 (Near-Optimal sample and communication complexities). Theorem $3 . 1$ suggests that when $I$ and $b$ are selected appropriately, then STEM achieves $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ and $\\tilde { \\mathcal { O } } ( \\overline { { \\epsilon ^ { - 1 } } } )$ sample and communication complexities. Taking them separately, these complexity bounds are the best achievable by the existing FL algorithms (upto logarithmic factors regardless of sample or batch Lipschitz smooth assumption) $[ [ 3 5 ] ]$ ; see Table $1 .$ We note that the $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ complexity is the best possible that can be achieved by centralized SGD with the sample Lipschitz gradient assumption; see $ { \\mathbb { I } } ^ {  { \\left[ 5 \\right] } }$ . On the other hand, the $\\bar { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ complexity bound is also likely to be the optimal, since in $\\mathbb { P }$ the authors showed that even when the local steps use a class of (deterministic) first-order algorithms, $\\mathcal { O } ( \\epsilon ^ { - 1 } )$ is the best achievable communication complexity. The only difference is that $\\bigstar \\bigstar$ does not explicitly assume the inter-node variance bound (i.e., the second relation in Assumption $2 \\cdot$ -(ii)). We leave the precise characterization of the communication lower bound with inter-node variance as future work. □ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark 2 (Large Batch Sizes and/or Local Updates). At first glance, it may seem that the requirement of STEM to compute large mini-batches and/or local updates (cf. Table $^ { 1 ) }$ to achieve this (near) optimal performance is a drawback, however, we note that it is in fact an advantage of STEM that it allows the WNs to perform larger number of local updates (or compute large minibatches) without communicating often. This follows from the fact that irrespective of the number of local updates (or batch sizes) STEM achieves near-optimal communication complexity while attaining optimal overall sample complexity. Moreover, note that even with $b = I = \\mathcal { O } ( 1 )$ (i.e., $b$ and $I$ are chosen as constants), STEM achieves the same (optimal) sample and communication complexities as achieved by FedGLOMO [18] and MIME [17]. We further note that to the best of our knowledge the algorithms that achieve the communication complexity of $\\mathcal { O } ( \\epsilon ^ { - 1 } )$ either require the number of local updates or the batch-sizes that depend on the solution accuracy $\\epsilon$ . For example, FedProx $\\mathbb { I O } ]$ , FedPD $\\bigstar \\bigstar$ , and FedDyn $\\pmb { \\Vert 3 6 \\Vert }$ rely on solving the “local problems\" to achieve an $\\epsilon$ -accuracy, which implies that the number of local updates (or the batch sizes) implicitly depends on the desired solution accuracy $\\epsilon$ , as is the case for STEM. Similarly, as shown in $\\bar { \\mathbb { E } 2 } \\mathbb { I }$ and $\\bar { \\textregistered 4 } \\bar { 1 }$ the communication complexity of FedAvg and its momentum version can be improved from $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ to $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ when the number of local updates (or batch size) is chosen as $\\mathcal { O } ( \\epsilon ^ { - 1 / 2 } )$ (cf. Section $3 . 2$ for a more detailed discussion). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter $\\nu \\in [ 0 , 1 ]$ is used to balance the local minibatch sizes $b$ , and the number of local updates $I$ . Eqs. in $\\textcircled { 3 }$ suggest that when $\\nu$ increases from 0 to 1, $b$ decreases and $I$ increases. Specifically, if $\\nu = 1$ , then $b$ is a constant but $I = \\mathcal { O } ( T ^ { 1 / 3 } / K ^ { 2 / 3 } )$ . In this case, each WN chooses a small minibatch while executing multiple local updates, and STEM resembles a FedAvg (a.k.a. Local SGD) algorithm but with double-sided momentum update directions, and is referred to as Fed STEM. In contrast, if $\\nu = 0$ , then $b = \\mathcal { O } ( T ^ { 1 / 2 } / K )$ but $I$ is a constant. In this case, each WN chooses a large batch size while executing only a few, or even one, local updates, and STEM resembles the Minibatch SGD, but again with different update directions, and is referred to as Minibatch STEM. Such a trade-off can be seen in Fig. 1b. Due to space limitation, these two special cases will be precisely stated in the supplementary materials as corollaries of Theorem 3.1. □ ",
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+    {
+        "type": "text",
+        "text": "Algorithm 2 The FedAvg Algorithm ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "1: Input: $\\{ \\eta _ { t } \\} _ { t = 0 } ^ { T } ; I$ , the # of local updates per communication round; $b$ , the minibatch sizes.   \n2: for $t = 1$ to $T$ do   \n3: 4: 5: for $\\begin{array} { r l } & { \\mathcal { \\kappa } _ { t } ^ { = } \\stackrel { \\mathrm { ~ L ~ U ~ O ~ } \\Lambda } { = } \\mathbf { 0 } } \\\\ & { d _ { t } ^ { ( k ) } = \\frac { 1 } { b } \\sum _ { \\xi _ { t } ^ { ( k ) } \\in \\mathcal { B } _ { t } ^ { ( k ) } } \\nabla f ^ { ( k ) } ( x _ { t } ^ { ( k ) } ; \\xi _ { t } ^ { ( k ) } ) \\mathrm { ~ w i t h ~ } | \\mathcal { B } _ { t } ^ { ( k ) } | = b } \\\\ & { x _ { t + 1 } ^ { ( k ) } = x _ { t } ^ { ( k ) } - \\eta _ { t } d _ { t } ^ { ( k ) } } \\\\ & { \\mathbf { i f } t \\operatorname* { m o d } I = 0 \\mathbf { \\Lambda } \\mathbf { t h e n } } \\\\ & { ~ x _ { t + 1 } ^ { ( k ) } = \\bar { x } _ { t + 1 } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } x _ { t + 1 } ^ { ( k ) } } \\\\ & { \\mathbf { e n d } \\mathbf { \\Phi } \\mathbf { i f } } \\end{array}$ $k = 1$ $K$   \n6:   \n7:   \n8:   \n9: end for   \n10: end for   \n11: Return: $\\scriptstyle { \\bar { x } } _ { a }$ where $a \\sim \\mathcal { U } \\{ 1 , . . . , T \\}$ . ",
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+        "type": "text",
+        "text": "Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem $\\left| \\overline { { \\mathbf { C . 1 0 } } } \\right|$ included in the supplemental material), we can see that STEM requires $\\bar { \\tilde { O } } ( \\operatorname* { m a x } \\big \\{ ( b \\cdot$ $I ) \\epsilon ^ { - 1 } , K ^ { - 1 } \\epsilon ^ { - 3 / 2 } \\rbrace )$ samples and $\\tilde { \\mathcal { O } } \\big ( \\operatorname* { m a x } \\big \\{ \\epsilon ^ { - 1 } , ( b \\cdot I ) ^ { - 1 } K ^ { - 1 } \\epsilon ^ { - 3 / 2 } \\big \\} \\big )$ \u0000 and communication rounds. According to the above expressions, if $b \\cdot I$ increases beyond $\\mathcal { O } ( K ^ { - 1 } \\epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\\tilde { \\mathcal O } ( \\epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \\cdot I$ decreases beyond $\\mathcal { O } ( K ^ { - 1 } \\epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ . For instance, if we choose $b = \\mathcal { O } ( 1 )$ and $I = { \\mathcal { O } } ( 1 )$ the communication complexity becomes $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\\boxed { 1 \\mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □ ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Remark 5 (Data Heterogeneity). The term $\\begin{array} { r } { \\tilde { \\mathcal { O } } \\biggl ( \\frac { \\zeta ^ { 2 } } { K ^ { 2 \\nu / 3 } T ^ { 1 - \\nu / 3 } } \\biggr ) } \\end{array}$ in the gradient bound $\\textcircled{4}$ captures the effect of the heterogeneity of data across WNs, where $\\zeta$ is the parameter characterizing the intra-node variance and has been defined in Assumption $\\bigstar$ (ii). Highly heterogeneous data with large $\\zeta ^ { 2 }$ can adversely impact the performance of STEM. Note that such a dependency on $\\zeta$ also appears in other existing FL algorithms, such as $[ \\bigcirc , \\bigcirc , \\bigcirc , \\bigcirc , \\bigcirc ]$ . However, there is one special case of STEM that does not depend on the parameter $\\zeta$ . This is the case where $I = 1$ , i.e., the minibatch SGD counterpart of STEM where only a single local iteration is performed between two communication rounds. We have the following corollary. □ ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1 (Minibatch STEM). Under Assumptions $\\bigstar \\bigstar \\bigstar | \\bigstar |$ , and choose the algorithm parameters as in Theorem $3 . I .$ At each WN, choose $I = 1$ , $b = ( T / K ^ { 2 } ) ^ { 1 / 2 }$ , and the initial batch size $B = \\boldsymbol { b } \\cdot \\boldsymbol { I }$ . Then STEM satisfies: ",
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+    {
+        "type": "text",
+        "text": "(i) For $\\bar { x } _ { a }$ chosen according to Algorithm $\\boldsymbol { l } ,$ we have ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/fe27c5ca3c886c333d5add660b12f630bb6023b83b7450c21e97035dfce670f9.jpg",
+        "text": "$$\n\\mathbb { E } \\| \\nabla f ( \\bar { x } _ { a } ) \\| ^ { 2 } = \\mathcal { O } \\Big ( \\frac { f ( \\bar { x } _ { 1 } ) - f ^ { * } } { T } \\Big ) + \\tilde { \\mathcal { O } } \\Big ( \\frac { \\sigma ^ { 2 } } { T } \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(ii) Minibatch STEM achieves $\\tilde { \\mathcal O } ( \\epsilon ^ { - 3 / 2 } )$ sample and $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ communication complexity. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we show that FedAvg also exhibits a trade-off similar to that of STEM but with worse sample and communication complexities. ",
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+    {
+        "type": "text",
+        "text": "3.2 Special cases: The FedAvg algorithm ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We briefly discuss another interesting special case of STEM, where the local momentum update is replaced by the conventional SGD (i.e., $a _ { t } = 1 , ~ \\forall ~ t )$ , while the server does not perform the momentum update (i.e., $\\bar { d } _ { t } = 0 , \\forall t )$ . This is essentially the classical FedAvg algorithm, just that it balances the number of local updates $I$ and the minibatch size $b$ . We show that this algorithm also exhibits a trade-off between $b$ and $I$ and on the trade-off curve it achieves $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ sample complexity and $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ communication complexity. ",
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+    {
+        "type": "table",
+        "img_path": "images/6bf0f1ad1737bddbd1a909e1c507f78edccdbe2e2cf8c63393f03d6b14335b18.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(a) Mild heterogeneity, $b = 6 4$ , and $I = 7$ . "
+        ],
+        "table_body": "<table><tr><td>Algorithm</td><td> Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>78.2</td><td>74.1</td></tr><tr><td>FedProx</td><td>79.2</td><td>74.8</td></tr><tr><td>FedDyn</td><td>68.9</td><td>66.0</td></tr><tr><td>SCAFFOLD</td><td>71.9</td><td>74.0</td></tr><tr><td>MIME</td><td>82.6</td><td>76.8</td></tr><tr><td>FedGLOMO</td><td>76.1</td><td>72.8</td></tr><tr><td> STEM</td><td>80.1</td><td>78.8</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b0513e337be144b5f857c24b8a10281f3fd33f9a14e65f9ee342f421e0353331.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Moderate heterogeneity, $b = 8$ , and $I = 6 1$ "
+        ],
+        "table_body": "<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>73.6</td><td>75.4</td></tr><tr><td>FedProx</td><td>80.0</td><td>75.2</td></tr><tr><td>FedDyn</td><td>76.1</td><td>71.3</td></tr><tr><td>SCAFFOLD</td><td>72.5</td><td>73.7</td></tr><tr><td>MIME</td><td>61.5</td><td>58.6</td></tr><tr><td>FedGLOMO</td><td>10.0</td><td>10.0</td></tr><tr><td>STEM</td><td>81.1</td><td>78.5</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 2: Training and testing accuracy of different algorithms on CIFAR-10 dataset for different batch-sizes, number of local updates, and heteregeneity settings. ",
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+    {
+        "type": "text",
+        "text": "Theorem 3.2 (The FedAvg Algorithm). Under Assumptions 1 and 2, suppose the stepsize is chosen as: $\\begin{array} { r } { \\eta = \\sqrt { \\frac { b K } { T } } } \\end{array}$ ; Let us set: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/614827bc2783d2a130099b5dcc25a039f7d6ca4f991f27f94b6b9f6e8cc8d82c.jpg",
+        "text": "$$\nI = \\mathcal { O } \\big ( ( T / K ^ { 3 } ) ^ { \\nu / 4 } \\big ) , \\quad b = \\mathcal { O } \\big ( ( T / K ^ { 3 } ) ^ { 1 / 3 - \\nu / 3 } \\big )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\nu \\in [ 0 , 1 ]$ is a constant. Then for FedAvg with $T \\geq 8 1 L ^ { 2 } I ^ { 2 } b K$ , the following holds ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(i) For $\\scriptstyle { \\bar { x } } _ { a }$ chosen according to Algorithm $\\perp$ we have ",
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fcbbd1eccd40989ffc376397736fd5950f8008a4dccca06a87727fb72022e909.jpg",
+        "text": "$$\n\\mathbb { E } \\Vert \\nabla f ( \\bar { x } _ { a } ) \\Vert ^ { 2 } = \\mathcal { O } \\Bigg ( \\frac { f ( \\bar { x } _ { 1 } ) - f ^ { * } } { K ^ { \\nu / 2 } T ^ { 2 / 3 - \\nu / 6 } } \\Bigg ) + \\mathcal { O } \\Bigg ( \\frac { \\sigma ^ { 2 } } { K ^ { \\nu / 2 } T ^ { 2 / 3 - \\nu / 6 } } \\Bigg ) + \\mathcal { O } \\Bigg ( \\frac { \\zeta ^ { 2 } } { K ^ { \\nu / 2 } T ^ { 2 / 3 - \\nu / 6 } } \\Bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(ii) For any choice of $\\nu \\in [ 0 , 1 ]$ we have: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Sample Complexity: The sample complexity of FedAvg is $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ . This implies that each WN requires at most $\\tilde { \\mathcal { O } } ( K ^ { - 1 } \\epsilon ^ { - 2 } )$ gradient computations, thereby achieving linear speedup with the number of WNs in the network. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Communication Complexity: The communication complexity of FedAvg is $\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that the requirement on $T$ being lower bounded is only relevant for theoretical purposes, a similar requirement was also imposed in $[ \\mathbb { 1 4 } ]$ to prove convergence. Again, the parameter $\\nu \\in [ 0 , 1 ]$ in the statement of Theorem $3 . 2$ balances $I$ and $b$ at each WN while maintaining state-of-the-art sample and communication complexities; please see Table $^ 1$ for a comparison of those bounds with existing FedAvg bounds. For $\\nu = 1$ , FedAvg (cf. Theorem $\\textcircled { 3 . 2 }$ reduces to FedAvg proposed in [12, 14] and for $\\nu = 0$ , the algorithm can be viewed as a large batch FedAvg with constant local updates [15, 16]. Note that similar to STEM, it is known that for $I = 1$ , the Minibatch SGD’s performance is independent of the heterogeneity parameter, $\\zeta \\equiv \\mathbb { I I } 3 \\mathbb { I }$ . We also point out that if Algorithm $^ 1$ uses Nesterov’s or Polyak’s momentum $[ \\textcircled { 1 4 } ]$ at local WNs instead of the recursive momentum estimator we get the same guarantees as in Theorem $3 . 2 .$ ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In summary, this section established that once the WN’s and the SN’s update directions (SGD in FedAvg and momentum based directions in STEM) are fixed, there exists a sequence of optimal choices of the number of local updates $I$ , and the batch sizes $b$ , which guarantees the best possible sample and communication complexities for the particular algorithm. The trade-off analysis presented in this section provides some useful guidelines for how to best select $b$ and $I$ in practice. Our subsequent numerical results will also verify that if $b$ or $I$ are not chosen judiciously, then the practical performance of the algorithms can degrade significantly. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 Numerical results ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we validate the proposed STEM algorithm and compare its performance with the de facto standard FedAvg [11], and the algorithms stated in Table $^ { 1 . }$ Note that instead of FedPD we include the performance comparison with FedDyn $\\pmb { \\mathbb { B } } 6 \\|$ since they are known to be very closely related. The goal of our experiments are three-fold: (1) To show that STEM performs on par, if not better, compared to other algorithms in different heterogeneity settings, (2) there are multiple ways to reach the desired solution accuracy, one can either choose a large batch size and perform only a few local updates or select a smaller batch size and perform multiple local updates, and finally, (3) if the local updates and the batch sizes are not chosen appropriately, the WNs might need to perform excessive computations to achieve the desired solution accuracy, thereby slowing down convergence. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/dcdc37cb3965642ea8fadecde08dc3a78e268deb8860ba74a94a052f25763bcd.jpg",
+        "table_caption": [
+            "Table 3: Training and testing accuracy on CIFAR-10 dataset for high heterogeneity, $b =$ 128 and $I = 6$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>57.6</td><td>57.1</td></tr><tr><td>FedProx</td><td>59.1</td><td>58.5</td></tr><tr><td>FedDyn</td><td>51.2</td><td>51.3</td></tr><tr><td>SCAFFOLD</td><td>53.1</td><td>54.7</td></tr><tr><td>MIME</td><td>56.1</td><td>55.1</td></tr><tr><td>FedGLOMO</td><td>56.8</td><td>56.1</td></tr><tr><td> STEM</td><td>58.5</td><td>57.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a8ee7b8ff04d3f8e652b073870044e3954f021e10e2ae94f2132f7ce72b56ed4.jpg",
+        "table_caption": [
+            "Table 4: Training and testing accuracy on Shakespeare dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>Training Acc.</td><td>Testing Acc.</td></tr><tr><td>FedAvg</td><td>40.1</td><td>39.2</td></tr><tr><td>FedProx</td><td>43.5</td><td>43.2</td></tr><tr><td>FedDyn</td><td>43.7</td><td>43.2</td></tr><tr><td>SCAFFOLD</td><td>40.3</td><td>41.3</td></tr><tr><td>MIME</td><td>32.1</td><td>32.1</td></tr><tr><td>FedGLOMO</td><td>40.3</td><td>40.1</td></tr><tr><td> STEM</td><td>44.5</td><td>43.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a3649642909a08bbdd852d92b8ec00817216ffd484e8ea8263d27aa9fb977f4.jpg",
+        "image_caption": [
+            "Figure 2: Training loss and the testing accuracy for classification on MNIST data set against the number of samples accessed at each WN for moderate heterogeneity setting with $b = 8$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data and Parameter Settings: We compare the algorithms for image classification tasks on CIFAR10 and MNIST data sets with 100 WNs, and for next character prediction task on Shakespeare dataset $\\pmb { \\Vert 3 7 } \\Vert$ with 143 WNs in the network. For both CIFAR-10 and MNIST, each WN implements a two-hidden-layer convolutional neural network (CNN) architecture followed by three linear layers for CIFAR-10 and two for MNIST. For CIFAR-10 (and MNIST) datset, we consider three settings with mild, moderate and high heterogeneity. For all the three settings, the data is partitioned into disjoint sets among the WNs. In the mild heterogeneity setting, the WNs have access to partitioned data from all the classes. In the moderate (resp. high) heterogeneity setting the data is partitioned such that each WN can access data from only 5 (resp. 2) out of 10 classes. For CIFAR-10 (resp. MNIST), each WN has access to 490 (resp. 540) samples for training and 90 (resp. 80) samples for testing purposes. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also compare the performance of algorithms on a popular FL benchmarking dataset, Shakespeare dataset $ { \\mathbb { I } } ^ { \\smash { \\sum } }$ . For this task, we adopt the settings from $\\dot { \\left[ \\left| 1 0 \\right| \\right] }$ and utilize a 2-Layer LSTM network with 100 hidden units and an 8-D embedding layer at each WN. Each WN has access to 3616 samples on average, and the samples are randomly split into an $80 \\%$ training set and a $20 \\%$ testing set. We randomly sample 10 nodes out of 143 for the training purpose. All the experiments are implemented on a single NVIDIA Quadro RTX 5000 GPU. More details are provided in appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the proposed STEM algorithm, recall that the step-size is $\\eta _ { t } = \\bar { \\kappa } / ( w _ { t } + \\sigma ^ { 2 } t ) ^ { 1 / 3 }$ with momentum parameter defined as $a _ { t } = { c } \\eta _ { t } ^ { 2 }$ . The step-size is used to update the iterates while the momentum parameter is used to construct the stochastic gradient estimate (cf. Algorithm 1 and Theorem $\\boxed { 3 . 1 }$ . For the experiments, we set $w _ { t } = \\sigma ^ { 2 } = 1$ and $c \\doteq \\bar { c } / \\bar { \\kappa } ^ { 2 }$ and tune for $\\bar { \\kappa } \\in [ 1 0 ^ { - 1 } , \\overline { { 1 0 } } ^ { - 2 } ]$ for the CIFAR-10 dataset and for ${ \\bar { \\kappa } } \\in \\{ 1 0 ^ { 1 } , 1 0 ^ { 0 } , 1 0 ^ { - 1 } , 1 0 ^ { - 2 } \\}$ for the Shakespeare dataset. For both the datasets we tune for $\\bar { c }$ in the range [1, 10]. For FedProx $[ \\equiv ]$ and FedDyn $\\lVert \\bar { 3 6 } \\rVert$ we choose the regularization constant to be 0.1. The momentum parameters for FedGLOMO $\\pm \\textcircled { 1 8 } \\textcircled { 1 }$ and MIME $\\mathbb { \\lVert 1 7 \\rVert }$ are set based on the choices given in the respective papers. Specifically, for FedGLOMO we choose the parameter $\\beta _ { k } = 0 . 2$ and design the momentum gradient using a damping factor given in Appendix A.4 of FedGLOMO $\\mathbb { \\left[ \\left[ 8 \\right] \\right] }$ . Moreover, for MIME we choose the momentum parameter as 0.9. For the rest of the algorithms (including FedAvg and SCAFFOLD), the step-size is tuned from the set $\\{ 1 0 ^ { 1 } , 1 0 ^ { 0 } , 1 0 ^ { - 1 } , \\hat { 1 0 } ^ { - 2 } \\}$ . ",
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+        "text": "Discussion: We evaluate the training and testing performance of STEM against multiple algorithms for different heterogeneity settings, minibatch sizes, and number of local updates. In Tables $\\bigstar$ $\\boxed { 2 \\mathbf { b } }$ and $\\bigstar ,$ we compare the training and testing accuracy of STEM to that of other algorithms on the CIFAR-10 dataset. Specific, heterogeneity settings, the choices of minibatches, and number of local updates are stated along with the tables. Note that STEM performs uniformly well under all the conditions. Moreover, note from Table $\\bigstar$ that FedGLOMO diverges once the number of local updates are high. Also, note from Table $\\textcircled { 3 }$ that FedProx and STEM adapt well to high heterogeneity. Finally, with the next set of experiments we emphasize the importance of choosing $b$ and $I$ carefully. In Figure $\\bigstar ,$ we compare the training and testing performance of STEM, FedAvg and SCAFFOLD, against the number of samples accessed at each WN for the classification task on MNIST dataset with moderate heterogeneity. We fix $b = 8$ and conduct experiments under two settings, one with $I = 6 7$ , and the other with $I = 5 3 6$ local updates at each WN. Note that although a large number of local updates might lead to fewer communication rounds but it can make the sample complexity extremely high as is demonstrated by Figure $2 .$ For example, Figure $\\bigtriangledown$ shows that to reach testing accuracy of $9 6 - 9 7 \\%$ with $I = 6 7$ , STEM requires approximately $5 0 0 0 - 6 0 0 0$ samples, in contrast with $I = 5 3 6$ it requires more than 25000 samples at each WN. Similar behavior can be observed if we fix $I > 1$ and increase the local batch sizes. This implies not choosing the local updates and the batch sizes judiciously might lead to increased sample complexity. Additional experiments are included in the supplementary material to further evaluate the performance of the proposed algorithms. ",
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+        "type": "text",
+        "text": "Conclusion ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "In this work, we proposed a novel algorithm STEM, for distributed stochastic non-convex optimization with applications to FL. We showed that STEM reaches an $\\epsilon$ -stationary point with $\\tilde { \\mathcal O } ( \\epsilon ^ { - 3 / 2 } )$ sample complexity while achieving linear speed-up with the number of WNs. Moreover, the algorithm achieves a communication complexity of $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 1 } )$ . We established a (optimal) trade-off that allows interpolation between varying choices of local updates and the batch sizes at each WN while maintaining (near optimal) sample and communication complexities. We showed that FedAvg (a.k.a LocalSGD) also exhibits a similar trade-off while achieving worse complexities. Our results provide guidelines to carefully choose the number of local updates, update directions, and minibatch sizes to achieve the best performance. The future directions of this work include developing lower bounds on communication complexity that establishes the tightness of the analysis conducted in this work. ",
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+        "type": "text",
+        "text": "Acknowledgement ",
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+        "text": "We thank the anonymous reviewers for their valuable comments and suggestions. The work of Prashant Khanduri and Mingyi Hong was supported by NSF grant CMMI-1727757, AFOSR grant 19RT0424 and ARO grant W911NF-19-1-0247. The work of Mingyi Hong was also supported by an IBM Faculty Research award. The work of Jia Liu has been supported in part by NSF grants CAREER CNS-2110259, CNS-2112471, CNS-2102233, CCF-2110252, ECCS-2140277, and a Google Faculty Research Award. ",
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+        "text": "References   \n[1] J. Konecnˇ y, H. B. McMahan, D. Ramage, and P. Richtárik, “Federated optimization: Distributed \\` machine learning for on-device intelligence,” arXiv preprint arXiv:1610.02527, 2016.   \n[2] M. Li, D. G. Andersen, A. J. Smola, and K. Yu, “Communication efficient distributed machine learning with the parameter server,” in Advances in Neural Information Processing Systems 27, Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, Eds. Curran Associates, Inc., 2014, pp. 19–27.   \n[3] J. Dean, G. Corrado, R. Monga, K. Chen, M. Devin, M. Mao, M. Ranzato, A. Senior, P. Tucker, K. Yang et al., “Large scale distributed deep networks,” in Advances in neural information processing systems, 2012, pp. 1223–1231.   \n[4] T. Léauté and B. Faltings, “Protecting privacy through distributed computation in multi-agent decision making,” Journal of Artificial Intelligence Research, vol. 47, pp. 649–695, 2013.   \n[5] C. Fang, C. J. Li, Z. 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+                                    "score": 1.0,
+                                    "content": "solution. To the best of our knowledge, this is the first FL algorithm that achieves",
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+                                    "content": "SGD, which is a momentum-less special case of the STEM), a similar trade-off",
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+                                    "score": 1.0,
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+                                    "content": "insights on this trade-off provides guidelines for choosing the four important design",
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+                                    "score": 1.0,
+                                    "content": "and the number of local updates.",
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+                                    "content": "be aggregated by the SN so to avoid being influenced too much by the local data. More importantly,",
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+                                    "score": 1.0,
+                                    "content": ". Overall, it is the judicious choice of all these",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "be aggregated by the SN so to avoid being influenced too much by the local data. More importantly,",
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+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ". Overall, it is the judicious choice of all these",
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+                                    "content": "achieved by centralized SGD with the sample Lipschitz gradient assumption; see",
+                                    "type": "text"
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+                                    "score": 0.75,
+                                    "content": " { \\mathbb { I } } ^ {  { \\left[ 5 \\right] } }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". On the other",
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+                        {
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+                                    "score": 1.0,
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+                                {
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+                                    "score": 1.0,
+                                    "content": "complexity bound is also likely to be the optimal, since in",
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+                                    "content": "\\mathbb { P }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "the authors showed",
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "that even when the local steps use a class of (deterministic) first-order algorithms,",
+                                    "type": "text"
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+                                {
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+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 1 } )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": "is the best",
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+                                    "score": 1.0,
+                                    "content": "achievable communication complexity. The only difference is that",
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+                                {
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+                                    "score": 0.66,
+                                    "content": "\\bigstar \\bigstar",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "does not explicitly assume",
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the inter-node variance bound (i.e., the second relation in Assumption",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.72,
+                                    "content": "2 \\cdot",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "-(ii)). We leave the precise",
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+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "characterization of the communication lower bound with inter-node variance as future work.",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.999,
+                                    "content": "□",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Remark 2 (Large Batch Sizes and/or Local Updates). At first glance, it may seem that the requirement",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "of STEM to compute large mini-batches and/or local updates (cf. Table",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.63,
+                                    "content": "^ { 1 ) }",
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+                                {
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+                                    "content": "to achieve this (near)",
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+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "optimal performance is a drawback, however, we note that it is in fact an advantage of STEM that it",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "allows the WNs to perform larger number of local updates (or compute large minibatches) without",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "communicating often. This follows from the fact that irrespective of the number of local updates",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(or batch sizes) STEM achieves near-optimal communication complexity while attaining optimal",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "overall sample complexity. Moreover, note that even with",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.93,
+                                    "content": "b = I = \\mathcal { O } ( 1 )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(i.e.,",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.71,
+                                    "content": "b",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.69,
+                                    "content": "I",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "are chosen as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "constants), STEM achieves the same (optimal) sample and communication complexities as achieved",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "by FedGLOMO [18] and MIME [17]. We further note that to the best of our knowledge the algorithms",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "that achieve the communication complexity of",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.93,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 1 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "either require the number of local updates or",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        106,
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+                                        541
+                                    ],
+                                    "score": 1.0,
+                                    "content": "the batch-sizes that depend on the solution accuracy",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.4,
+                                    "content": "\\epsilon",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        324,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ". For example, FedProx",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        420,
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+                                        541
+                                    ],
+                                    "score": 0.78,
+                                    "content": "\\mathbb { I O } ]",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ", FedPD",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.67,
+                                    "content": "\\bigstar \\bigstar",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ", and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        552
+                                    ],
+                                    "score": 1.0,
+                                    "content": "FedDyn",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.63,
+                                    "content": "\\pmb { \\Vert 3 6 \\Vert }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        160,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "rely on solving the “local problems\" to achieve an",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.68,
+                                    "content": "\\epsilon",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "-accuracy, which implies that the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                564
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        564
+                                    ],
+                                    "score": 1.0,
+                                    "content": "number of local updates (or the batch sizes) implicitly depends on the desired solution accuracy",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.42,
+                                    "content": "\\epsilon",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "is the case for STEM. Similarly, as shown in",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.83,
+                                    "content": "\\bar { \\mathbb { E } 2 } \\mathbb { I }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        302,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        573
+                                    ],
+                                    "score": 0.81,
+                                    "content": "\\bar { \\textregistered 4 } \\bar { 1 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        336,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "the communication complexity of FedAvg",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                                586
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        586
+                                    ],
+                                    "score": 1.0,
+                                    "content": "and its momentum version can be improved from",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        343,
+                                        586
+                                    ],
+                                    "score": 0.91,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 2 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "to",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        586
+                                    ],
+                                    "score": 0.92,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "when the number of local",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "updates (or batch size) is chosen as",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.93,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 1 / 2 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "; see Table",
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+                                    "content": " { \\mathbb { I } } ^ {  { \\left[ 5 \\right] } }",
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+                                    "score": 1.0,
+                                    "content": "the inter-node variance bound (i.e., the second relation in Assumption",
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+                                    "content": "characterization of the communication lower bound with inter-node variance as future work.",
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+                                    "score": 1.0,
+                                    "content": "optimal performance is a drawback, however, we note that it is in fact an advantage of STEM that it",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "communicating often. This follows from the fact that irrespective of the number of local updates",
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+                                    "score": 1.0,
+                                    "content": "(or batch sizes) STEM achieves near-optimal communication complexity while attaining optimal",
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "overall sample complexity. Moreover, note that even with",
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+                                    "score": 0.93,
+                                    "content": "b = I = \\mathcal { O } ( 1 )",
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+                                {
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+                                    "score": 0.69,
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+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "are chosen as",
+                                    "type": "text"
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+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "constants), STEM achieves the same (optimal) sample and communication complexities as achieved",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "by FedGLOMO [18] and MIME [17]. We further note that to the best of our knowledge the algorithms",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "that achieve the communication complexity of",
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+                                    "score": 0.93,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 1 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "either require the number of local updates or",
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+                            ],
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+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "the batch-sizes that depend on the solution accuracy",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.4,
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+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        324,
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+                                    "score": 1.0,
+                                    "content": ". For example, FedProx",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.78,
+                                    "content": "\\mathbb { I O } ]",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", FedPD",
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+                                {
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+                                    "content": "\\bigstar \\bigstar",
+                                    "type": "inline_equation"
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+                                    "content": ", and",
+                                    "type": "text"
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+                            ],
+                            "index": 35
+                        },
+                        {
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "FedDyn",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.63,
+                                    "content": "\\pmb { \\Vert 3 6 \\Vert }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "rely on solving the “local problems\" to achieve an",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.68,
+                                    "content": "\\epsilon",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "-accuracy, which implies that the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                564
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        564
+                                    ],
+                                    "score": 1.0,
+                                    "content": "number of local updates (or the batch sizes) implicitly depends on the desired solution accuracy",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.42,
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+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is the case for STEM. Similarly, as shown in",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.83,
+                                    "content": "\\bar { \\mathbb { E } 2 } \\mathbb { I }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.81,
+                                    "content": "\\bar { \\textregistered 4 } \\bar { 1 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the communication complexity of FedAvg",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "and its momentum version can be improved from",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.91,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 2 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "to",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "\\mathcal { O } ( \\epsilon ^ { - 3 / 2 } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
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+                            ],
+                            "index": 39
+                        },
+                        {
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+                                {
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+                                    "content": "for a more detailed discussion).",
+                                    "type": "text"
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+                        }
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+                                    "score": 1.0,
+                                    "content": "Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter",
+                                    "type": "text"
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+                                    "content": "adversely impact the performance of STEM. Note that such a dependency on",
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+                                    "content": "[ \\bigcirc , \\bigcirc , \\bigcirc , \\bigcirc , \\bigcirc ]",
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+                                    "content": ". However, there is one special case of STEM that does",
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+                                    "score": 1.0,
+                                    "content": ". This is the case where",
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+                                    "score": 1.0,
+                                    "content": ", i.e., the minibatch SGD counterpart",
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+                                    "score": 1.0,
+                                    "content": "of STEM where only a single local iteration is performed between two communication rounds. We",
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+                                    "score": 1.0,
+                                    "content": "have the following corollary.",
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+                                    "score": 1.0,
+                                    "content": "as in Theorem",
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+                                    "score": 1.0,
+                                    "content": ",",
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+                                    "score": 0.9,
+                                    "content": "b = ( T / K ^ { 2 } ) ^ { 1 / 2 }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", and the initial batch size",
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+                                    "content": "B = \\boldsymbol { b } \\cdot \\boldsymbol { I }",
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+                                    "score": 1.0,
+                                    "content": "Then STEM satisfies:",
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+                        }
+                    ],
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+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "and communication complexities; please see Table",
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+                                    "score": 1.0,
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+                                {
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+                                    "score": 1.0,
+                                    "content": "reduces to FedAvg proposed in [12, 14]",
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+                                    "content": ", the algorithm can be viewed as a large batch FedAvg with constant local updates",
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "[15, 16]. Note that similar to STEM, it is known that for",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.9,
+                                    "content": "I = 1",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", the Minibatch SGD’s performance",
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+                            ],
+                            "index": 39
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "is independent of the heterogeneity parameter,",
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+                                    "score": 0.38,
+                                    "content": "\\zeta \\equiv \\mathbb { I I } 3 \\mathbb { I }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". We also point out that if Algorithm",
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+                                    "score": 0.56,
+                                    "content": "^ 1",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "uses",
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+                        {
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+                                    "score": 0.77,
+                                    "content": "[ \\textcircled { 1 4 } ]",
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+                                    "score": 1.0,
+                                    "content": "we get the same guarantees as in Theorem",
+                                    "type": "text"
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+                                    "score": 0.33,
+                                    "content": "3 . 2 .",
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+                            ],
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+                        }
+                    ],
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+                                    "content": "In summary, this section established that once the WN’s and the SN’s update directions (SGD in",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "and communication complexities; please see Table",
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+                                    "score": 1.0,
+                                    "content": ". We also point out that if Algorithm",
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+                                    "score": 1.0,
+                                    "content": "uses",
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+                                    "score": 1.0,
+                                    "content": "we get the same guarantees as in Theorem",
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+                                    "score": 1.0,
+                                    "content": "the CIFAR-10 dataset. Specific, heterogeneity settings, the choices of minibatches, and number of",
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+                                    "score": 1.0,
+                                    "content": "local updates are stated along with the tables. Note that STEM performs uniformly well under all",
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+                                    "score": 1.0,
+                                    "content": "that FedGLOMO diverges once the number of local",
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+                                    "score": 1.0,
+                                    "content": "that FedGLOMO diverges once the number of local",
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+                                    "content": "it requires more than 25000 samples at each WN. Similar behavior can be observed if we fix",
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+                                    "score": 1.0,
+                                    "content": "and increase the local batch sizes. This implies not choosing the local updates and the batch",
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+                    ],
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+                                    "content": "In this work, we proposed a novel algorithm STEM, for distributed stochastic non-convex optimiza-",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "tion with applications to FL. We showed that STEM reaches an",
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+                                    "score": 0.93,
+                                    "content": "\\tilde { \\mathcal O } ( \\epsilon ^ { - 3 / 2 } )",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "sample complexity while achieving linear speed-up with the number of WNs. Moreover, the algo-",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". We established a (optimal) trade-off that",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "maintaining (near optimal) sample and communication complexities. We showed that FedAvg (a.k.a",
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+                                    "score": 1.0,
+                                    "content": "LocalSGD) also exhibits a similar trade-off while achieving worse complexities. Our results provide",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "We thank the anonymous reviewers for their valuable comments and suggestions. The work of",
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+                                    "content": "Prashant Khanduri and Mingyi Hong was supported by NSF grant CMMI-1727757, AFOSR grant",
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+                    ],
+                    "lines": [
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diff --git a/parse/train/J28lNO4p3ki/J28lNO4p3ki_model.json b/parse/train/J28lNO4p3ki/J28lNO4p3ki_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..e882c3777a2a08c0ac622b2924bf146977d1074b
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diff --git a/parse/train/KBnXrODoBW/KBnXrODoBW.md b/parse/train/KBnXrODoBW/KBnXrODoBW.md
new file mode 100644
index 0000000000000000000000000000000000000000..2935ab5b871b4ccf325292a5f962ad9ef0905eaf
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@@ -0,0 +1,242 @@
+# Pay Attention to MLPs
+
+Hanxiao Liu, Zihang Dai, David R. So, Quoc V. Le Google Research, Brain Team {hanxiaol,zihangd,davidso,qvl}@google.com
+
+# Abstract
+
+Transformers [1] have become one of the most important architectural innovations in deep learning and have enabled many breakthroughs over the past few years. Here we propose a simple network architecture, gMLP, based on MLPs with gating, and show that it can perform as well as Transformers in key language and vision applications. Our comparisons show that self-attention is not critical for Vision Transformers, as $\mathrm { g M L P }$ can achieve the same accuracy. For BERT, our model achieves parity with Transformers on pretraining perplexity and is better on some downstream NLP tasks. On finetuning tasks where gMLP performs worse, making the $\mathrm { g M L P }$ model substantially larger can close the gap with Transformers. In general, our experiments show that gMLP can scale as well as Transformers over increased data and compute.
+
+# 1 Introduction
+
+Transformers [1] have enabled many breakthroughs in natural language processing (e.g., [2, 3, 4, 5, 6]) and have been shown to work well for computer vision (e.g., [7, 8, 9, 10]). Thanks to this success, Transformers have largely replaced LSTM-RNN [11] as the default architecture in NLP, and have become an appealing alternative to ConvNets [12, 13, 14, 15, 16, 17] in computer vision.
+
+The Transformer architecture combines two important concepts: (1) a recurrent-free architecture which computes the representations for each individual token in parallel, and (2) multi-head selfattention blocks which aggregate spatial information across tokens. On one hand, the attention mechanism [18] introduces the inductive bias that the spatial interactions should be dynamically parameterized based on the input representations. On the other hand, it is known that MLPs with static parameterization can represent arbitrary functions [19]. It therefore remains an open question whether the inductive bias in self-attention is essential to the remarkable effectiveness of Transformers.
+
+Here we study the necessity of self-attention modules in key language and vision applications of Transformers. Specifically, we propose an MLP-based alternative to Transformers without self-attention, which simply consists of channel projections and spatial projections with static parameterization. We experiment with several design choices for this architecture and find spatial projections work well when they are linear and paired with multiplicative gating (Figure 1). We name the model gMLP because it is built out of basic MLP layers with gating.
+
+We apply $\mathrm { g M L P }$ to image classification and obtain strong results on ImageNet. $\mathrm { g M L P }$ achieves comparable performance with DeiT [8], namely Vision Transformer (ViT) [7] with improved regularization, in a similar training setup. With $66 \%$ less parameters, a gMLP model is $3 \%$ more accurate than MLP-Mixer [20]. Together with Tolstikhin et al. [20], Melas-Kyriazi [21], Touvron et al. [22] and Ding et. al. [23], our results question the necessity of self-attention layers in Vision Transformers.
+
+We apply gMLP to masked language modeling (MLM) in the BERT [2] setup, one of the most wellestablished applications of Transformers, and find that it is as good as Transformers at minimizing perplexity during pretraining. Our experiments indicate that perplexity is only correlated with model capacity and is insensitive to the presence of self-attention. As capacity increases, we observe that
+
+def gmlp_block(x, d_model, d_ffn): shortcut $=$ x $\mathrm { ~ x ~ } =$ norm(x, axis="channel") $\mathrm { ~ x ~ } =$ proj(x, d_ffn, axis="channel") $\mathrm { ~ x ~ } =$ gelu(x) $\mathrm { ~ x ~ } =$ spatial_gating_unit(x) $\mathrm { ~ x ~ } =$ proj(x, d_model, axis $= ^ { 1 1 }$ channel") return $\texttt { x + }$ shortcut
+
+![](images/018d82d7640366b577f21adb6e4e6c60f398a8eb4976b569ff0a2114a750edde.jpg)  
+Figure 1: Overview of the $\mathrm { g M L P }$ architecture with Spatial Gating Unit (SGU). The model consists of a stack of $L$ blocks with identical structure and size. All projection operations are linear and “ $\odot$ ” refers to element-wise multiplication (linear gating). The input and output protocols follow BERT for NLP and ViT for vision. Unlike Transformers, gMLPs do not require positional encodings, nor is it necessary to mask out the paddings during NLP finetuning.
+
+def spatial_gating_unit $\mathbf { \Psi } ( \mathbf { x } )$ : u, $\tt { v } =$ split(x, axis="channel") $\tt { v } =$ norm(v, axis="channel") n = get_dim(v, axis $\mathrel { \mathop : } = \mathrel { \mathop : }$ spatial") v = proj(v, n, axis="spatial", init_bias $\mathrel { \mathop : } = 1$ ) return u ∗ v
+
+both pretraining and finetuning metrics for gMLPs improve as quickly as for Transformers. This is remarkable because it indicates ${ \mathrm { g M L P s } }$ scale just as well as Transformers despite the absence of self-attention, and any performance gap can always be offset by training a larger model with increased data and compute. With a standard 256-batch size $\times \ 1 \mathbf { M }$ -step training setup as in original BERT, a large $\mathrm { g M L P }$ model achieves $8 7 . 7 \%$ accuracy on MNLI and $8 2 . 1 \%$ F1 on SQuAD v2.0. Note, these are better than the $\mathbf { B E R T _ { l a r g e } }$ results reported in Devlin et al. [2] obtained using Transformers.
+
+For BERT’s finetuning, Transformers can be more practically advantageous over gMLPs on tasks that require cross-sentence alignment (e.g., by $0 . 8 \%$ on MNLI-m in the 300M-param regime), even with similar pretraining perplexity. This problem can be addressed by making gMLPs substantially larger— $3 \times$ as large as Transformers. A more practical solution is to blend in only a tiny bit of selfattention—a single-head self-attention with size up to 128 is sufficient to make $\mathrm { g M L P s }$ outperform Transformers on all NLP tasks we evaluated with even better parameter efficiency. The improvement is sometimes very significant (e.g., $+ 4 . 4 \%$ on SQuAD $\mathrm { v } 2 . 0$ over $\mathbf { B E R T _ { l a r g e } }$ ).
+
+Overall, the surprising effectiveness of gMLPs in both vision and NLP domains suggests that selfattention is not a necessary ingredient for scaling up machine learning models, although it can be a useful addition depending on the task. With increased data and compute, models with simpler spatial interaction mechanisms such as $\mathrm { g M L P }$ can be as powerful as Transformers and the capacity allocated to self-attention can be either removed or substantially reduced.
+
+# 2 Model
+
+Our model, gMLP, consists of a stack of $L$ blocks with identical size and structure. Let $\ b { X } \in \mathbb { R } ^ { n \times d }$ be the token representations with sequence length $n$ and dimension $d$ . Each block is defined as:
+
+$$
+Z = \sigma ( X U ) , \qquad \tilde { Z } = s ( Z ) , \qquad Y = \tilde { Z } V
+$$
+
+where $\sigma$ is an activation function such as GeLU [24]. $U$ and $V$ define linear projections along the channel dimension—the same as those in the FFNs of Transformers (e.g., their shapes are $7 6 8 \times 3 0 7 2$ and $3 0 7 2 \times 7 6 8$ for $\mathbf { B E R T _ { b a s e } }$ ). Shortcuts, normalizations and biases are omitted for brevity.
+
+A key ingredient in the aforementioned formulation is $s ( \cdot )$ , a layer which captures spatial interactions (see below). When $s$ is an identity mapping, the above transformation degenerates to a regular FFN, where individual tokens are processed independently without any cross-token communication. One of our major focuses is therefore to design a good $s$ capable of capturing complex spatial interactions across tokens. The overall block layout is inspired by inverted bottlenecks [25] which define $s ( \cdot )$ as a spatial depthwise convolution. Note, unlike Transformers, our model does not require position embeddings because such information will be captured in $s ( \cdot )$ .
+
+Our model uses exactly the same input and output protocols as BERT (for NLP) and ViT (for vision). For example, when finetuning on language tasks, we concatenate together multiple text segments followed by paddings, and the predictions are deduced from the last-layer representation of a reserved <cls> symbol. Although many of these protocols were introduced for Transformers and hence can be suboptimal for gMLPs, strictly following them helps avoid confounding factors in our experiments and makes our layers more compatible with existing Transformer implementations.
+
+# 2.1 Spatial Gating Unit
+
+To enable cross-token interactions, it is necessary for the layer $s ( \cdot )$ to contain a contraction operation over the spatial dimension. The simplistic option would be a linear projection:
+
+$$
+f _ { W , b } ( Z ) = W Z + b
+$$
+
+where $W \in \mathbb { R } ^ { n \times n }$ is a matrix for which the size is the same as the sequence length, $n$ , and $b$ refers token-specific biases. For example, if the padded input sequence has 128 tokens, the shape for $W$ will be $1 2 8 \times 1 2 8$ . Unlike self-attention where $W ( Z )$ is dynamically generated from $Z$ , the spatial projection matrix $W$ here in Equation (2) is independent from the input representations.
+
+In this work, we formulate layer $s ( \cdot )$ as the output of linear gating:
+
+$$
+s ( Z ) = Z \odot f _ { W , b } ( Z )
+$$
+
+where $\odot$ denotes element-wise multiplication. For training stability, we find it critical to initialize $W$ as near-zero values and $b$ as ones, meaning that $f _ { W , b } ( Z ) \approx { \bf 1 }$ and therefore $s ( Z ) \approx Z$ at the beginning of training. This initialization ensures each gMLP block behaves like a regular FFN at the early stage of training, where each token is processed independently, and only gradually injects spatial information across tokens during the course of learning.
+
+We further find it effective to split $Z$ into two independent parts $( Z _ { 1 } , Z _ { 2 } )$ along the channel dimension for the gating function and for the multiplicative bypass:
+
+$$
+s ( Z ) = Z _ { 1 } \odot f _ { W , b } ( Z _ { 2 } )
+$$
+
+We also normalize the input to $f _ { W , b }$ which empirically improves stability of large NLP models. This gives us the unit illustrated in Figure 1, which we refer to as the Spatial Gating Unit (SGU) in the rest of the paper. In Table 3, we provide ablation studies to compare SGU with several other variants of $s ( \cdot )$ , showing that it works better and narrows the performance gap with self-attention.
+
+Connections to Existing Layers. The overall formulation of SGU resembles Gated Linear Units (GLUs) [26, 27, 28] as well as earlier works including Highway Networks [29] and LSTM-RNNs [11]. A key distinction is that our gating is computed based on a projection over the spatial (cross-token) dimension rather than the channel (hidden) dimension. SGU is also related to Squeeze-and-Excite (SE) blocks [30] in terms of element-wise multiplication. However, different from SE blocks, SGU does not contain cross-channel projections at all, nor does it enforce permutation invariance (a key feature for content-based attentive modules) due to its static parameterization for the spatial transformation. The spatial projection in SGU could in theory learn to express superficial depthwise convolutions—unlike typical depthwise convolutions with channel-specific filters, SGU learns only a single transformation shared across channels. Finally, we note SGUs offer an alternative mechanism to capture high-order relationships other than self-attention. Specifically, the output for Equation (3) contains up to 2nd-order interactions (e.g., $z _ { i } z _ { j }$ ) whereas output for self-attention (assuming no nonlinearity) contains up to 3rd-order interactions (e.g., $q _ { i } k _ { j } v _ { k } )$ . In terms of computation cost, SGU has $n ^ { 2 } e / 2$ multiply-adds which is comparable to the $2 n ^ { 2 } d$ of dot-product self-attention.1 Both are linear over the input channel size and quadratic over the sequence length $n$ .
+
+# 3 Image Classification
+
+Here we examine $\mathrm { g M L P }$ in the vision domain by applying it to the image classification task on ImageNet [31] without using extra data. We compare our MLP-like models with recent attentive models based on vanilla Transformers, including Vision Transformer (ViT) [7], DeiT [8] (ViT with improved regularization), and several other representative convolutional networks.
+
+Table 1 summarizes the configurations of our gMLP image classification models. The input and output protocols follow ViT/B16 where the raw image is converted into $1 6 \times 1 6$ patches at the stem. The depth and width are chosen so that the models are comparable with ViT/DeiT in capacity. Like Transformers, we find gMLPs tend to drastically overfit the training data. We therefore apply a similar regularization recipe as the one used in DeiT.2 To avoid extensive tuning, we adjust only the strengths of stochastic depth [32] as we move from smaller to larger models in Table 1. All the other hyperparameters remain shared across our three models. See Appendix A.1 for details.
+
+Table 1: Architecture specifications of $\mathrm { g M L P }$ models for vision.   
+
+<table><tr><td></td><td>#L</td><td>dmodel</td><td>dffn</td><td>Params (M)</td><td>FLOPs (B)</td><td>Survival Prob</td></tr><tr><td>gMLP-Ti</td><td>30</td><td>128</td><td>768</td><td>5.9</td><td>2.7</td><td>1.00</td></tr><tr><td>gMLP-S</td><td>30</td><td>256</td><td>1536</td><td>19.5</td><td>8.9</td><td>0.95</td></tr><tr><td>gMLP-B</td><td>30</td><td>512</td><td>3072</td><td>73.4</td><td>31.6</td><td>0.80</td></tr></table>
+
+Our ImageNet results are summarized in Table 1 and Figure 2.3 It is interesting to see that gMLPs are comparable with DeiT [8], namely ViT [7] trained using improved regularization. The results suggest that models without self-attention can be as data-efficient as Transformers for image classification. In fact, when the models are properly regularized, their accuracies seem better correlated with capacity instead of the presence of self-attention. Moreover, the accuracy-parameter/FLOPs tradeoff of gMLPs surpasses all concurrently proposed MLP-like architectures [20, 21, 22], which we attribute to the effectiveness of our Spatial Gating Unit (see Table 3 in the next section for an ablation). We also note while ${ \mathrm { g M L P s } }$ are competitive with vanilla Transformers, their performance is behind the best existing ConvNet models (e.g., [33, 34]) or hybrid models (e.g., [35, 36, 37, 38, 10]).
+
+Table 2: ImageNet-1K results without extra data.   
+
+<table><tr><td rowspan="2">Model</td><td>ImageNet Top-1 (%)*</td><td>Input Resolution</td><td>Params (M)</td><td>MAdds (B)</td></tr><tr><td colspan="4">ConvNets</td></tr><tr><td>ResNet-152 [16]</td><td>78.3</td><td>224</td><td>60</td><td>11.3</td></tr><tr><td>RegNetY-8GF[39]</td><td>81.7</td><td>224</td><td>39</td><td>8.0</td></tr><tr><td>EfficientNet-B0 [17]</td><td>77.1</td><td>224</td><td>5</td><td>0.39</td></tr><tr><td>EfficientNet-B3[17]</td><td>81.6</td><td>300</td><td>12</td><td>1.8</td></tr><tr><td>EfficientNet-B7 [17]</td><td>84.3</td><td>600</td><td>66</td><td>37.0</td></tr><tr><td>NFNet-F0 [33]</td><td>83.6</td><td>192</td><td>72</td><td>12.4</td></tr><tr><td></td><td colspan="4">Transformers</td></tr><tr><td>ViT-B/16 [7]</td><td>77.9</td><td>384</td><td>86</td><td>55.4</td></tr><tr><td>ViT-L/16 [7]</td><td>76.5</td><td>384</td><td>307</td><td>190.7</td></tr><tr><td>DeiT-Ti [8] (ViT+reg)</td><td>72.2</td><td>224</td><td>5</td><td>1.3</td></tr><tr><td>DeiT-S [8](ViT+reg)</td><td>79.8</td><td>224</td><td>22</td><td>4.6</td></tr><tr><td>DeiT-B [8] (ViT+reg)</td><td>81.8</td><td>224</td><td>86</td><td>17.5</td></tr><tr><td></td><td colspan="4">MLP-like†</td></tr><tr><td>Mixer-B/16 [20]</td><td>76.4</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-B/16 (our setup)</td><td>77.3</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-L/16 [20]</td><td>71.8</td><td>224</td><td>207</td><td>44.8</td></tr><tr><td>ResMLP-12 [22]</td><td>76.6</td><td>224</td><td>15</td><td>3.0</td></tr><tr><td>ResMLP-24 [22]</td><td>79.4</td><td>224</td><td>30</td><td>6.0</td></tr><tr><td>ResMLP-36[22]</td><td>79.7</td><td>224</td><td>45</td><td>8.9</td></tr><tr><td>gMLP-Ti (ours)</td><td>72.3</td><td>224</td><td>6</td><td>1.4</td></tr><tr><td>gMLP-S (ours)</td><td>79.6</td><td>224</td><td>20</td><td>4.5</td></tr><tr><td>gMLP-B (ours)</td><td>81.6</td><td>224</td><td>73</td><td>15.8</td></tr></table>
+
+\* Standard deviation across multiple independent runs is around 0.1. † Tokenization & embedding process at the stem can be viewed as a convolution.
+
+Figure 3 visualizes the spatial projection matrices in gMLP-B. Remarkably, the spatial weights after learning exhibit both locality and spatial invariance. In other words, each spatial projection matrix effectively learns to perform convolution with a data-driven, irregular (non-square) kernel shape.
+
+![](images/4ce114a6aa7138cbd566c0e0b5ee0dbdefe6fbfc6e3bb7bd0ddb7399d3c6d83b.jpg)  
+Figure 2: ImageNet accuracy vs model capacity.
+
+![](images/e9c6f27dad6f981df32081593ce223c652c9c19a1ac8f3a7e6353aba3fdced9b.jpg)  
+Figure 3: Spatial projection weights in gMLPB. Each row shows the filters (reshaped into 2D) for a selected set of tokens in the same layer.
+
+# 4 Masked Language Modeling with BERT
+
+Here we conduct empirical studies over the masked language modeling (MLM) task. The input/output protocol for both pretraining and finetuning follows BERT [2]. Different from Transformer-based models, we do not use positional encodings. We also find it unnecessary to mask out <pad> tokens in gMLP blocks during finetuning as the model can quickly learn to ignore them. For ablations and case studies, all models are trained with batch size 2048, max length 128 for 125K steps over the RealNews-like subset of C4 [5]. For main results, models are trained with batch size 256, max length 512 for 1M steps over the full English C4 dataset. See Appendix A.2 for details.
+
+Our preliminary MLM experiments show that gMLPs always learn Toeplitz-like matrices as the spatial weights (Appendix C). This means ${ \mathrm { g M L P s } }$ are able to learn the notion of shift invariance from data, a property naturally implied by the MLM task where any offset of the input sequence does not affect the slot filling outcome. In this case, the learned $f _ { W , b } ( \cdot )$ acts like a 1-d convolution whose kernel size equals the entire sequence length (unlike depthwise convolution with channel-specific filters, here the same $W$ is shared across channels). In the following MLM experiments, we restrict $W$ to be a Toeplitz matrix to avoid redundant model parameterization (since $W$ will be Toeplitz-like regardless after learning). Note this constraint is empirically quality-neutral.
+
+# 4.1 Ablation: The Importance of Gating in gMLP for BERT’s Pretraining
+
+In Table 3 below, we establish baselines for our ablation studies. These include:
+
+1. BERT with a Transformer architecture and learnable absolute position embeddings.   
+2. BERT with a Transformer architecture and T5-style learnable relative position biases [5]. The biases are both layer- and head-specific as we find this yields the best results.   
+3. Same as above, but we remove all content-dependent terms inside the softmax and only retain the relative positional biases. This baseline is a straightforward variant of Transformers without self-attention, which can also be viewed as a Random Synthesizer [40].   
+4. MLP-Mixer [20] which replaces the multi-head self-attention module in Transformers with a two-layer spatial MLP. This model was developed for image classification and here we investigate it on MLM tasks using the same training setup with BERT and gMLP.
+
+We compare these baselines against several versions of ${ \mathrm { g M L P s } }$ with similar sizes in Table 3. Note that Multiplicative, Split (last row) is the Spatial Gating Unit we describe in the method section and use in the rest of the paper. First, SGU outperforms other variants in perplexity. Secondly and remarkably, gMLP with SGU also achieves perplexity comparable to Transformer. Note the difference between the strongest baseline (perplexity ${ = } 4 . 2 6 $ ) and ours (perplexity ${ = } 4 . 3 5$ ) is insignificant relative to the perplexity change when the models are scaled (see Table 4 in the next section). Spatial projection weights learned by ${ \mathrm { g M L P s } }$ are visualized in Figure 4.
+
+Table 3: MLM validation perplexities of Transformer baselines and four versions of ${ \mathrm { g M L P s } }$ . $f$ refers to the spatial linear projection in Equation (2) with input normalization. The MLP-Mixer baseline model has $_ { \mathrm { L } = 2 4 }$ layers with $d _ { \mathrm { m o d e l } } = 7 6 8$ , $d _ { \mathrm { s p a t i a l } } { = } 3 8 4$ and $d _ { \mathrm { f f n } } { = } 3 0 7 2$ . Each gMLP model has $_ { \mathrm { L } = 3 6 }$ layers with $d _ { \mathrm { m o d e l } } = 5 1 2$ and $d _ { \mathrm { f f n } } = 3 0 7 2$ . No positional encodings are used for Mixer or gMLPs.   
+
+<table><tr><td>Model</td><td>Perplexity*</td><td>Params (M)</td></tr><tr><td>BERTbase</td><td>4.37</td><td>110</td></tr><tr><td>BERTbase + rel pos</td><td>4.26</td><td>110</td></tr><tr><td>BERTbase + rel pos - attn</td><td>5.64</td><td>96</td></tr><tr><td>MLP-Mixer</td><td>5.34</td><td>112</td></tr><tr><td>Linear gMLP,s(Z)= f(Z)</td><td>5.14</td><td>92</td></tr><tr><td>Additive gMLP,s(Z)= Z+ f(Z)</td><td>4.97</td><td>92</td></tr><tr><td>Multiplicative gMLP,s(Z) = Z  f(Z)</td><td>4.53</td><td>92</td></tr><tr><td>Multiplicative,Split gMLP,s(Z) = Z1  f(Z2),Z= Zi||Z2</td><td>4.35</td><td>102</td></tr></table>
+
+Standard deviation across multiple independent runs is around 0.01.
+
+![](images/731fe5bbb1001c5b9f64529e6e9ed073521439ba52c88469a01d1124d0dfaa34.jpg)  
+Figure 4: Visualization of the spatial filters in $\mathrm { g M L P }$ learned on the MLM task. For each layer in the model we plot the row in $W$ associated with the token in the middle of the sequence. The $\mathbf { X }$ -axis of each subplot has a length of 128 which equal the number of tokens in the sequence. The learned filters appear to be smooth and have several types: forward-looking (e.g., 1st in 2nd row), backward-looking (e.g., 5th in 2nd row) and bi-directional (e.g., 2nd last in the last row).
+
+# 4.2 Case Study: The Behavior of $\mathbf { g } \mathbf { M L P }$ as Model Size Increases
+
+In Table 4, we investigate the scaling properties of Transformers and gMLPs in BERT as their model capacity grows. Specifically, we scale the depth of these models by a factor of $\{ 0 . 5 , 1 , 2 , 4 \} \times$ and report the their pretraining MLM perplexities on the validation set as well as finetuning results on the dev sets of two tasks in GLUE [41]. Note each individual Transformer layer is effectively two consecutive blocks: one for self-attention and one for FFN. In the table below we use the notation of $1 2 + 1 2$ to refer to 12 of self-attention blocks plus 12 of FFN blocks in the Transformer baselines.
+
+Table 4: Pretraining and dev-set finetuning results over increased model capacity. We use the relative positional encoding scheme for Transformers which performs the best in Table 3.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>#L</td><td rowspan=1 colspan=2>Params (M) Perplexity</td><td rowspan=1 colspan=1>SST-2 MNLI-m</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>6+618</td><td rowspan=1 colspan=1>6759</td><td rowspan=1 colspan=1>4.915.25</td><td rowspan=1 colspan=1>90.4    81.591.2     77.7</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>12+1236</td><td rowspan=1 colspan=1>110102</td><td rowspan=1 colspan=1>4.264.35</td><td rowspan=1 colspan=1>91.3     83.392.3     80.9</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>24+2472</td><td rowspan=1 colspan=1>195187</td><td rowspan=1 colspan=1>3.833.79</td><td rowspan=1 colspan=1>92.1     85.293.5     82.8</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>48+48144</td><td rowspan=1 colspan=1>365357</td><td rowspan=1 colspan=1>3.473.43</td><td rowspan=1 colspan=1>92.8     86.395.1     84.6</td></tr></table>
+
+The results above show that a deep enough gMLP is able to match and even outperform the perplexity of Transformers with comparable capacity.4 In addition, the perplexity-parameter relationships for both architecture families approximately follow a power law (left of Figure 5). This implies the empirical scaling laws originally observed for Transformer-based language models [42] might be broadly applicable across different model families.
+
+![](images/03ebb35b54fcbc8c3b3a364ab001ed17abb00c96397df6daed10ed232e24238b.jpg)  
+Figure 5: Scaling properties with respect to perplexity and finetuning accuracies. The figures show that for pretraining, gMLPs are equally good at optimizing perplexity as Transformers. For finetuning, the two model families exhibit comparable scalability despite task-specific offsets.
+
+Table 4 also leads to an interesting observation that the pretraining perplexities across different model families are not equal in terms of finetuning. While gMLPs outperform Transformers on SST-2, they are worse on MNLI. The results imply that the finetuning performance for NLP tasks is a function of not only the perplexity but also the inductive bias in the architecture. Figure 5 shows that despite the architecture-specific discrepancies between pretraining and finetuning, gMLPs and Transformers exhibit comparable scalability (slope) on both finetuning tasks. This means one can always offset the gap by enlarging the model capacity. In other words, the results indicate that model scalability with respect to downstream metrics can be independent from the presence of self-attention.
+
+# 4.3 Ablation: The Usefulness of Tiny Attention in BERT’s Finetuning
+
+So far we have found that self-attention is not a required component to achieve strong MLM perplexity or scalability. At the meantime, we also identified NLP finetuning tasks where gMLPs transfer less well than Transformers (Table 4). The fact that our MLP-like model is advantageous on SST-2 but worse on MNLI is particularly informative—the former is a single-sentence task whereas the latter involves sentence pairs (premise and hypothesis) [43]. We suspect the role of self-attention during finetuning is related to cross-sentence alignment.
+
+To isolate the effect of self-attention, we experiment with a hybrid model where a tiny self-attention block is attached to the gating function of gMLP (Figure 6). Since gMLP itself is already capable in capturing spatial relationships, we hypothesize that this extra self-attention module does not have to be heavy, and that its presence is more relevant than its capacity. A typical tiny attention module in our experiments has only a single head with size 64, significantly smaller than a typical multi-head self-attention in Transformers with 12 heads and a total size of 768. In the following, we refer to the hybrid model, namely $\mathrm { g M L P }$ with a tiny self-attention, as aMLP (“a” for attention).
+
+# Pseudo-code for the tiny attention module
+
+def tiny_attn(x, d_out, d_attn=64): qkv $=$ proj(x, 3 ∗ d_attn, axis="channel") q, k, v $=$ split(qkv, 3, axis="channel") $\kappa =$ einsum("bnd,bmd−>bnm", q, k) a = softmax(w $^ *$ rsqrt(d_attn)) x = einsum("bnm,bmd−>bnd", a, v) return proj(x, d_out, axis="channel")
+
+![](images/e228c75c14097e90bf3596da75b5d6c0ccbec039247e6155396cb13236ea093c.jpg)  
+Figure 6: Hybrid spatial gating unit with a tiny self-attention module. We use the normalized input of the $\mathrm { g M L P }$ block (endpoint after the input normalization and right before the channel expansion) as the input to the tiny self-attention. For SGU we have ${ d _ { \mathrm { o u t } } = d _ { \mathrm { f f n } } / 2 }$ due to the channel split.
+
+In Figure 7, we investigate the transferability of MLM models via the calibration plots between their pretraining perplexities and finetuning metrics. Models evaluated include $\mathbf { B E R T _ { b a s e } }$ , gMLP and its hybrid version aMLP with a 64-d single-head self-attention (Figure 6). The data points were collected by varying the model depth by $\{ 0 . 5 , 1 , 2 \} \times$ or data by $\{ 1 , 2 , 4 , 8 \} \times$ . It can be seen that gMLPs transfer better to SST-2 than Transformers regardless of the presence of self-attention, While gMLP performs worse on MNLI, attaching a tiny bit of self-attention is sufficient to close the gap. In Appendix D we visualize the tiny self-attention modules in aMLP over MNLI examples, showing that they are primarily responsible for the alignment between sentence pairs.
+
+![](images/263412114ce248c9b9dccbc4dbdcbba36ac2aa648dce820d76c34069690a0cd1.jpg)  
+Figure 7: Transferability from MLM pretraining perpexity to finetuning accuracies on GLUE. aMLP refers to $\mathrm { g M L P }$ enhanced with a 64-d single-head self-attention, as illustrated in Figure 6. In contrast, each self-attention module in the BERT baseline contains 12 heads with a total size of 768.
+
+In Figure 8 we put together the scaling properties of the three models, showing that aMLP $\mathrm { ( g M L P + }$ tiny attention) consistently outperforms Transformer on both finetuning tasks.
+
+![](images/395d749d892cbad8add03840c7c07a237a2c72cef704c6aff4263e86c3de1d09.jpg)  
+Figure 8: Comparing the scaling properties of Transformers, $\mathrm { g M L P s }$ and aMLPs (with 64-d, singlehead attention). Results were obtained using the same setup in Section 4.2.
+
+# 4.4 Main Results for MLM in the BERT Setup
+
+Below we present pretraining and finetuning results in the full BERT setup. Different from ablation and case studies, here we use the full English C4 dataset and adopt a common MLM setup with batch size 256, max length 512 and 1M training steps. For fair comparison, we adjust the depth and width of $\mathrm { g M L P s }$ to ensure comparable model capacity with the Transformer baselines. The model specifications are given in Table 5 and hyperparameters are detailed in Appendix A.2. For finetuning, we report the dev-set performance for SST-2 and MNLI in GLUE [41] and each result entry was obtained by taking the median of five independent runs. In addition, we report finetuning results on SQuAD [44, 45] to test the models’ ability in reasoning over a longer context.
+
+Results are presented in Table 6. Consistent with our findings earlier in Section 4.1 and Section 4.2, gMLPs are competitive with Transformers in terms of perplexity, especially in the larger scale setup. There are several observations related to the finetuning results:
+
+First, on finetuning tasks where gMLPs underperform Transformers, the performance gap tends to narrow as the model capacity increases. For example, while $\mathrm { g M L P }$ performs worse by $8 . 5 \%$ on SQuAD-v2.0 in the base scale, the performance gap relative to the baseline decreases to $2 . 7 \%$ at the larger scale. Notably, our ${ \mathrm { g M L P } } _ { \mathrm { l a r g e } }$ achieves $8 9 . 5 \%$ F1 on SQuAD-v1.1 without any self-attention or dynamic spatial parameterization [28], which is well above the $8 8 . 5 \%$ reported for $\mathbf { B E R T _ { b a s e } }$ in Devlin et al. [2] and is only $1 . 4 \%$ away from the original result for $\mathbf { B E R T _ { l a r g e } }$ . We also include one additional data point by scaling up $\mathrm { g M L P }$ even further. The resulting model, $\mathrm { g } \mathrm { \bar { M } L P _ { \mathrm { x l a r g e } } }$ , outperforms $\mathbf { B E R T _ { l a r g e } }$ on SQuAD- $\mathbf { \nabla } \cdot \mathbf { v } 2 . 0 $ —a difficult task involving question-answer pairs—without any self-attention. While this is not a fair comparison due to different model sizes, it is an existence proof that MLP-like models can be competitive with Transformers on challenging NLP tasks.
+
+Table 5: Model specifications in the full BERT setup.   
+
+<table><tr><td></td><td>Params (M)</td><td>FLOPs (B)</td><td>#L</td><td>dmodel</td><td>dffn</td></tr><tr><td rowspan="3">BERTbase gMLPbase aMLPbase</td><td>110</td><td>100.8</td><td>12+12</td><td>768</td><td>3072</td></tr><tr><td>130</td><td>158.0</td><td>48</td><td>512</td><td>3072</td></tr><tr><td>109</td><td>128.9</td><td>36</td><td>512</td><td>3072</td></tr><tr><td rowspan="3">BERTlarge gMLPlarge aMLPlarge</td><td>336</td><td>341.2</td><td>24+24</td><td>1024</td><td>4096</td></tr><tr><td>365</td><td>430.1</td><td>96</td><td>768</td><td>3072</td></tr><tr><td>316</td><td>370.3</td><td>72</td><td>768</td><td>3072</td></tr><tr><td> gMLPxlarge</td><td>941</td><td>1091.3</td><td>144</td><td>1024</td><td>4096</td></tr></table>
+
+Table 6: Pretraining perplexities and dev-set results for finetuning. “ours” indicates models trained using our setup. We report accuracies for SST-2 and MNLI, and F1 scores for SQuAD v1.1/2.0.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Perplexity</td><td rowspan="2">SST-2</td><td rowspan="2">MNLI (m/mm)</td><td colspan="2">SQuAD</td><td rowspan="2">Attn Size</td><td rowspan="2">Params (M)</td></tr><tr><td>v1.1</td><td>v2.0</td></tr><tr><td>BERTbase [2]</td><td>1</td><td>92.7</td><td>84.4/-</td><td>88.5</td><td>76.3</td><td>768 (64 × 12)</td><td>110</td></tr><tr><td>BERTbase (ours)</td><td>4.17</td><td>93.8</td><td>85.6/85.7</td><td>90.2</td><td>78.6</td><td>768 (64 × 12)</td><td>110</td></tr><tr><td>gMLPbase</td><td>4.28</td><td>94.2</td><td>83.7/84.1</td><td>86.7</td><td>70.1</td><td>1</td><td>130</td></tr><tr><td>aMLPbase</td><td>3.95</td><td>93.4</td><td>85.9/85.8</td><td>90.7</td><td>80.9</td><td>64</td><td>109</td></tr><tr><td>BERTlarge [2]</td><td>1</td><td>93.7</td><td>86.6/-</td><td>90.9</td><td>81.8</td><td>1024 (64 × 16)</td><td>336</td></tr><tr><td>BERTlarge (ours)</td><td>3.35</td><td>94.3</td><td>87.0/87.4</td><td>92.0</td><td>81.0</td><td>1024 (64 × 16)</td><td>336</td></tr><tr><td>gMLPlarge</td><td>3.32</td><td>94.8</td><td>86.2/86.5</td><td>89.5</td><td>78.3</td><td>1</td><td>365</td></tr><tr><td>aMLPlarge</td><td>3.19</td><td>94.8</td><td>88.4/88.4</td><td>92.2</td><td>85.4</td><td>128</td><td>316</td></tr><tr><td>gMLPxlarge</td><td>2.89</td><td>95.6</td><td>87.7/87.7</td><td>90.9</td><td>82.1</td><td>1</td><td>941</td></tr></table>
+
+Furthermore, we show that blending in a tiny single-head self-attention of size either 64 or 128 is sufficient to make gMLPs outperform Transformers of similar capacity, sometimes by a significant margin. For example, our hybrid model $\mathrm { a M L P _ { l a r g e } }$ achieves $4 . 4 \%$ higher F1 than Transformers on SQuAD-v2.0. The results suggest that the capacity in the multi-head self-attention of Transformers can be largely redundant, and that the majority of its functionalities can be captured by the spatial gating unit in gMLPs. The results also imply that the inductive biases in the spatial gating unit of gMLPs and the tiny attention are complementary to each other. While the benefits of architectural inductive bias may vanish over increased compute, tiny attention does improve the practical value of gMLPs in the regime that we investigate in this work.
+
+# 5 Conclusion
+
+Since the seminal work of Vaswani et al. [1], Transformers have been widely adopted across NLP and computer vision. This adoption has enabled many impressive results especially in NLP. To date, it is still unclear what empowers such success: is it the feedforward nature of Transformers or is it the multi-head self-attention layers in Transformers?
+
+Our work suggests a simpler alternative to the multi-head self-attention layers in Transformers. We show that gMLPs, a simple variant of MLPs with gating, can be competitive with Transformers in terms of BERT’s pretraining perplexity and ViT’s accuracy. gMLPs are also comparable with Transformers in terms of the scalability over increased data and compute. As for BERT finetuning, we find gMLPs can achieve appealing results on challenging tasks such as SQuAD without self-attention, and can significantly outperform Transformers in certain cases. We also find the inductive bias in Transformer’s multi-head self-attention useful on downstream tasks that require cross-sentence alignment. However in those cases, making gMLP substantially larger closes the gap with Transformers. More practically, blending a small single-head self-attention into gMLP allows for an even better architecture without the need for increasing model size.
+
+# Acknowledgements
+
+We thank Gabriel Bender, Neil Houlsby, Thang Luong, Niki Parmar, Hieu Pham, Jascha SohlDickstein, Noam Shazeer, Ilya Sutskever, Jakob Uszkoreit and Ashish Vaswani for their feedback.
+
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+[36] Ashish Vaswani, Prajit Ramachandran, Aravind Srinivas, Niki Parmar, Blake Hechtman, and Jonathon Shlens. Scaling local self-attention for parameter efficient visual backbones. In CVPR, 2021.   
+[37] Aravind Srinivas, Tsung-Yi Lin, Niki Parmar, Jonathon Shlens, Pieter Abbeel, and Ashish Vaswani. Bottleneck transformers for visual recognition. arXiv preprint arXiv:2101.11605, 2021.   
+[38] Haiping Wu, Bin Xiao, Noel Codella, Mengchen Liu, Xiyang Dai, Lu Yuan, and Lei Zhang. Cvt: Introducing convolutions to vision transformers. arXiv preprint arXiv:2103.15808, 2021.   
+[39] Ilija Radosavovic, Raj Prateek Kosaraju, Ross Girshick, Kaiming He, and Piotr Dollár. Designing network design spaces. In CVPR, 2020.   
+[40] Yi Tay, Dara Bahri, Donald Metzler, Da-Cheng Juan, Zhe Zhao, and Che Zheng. Synthesizer: Rethinking self-attention in transformer models. arXiv preprint arXiv:2005.00743, 2020.   
+[41] Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R Bowman. Glue: A multi-task benchmark and analysis platform for natural language understanding. In ICLR, 2019.   
+[42] Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361, 2020.   
+[43] Adina Williams, Nikita Nangia, and Samuel R Bowman. A broad-coverage challenge corpus for sentence understanding through inference. In NAACL, 2017.   
+[44] Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. Squad: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In EMNLP, 2016.   
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\ No newline at end of file
diff --git a/parse/train/KBnXrODoBW/KBnXrODoBW_content_list.json b/parse/train/KBnXrODoBW/KBnXrODoBW_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..555ea3106704e7d1dc52ca779eecc832d7001ffb
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+++ b/parse/train/KBnXrODoBW/KBnXrODoBW_content_list.json
@@ -0,0 +1,1070 @@
+[
+    {
+        "type": "text",
+        "text": "Pay Attention to MLPs ",
+        "text_level": 1,
+        "bbox": [
+            359,
+            122,
+            640,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hanxiao Liu, Zihang Dai, David R. So, Quoc V. Le Google Research, Brain Team {hanxiaol,zihangd,davidso,qvl}@google.com ",
+        "bbox": [
+            323,
+            200,
+            676,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            279,
+            535,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers [1] have become one of the most important architectural innovations in deep learning and have enabled many breakthroughs over the past few years. Here we propose a simple network architecture, gMLP, based on MLPs with gating, and show that it can perform as well as Transformers in key language and vision applications. Our comparisons show that self-attention is not critical for Vision Transformers, as $\\mathrm { g M L P }$ can achieve the same accuracy. For BERT, our model achieves parity with Transformers on pretraining perplexity and is better on some downstream NLP tasks. On finetuning tasks where gMLP performs worse, making the $\\mathrm { g M L P }$ model substantially larger can close the gap with Transformers. In general, our experiments show that gMLP can scale as well as Transformers over increased data and compute. ",
+        "bbox": [
+            233,
+            309,
+            766,
+            460
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            484,
+            310,
+            502
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers [1] have enabled many breakthroughs in natural language processing (e.g., [2, 3, 4, 5, 6]) and have been shown to work well for computer vision (e.g., [7, 8, 9, 10]). Thanks to this success, Transformers have largely replaced LSTM-RNN [11] as the default architecture in NLP, and have become an appealing alternative to ConvNets [12, 13, 14, 15, 16, 17] in computer vision. ",
+        "bbox": [
+            174,
+            516,
+            825,
+            571
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The Transformer architecture combines two important concepts: (1) a recurrent-free architecture which computes the representations for each individual token in parallel, and (2) multi-head selfattention blocks which aggregate spatial information across tokens. On one hand, the attention mechanism [18] introduces the inductive bias that the spatial interactions should be dynamically parameterized based on the input representations. On the other hand, it is known that MLPs with static parameterization can represent arbitrary functions [19]. It therefore remains an open question whether the inductive bias in self-attention is essential to the remarkable effectiveness of Transformers. ",
+        "bbox": [
+            174,
+            578,
+            825,
+            675
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Here we study the necessity of self-attention modules in key language and vision applications of Transformers. Specifically, we propose an MLP-based alternative to Transformers without self-attention, which simply consists of channel projections and spatial projections with static parameterization. We experiment with several design choices for this architecture and find spatial projections work well when they are linear and paired with multiplicative gating (Figure 1). We name the model gMLP because it is built out of basic MLP layers with gating. ",
+        "bbox": [
+            174,
+            681,
+            825,
+            765
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We apply $\\mathrm { g M L P }$ to image classification and obtain strong results on ImageNet. $\\mathrm { g M L P }$ achieves comparable performance with DeiT [8], namely Vision Transformer (ViT) [7] with improved regularization, in a similar training setup. With $66 \\%$ less parameters, a gMLP model is $3 \\%$ more accurate than MLP-Mixer [20]. Together with Tolstikhin et al. [20], Melas-Kyriazi [21], Touvron et al. [22] and Ding et. al. [23], our results question the necessity of self-attention layers in Vision Transformers. ",
+        "bbox": [
+            174,
+            771,
+            825,
+            840
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We apply gMLP to masked language modeling (MLM) in the BERT [2] setup, one of the most wellestablished applications of Transformers, and find that it is as good as Transformers at minimizing perplexity during pretraining. Our experiments indicate that perplexity is only correlated with model capacity and is insensitive to the presence of self-attention. As capacity increases, we observe that ",
+        "bbox": [
+            174,
+            847,
+            825,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "def gmlp_block(x, d_model, d_ffn): shortcut $=$ x $\\mathrm { ~ x ~ } =$ norm(x, axis=\"channel\") $\\mathrm { ~ x ~ } =$ proj(x, d_ffn, axis=\"channel\") $\\mathrm { ~ x ~ } =$ gelu(x) $\\mathrm { ~ x ~ } =$ spatial_gating_unit(x) $\\mathrm { ~ x ~ } =$ proj(x, d_model, axis $= ^ { 1 1 }$ channel\") return $\\texttt { x + }$ shortcut ",
+        "bbox": [
+            508,
+            118,
+            753,
+            203
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/018d82d7640366b577f21adb6e4e6c60f398a8eb4976b569ff0a2114a750edde.jpg",
+        "image_caption": [
+            "Figure 1: Overview of the $\\mathrm { g M L P }$ architecture with Spatial Gating Unit (SGU). The model consists of a stack of $L$ blocks with identical structure and size. All projection operations are linear and “ $\\odot$ ” refers to element-wise multiplication (linear gating). The input and output protocols follow BERT for NLP and ViT for vision. Unlike Transformers, gMLPs do not require positional encodings, nor is it necessary to mask out the paddings during NLP finetuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            97,
+            486,
+            284
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "def spatial_gating_unit $\\mathbf { \\Psi } ( \\mathbf { x } )$ : u, $\\tt { v } =$ split(x, axis=\"channel\") $\\tt { v } =$ norm(v, axis=\"channel\") n = get_dim(v, axis $\\mathrel { \\mathop : } = \\mathrel { \\mathop : }$ spatial\") v = proj(v, n, axis=\"spatial\", init_bias $\\mathrel { \\mathop : } = 1$ ) return u ∗ v ",
+        "bbox": [
+            511,
+            214,
+            805,
+            280
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "both pretraining and finetuning metrics for gMLPs improve as quickly as for Transformers. This is remarkable because it indicates ${ \\mathrm { g M L P s } }$ scale just as well as Transformers despite the absence of self-attention, and any performance gap can always be offset by training a larger model with increased data and compute. With a standard 256-batch size $\\times \\ 1 \\mathbf { M }$ -step training setup as in original BERT, a large $\\mathrm { g M L P }$ model achieves $8 7 . 7 \\%$ accuracy on MNLI and $8 2 . 1 \\%$ F1 on SQuAD v2.0. Note, these are better than the $\\mathbf { B E R T _ { l a r g e } }$ results reported in Devlin et al. [2] obtained using Transformers. ",
+        "bbox": [
+            173,
+            392,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For BERT’s finetuning, Transformers can be more practically advantageous over gMLPs on tasks that require cross-sentence alignment (e.g., by $0 . 8 \\%$ on MNLI-m in the 300M-param regime), even with similar pretraining perplexity. This problem can be addressed by making gMLPs substantially larger— $3 \\times$ as large as Transformers. A more practical solution is to blend in only a tiny bit of selfattention—a single-head self-attention with size up to 128 is sufficient to make $\\mathrm { g M L P s }$ outperform Transformers on all NLP tasks we evaluated with even better parameter efficiency. The improvement is sometimes very significant (e.g., $+ 4 . 4 \\%$ on SQuAD $\\mathrm { v } 2 . 0$ over $\\mathbf { B E R T _ { l a r g e } }$ ). ",
+        "bbox": [
+            173,
+            483,
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+            582
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, the surprising effectiveness of gMLPs in both vision and NLP domains suggests that selfattention is not a necessary ingredient for scaling up machine learning models, although it can be a useful addition depending on the task. With increased data and compute, models with simpler spatial interaction mechanisms such as $\\mathrm { g M L P }$ can be as powerful as Transformers and the capacity allocated to self-attention can be either removed or substantially reduced. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            674,
+            259,
+            691
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model, gMLP, consists of a stack of $L$ blocks with identical size and structure. Let $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ be the token representations with sequence length $n$ and dimension $d$ . Each block is defined as: ",
+        "bbox": [
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+            705,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c7f7fd02ef54ad58deab3b63dffb850ea35636c7e3bb6788f9e24746d918a055.jpg",
+        "text": "$$\nZ = \\sigma ( X U ) , \\qquad \\tilde { Z } = s ( Z ) , \\qquad Y = \\tilde { Z } V\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            741,
+            647,
+            760
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is an activation function such as GeLU [24]. $U$ and $V$ define linear projections along the channel dimension—the same as those in the FFNs of Transformers (e.g., their shapes are $7 6 8 \\times 3 0 7 2$ and $3 0 7 2 \\times 7 6 8$ for $\\mathbf { B E R T _ { b a s e } }$ ). Shortcuts, normalizations and biases are omitted for brevity. ",
+        "bbox": [
+            173,
+            765,
+            825,
+            809
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A key ingredient in the aforementioned formulation is $s ( \\cdot )$ , a layer which captures spatial interactions (see below). When $s$ is an identity mapping, the above transformation degenerates to a regular FFN, where individual tokens are processed independently without any cross-token communication. One of our major focuses is therefore to design a good $s$ capable of capturing complex spatial interactions across tokens. The overall block layout is inspired by inverted bottlenecks [25] which define $s ( \\cdot )$ as a spatial depthwise convolution. Note, unlike Transformers, our model does not require position embeddings because such information will be captured in $s ( \\cdot )$ . ",
+        "bbox": [
+            173,
+            814,
+            825,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model uses exactly the same input and output protocols as BERT (for NLP) and ViT (for vision). For example, when finetuning on language tasks, we concatenate together multiple text segments followed by paddings, and the predictions are deduced from the last-layer representation of a reserved <cls> symbol. Although many of these protocols were introduced for Transformers and hence can be suboptimal for gMLPs, strictly following them helps avoid confounding factors in our experiments and makes our layers more compatible with existing Transformer implementations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Spatial Gating Unit ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            190,
+            349,
+            205
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To enable cross-token interactions, it is necessary for the layer $s ( \\cdot )$ to contain a contraction operation over the spatial dimension. The simplistic option would be a linear projection: ",
+        "bbox": [
+            173,
+            215,
+            823,
+            244
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e5c4dae68c79d1010c2b7fd3237f11b35cf0b787c253c05fa4e0267d371f418.jpg",
+        "text": "$$\nf _ { W , b } ( Z ) = W Z + b\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            251,
+            568,
+            268
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W \\in \\mathbb { R } ^ { n \\times n }$ is a matrix for which the size is the same as the sequence length, $n$ , and $b$ refers token-specific biases. For example, if the padded input sequence has 128 tokens, the shape for $W$ will be $1 2 8 \\times 1 2 8$ . Unlike self-attention where $W ( Z )$ is dynamically generated from $Z$ , the spatial projection matrix $W$ here in Equation (2) is independent from the input representations. ",
+        "bbox": [
+            173,
+            275,
+            825,
+            332
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this work, we formulate layer $s ( \\cdot )$ as the output of linear gating: ",
+        "bbox": [
+            173,
+            337,
+            611,
+            352
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b084e72db12a7f0a08d1156d8e0af26ab807cbf9481d25ba50788a491cfa7ca0.jpg",
+        "text": "$$\ns ( Z ) = Z \\odot f _ { W , b } ( Z )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            358,
+            571,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes element-wise multiplication. For training stability, we find it critical to initialize $W$ as near-zero values and $b$ as ones, meaning that $f _ { W , b } ( Z ) \\approx { \\bf 1 }$ and therefore $s ( Z ) \\approx Z$ at the beginning of training. This initialization ensures each gMLP block behaves like a regular FFN at the early stage of training, where each token is processed independently, and only gradually injects spatial information across tokens during the course of learning. ",
+        "bbox": [
+            173,
+            381,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We further find it effective to split $Z$ into two independent parts $( Z _ { 1 } , Z _ { 2 } )$ along the channel dimension for the gating function and for the multiplicative bypass: ",
+        "bbox": [
+            174,
+            457,
+            820,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6a604c6e6d1a2b5953c15b5a23e51838a39cd7db61dd6c11719cc258562c35a.jpg",
+        "text": "$$\ns ( Z ) = Z _ { 1 } \\odot f _ { W , b } ( Z _ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            492,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also normalize the input to $f _ { W , b }$ which empirically improves stability of large NLP models. This gives us the unit illustrated in Figure 1, which we refer to as the Spatial Gating Unit (SGU) in the rest of the paper. In Table 3, we provide ablation studies to compare SGU with several other variants of $s ( \\cdot )$ , showing that it works better and narrows the performance gap with self-attention. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Connections to Existing Layers. The overall formulation of SGU resembles Gated Linear Units (GLUs) [26, 27, 28] as well as earlier works including Highway Networks [29] and LSTM-RNNs [11]. A key distinction is that our gating is computed based on a projection over the spatial (cross-token) dimension rather than the channel (hidden) dimension. SGU is also related to Squeeze-and-Excite (SE) blocks [30] in terms of element-wise multiplication. However, different from SE blocks, SGU does not contain cross-channel projections at all, nor does it enforce permutation invariance (a key feature for content-based attentive modules) due to its static parameterization for the spatial transformation. The spatial projection in SGU could in theory learn to express superficial depthwise convolutions—unlike typical depthwise convolutions with channel-specific filters, SGU learns only a single transformation shared across channels. Finally, we note SGUs offer an alternative mechanism to capture high-order relationships other than self-attention. Specifically, the output for Equation (3) contains up to 2nd-order interactions (e.g., $z _ { i } z _ { j }$ ) whereas output for self-attention (assuming no nonlinearity) contains up to 3rd-order interactions (e.g., $q _ { i } k _ { j } v _ { k } )$ . In terms of computation cost, SGU has $n ^ { 2 } e / 2$ multiply-adds which is comparable to the $2 n ^ { 2 } d$ of dot-product self-attention.1 Both are linear over the input channel size and quadratic over the sequence length $n$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Image Classification ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we examine $\\mathrm { g M L P }$ in the vision domain by applying it to the image classification task on ImageNet [31] without using extra data. We compare our MLP-like models with recent attentive models based on vanilla Transformers, including Vision Transformer (ViT) [7], DeiT [8] (ViT with improved regularization), and several other representative convolutional networks. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the configurations of our gMLP image classification models. The input and output protocols follow ViT/B16 where the raw image is converted into $1 6 \\times 1 6$ patches at the stem. The depth and width are chosen so that the models are comparable with ViT/DeiT in capacity. Like Transformers, we find gMLPs tend to drastically overfit the training data. We therefore apply a similar regularization recipe as the one used in DeiT.2 To avoid extensive tuning, we adjust only the strengths of stochastic depth [32] as we move from smaller to larger models in Table 1. All the other hyperparameters remain shared across our three models. See Appendix A.1 for details. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/27f42f69254be8b6e8eeff8fb7a1820daeca371c25a2a48691d8d74cf19caa2f.jpg",
+        "table_caption": [
+            "Table 1: Architecture specifications of $\\mathrm { g M L P }$ models for vision. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>#L</td><td>dmodel</td><td>dffn</td><td>Params (M)</td><td>FLOPs (B)</td><td>Survival Prob</td></tr><tr><td>gMLP-Ti</td><td>30</td><td>128</td><td>768</td><td>5.9</td><td>2.7</td><td>1.00</td></tr><tr><td>gMLP-S</td><td>30</td><td>256</td><td>1536</td><td>19.5</td><td>8.9</td><td>0.95</td></tr><tr><td>gMLP-B</td><td>30</td><td>512</td><td>3072</td><td>73.4</td><td>31.6</td><td>0.80</td></tr></table>",
+        "bbox": [
+            258,
+            256,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our ImageNet results are summarized in Table 1 and Figure 2.3 It is interesting to see that gMLPs are comparable with DeiT [8], namely ViT [7] trained using improved regularization. The results suggest that models without self-attention can be as data-efficient as Transformers for image classification. In fact, when the models are properly regularized, their accuracies seem better correlated with capacity instead of the presence of self-attention. Moreover, the accuracy-parameter/FLOPs tradeoff of gMLPs surpasses all concurrently proposed MLP-like architectures [20, 21, 22], which we attribute to the effectiveness of our Spatial Gating Unit (see Table 3 in the next section for an ablation). We also note while ${ \\mathrm { g M L P s } }$ are competitive with vanilla Transformers, their performance is behind the best existing ConvNet models (e.g., [33, 34]) or hybrid models (e.g., [35, 36, 37, 38, 10]). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/123f13f92d912e05ece7dd96c75519ea83477124b08ac3939ad58b8b69f2b23d.jpg",
+        "table_caption": [
+            "Table 2: ImageNet-1K results without extra data. "
+        ],
+        "table_footnote": [
+            "\\* Standard deviation across multiple independent runs is around 0.1. † Tokenization & embedding process at the stem can be viewed as a convolution. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td>ImageNet Top-1 (%)*</td><td>Input Resolution</td><td>Params (M)</td><td>MAdds (B)</td></tr><tr><td colspan=\"4\">ConvNets</td></tr><tr><td>ResNet-152 [16]</td><td>78.3</td><td>224</td><td>60</td><td>11.3</td></tr><tr><td>RegNetY-8GF[39]</td><td>81.7</td><td>224</td><td>39</td><td>8.0</td></tr><tr><td>EfficientNet-B0 [17]</td><td>77.1</td><td>224</td><td>5</td><td>0.39</td></tr><tr><td>EfficientNet-B3[17]</td><td>81.6</td><td>300</td><td>12</td><td>1.8</td></tr><tr><td>EfficientNet-B7 [17]</td><td>84.3</td><td>600</td><td>66</td><td>37.0</td></tr><tr><td>NFNet-F0 [33]</td><td>83.6</td><td>192</td><td>72</td><td>12.4</td></tr><tr><td></td><td colspan=\"4\">Transformers</td></tr><tr><td>ViT-B/16 [7]</td><td>77.9</td><td>384</td><td>86</td><td>55.4</td></tr><tr><td>ViT-L/16 [7]</td><td>76.5</td><td>384</td><td>307</td><td>190.7</td></tr><tr><td>DeiT-Ti [8] (ViT+reg)</td><td>72.2</td><td>224</td><td>5</td><td>1.3</td></tr><tr><td>DeiT-S [8](ViT+reg)</td><td>79.8</td><td>224</td><td>22</td><td>4.6</td></tr><tr><td>DeiT-B [8] (ViT+reg)</td><td>81.8</td><td>224</td><td>86</td><td>17.5</td></tr><tr><td></td><td colspan=\"4\">MLP-like†</td></tr><tr><td>Mixer-B/16 [20]</td><td>76.4</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-B/16 (our setup)</td><td>77.3</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-L/16 [20]</td><td>71.8</td><td>224</td><td>207</td><td>44.8</td></tr><tr><td>ResMLP-12 [22]</td><td>76.6</td><td>224</td><td>15</td><td>3.0</td></tr><tr><td>ResMLP-24 [22]</td><td>79.4</td><td>224</td><td>30</td><td>6.0</td></tr><tr><td>ResMLP-36[22]</td><td>79.7</td><td>224</td><td>45</td><td>8.9</td></tr><tr><td>gMLP-Ti (ours)</td><td>72.3</td><td>224</td><td>6</td><td>1.4</td></tr><tr><td>gMLP-S (ours)</td><td>79.6</td><td>224</td><td>20</td><td>4.5</td></tr><tr><td>gMLP-B (ours)</td><td>81.6</td><td>224</td><td>73</td><td>15.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Figure 3 visualizes the spatial projection matrices in gMLP-B. Remarkably, the spatial weights after learning exhibit both locality and spatial invariance. In other words, each spatial projection matrix effectively learns to perform convolution with a data-driven, irregular (non-square) kernel shape. ",
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+        "img_path": "images/4ce114a6aa7138cbd566c0e0b5ee0dbdefe6fbfc6e3bb7bd0ddb7399d3c6d83b.jpg",
+        "image_caption": [
+            "Figure 2: ImageNet accuracy vs model capacity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/e9c6f27dad6f981df32081593ce223c652c9c19a1ac8f3a7e6353aba3fdced9b.jpg",
+        "image_caption": [
+            "Figure 3: Spatial projection weights in gMLPB. Each row shows the filters (reshaped into 2D) for a selected set of tokens in the same layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 Masked Language Modeling with BERT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here we conduct empirical studies over the masked language modeling (MLM) task. The input/output protocol for both pretraining and finetuning follows BERT [2]. Different from Transformer-based models, we do not use positional encodings. We also find it unnecessary to mask out <pad> tokens in gMLP blocks during finetuning as the model can quickly learn to ignore them. For ablations and case studies, all models are trained with batch size 2048, max length 128 for 125K steps over the RealNews-like subset of C4 [5]. For main results, models are trained with batch size 256, max length 512 for 1M steps over the full English C4 dataset. See Appendix A.2 for details. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our preliminary MLM experiments show that gMLPs always learn Toeplitz-like matrices as the spatial weights (Appendix C). This means ${ \\mathrm { g M L P s } }$ are able to learn the notion of shift invariance from data, a property naturally implied by the MLM task where any offset of the input sequence does not affect the slot filling outcome. In this case, the learned $f _ { W , b } ( \\cdot )$ acts like a 1-d convolution whose kernel size equals the entire sequence length (unlike depthwise convolution with channel-specific filters, here the same $W$ is shared across channels). In the following MLM experiments, we restrict $W$ to be a Toeplitz matrix to avoid redundant model parameterization (since $W$ will be Toeplitz-like regardless after learning). Note this constraint is empirically quality-neutral. ",
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+    {
+        "type": "text",
+        "text": "4.1 Ablation: The Importance of Gating in gMLP for BERT’s Pretraining ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Table 3 below, we establish baselines for our ablation studies. These include: ",
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+    {
+        "type": "text",
+        "text": "1. BERT with a Transformer architecture and learnable absolute position embeddings.   \n2. BERT with a Transformer architecture and T5-style learnable relative position biases [5]. The biases are both layer- and head-specific as we find this yields the best results.   \n3. Same as above, but we remove all content-dependent terms inside the softmax and only retain the relative positional biases. This baseline is a straightforward variant of Transformers without self-attention, which can also be viewed as a Random Synthesizer [40].   \n4. MLP-Mixer [20] which replaces the multi-head self-attention module in Transformers with a two-layer spatial MLP. This model was developed for image classification and here we investigate it on MLM tasks using the same training setup with BERT and gMLP. ",
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+    {
+        "type": "text",
+        "text": "We compare these baselines against several versions of ${ \\mathrm { g M L P s } }$ with similar sizes in Table 3. Note that Multiplicative, Split (last row) is the Spatial Gating Unit we describe in the method section and use in the rest of the paper. First, SGU outperforms other variants in perplexity. Secondly and remarkably, gMLP with SGU also achieves perplexity comparable to Transformer. Note the difference between the strongest baseline (perplexity ${ = } 4 . 2 6 $ ) and ours (perplexity ${ = } 4 . 3 5$ ) is insignificant relative to the perplexity change when the models are scaled (see Table 4 in the next section). Spatial projection weights learned by ${ \\mathrm { g M L P s } }$ are visualized in Figure 4. ",
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+    {
+        "type": "table",
+        "img_path": "images/230eb41e660952154dd9abe33bc6dbc3d8c04a14f867dc0c2c621db29dfc52f1.jpg",
+        "table_caption": [
+            "Table 3: MLM validation perplexities of Transformer baselines and four versions of ${ \\mathrm { g M L P s } }$ . $f$ refers to the spatial linear projection in Equation (2) with input normalization. The MLP-Mixer baseline model has $_ { \\mathrm { L } = 2 4 }$ layers with $d _ { \\mathrm { m o d e l } } = 7 6 8$ , $d _ { \\mathrm { s p a t i a l } } { = } 3 8 4$ and $d _ { \\mathrm { f f n } } { = } 3 0 7 2$ . Each gMLP model has $_ { \\mathrm { L } = 3 6 }$ layers with $d _ { \\mathrm { m o d e l } } = 5 1 2$ and $d _ { \\mathrm { f f n } } = 3 0 7 2$ . No positional encodings are used for Mixer or gMLPs. "
+        ],
+        "table_footnote": [
+            "Standard deviation across multiple independent runs is around 0.01. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Perplexity*</td><td>Params (M)</td></tr><tr><td>BERTbase</td><td>4.37</td><td>110</td></tr><tr><td>BERTbase + rel pos</td><td>4.26</td><td>110</td></tr><tr><td>BERTbase + rel pos - attn</td><td>5.64</td><td>96</td></tr><tr><td>MLP-Mixer</td><td>5.34</td><td>112</td></tr><tr><td>Linear gMLP,s(Z)= f(Z)</td><td>5.14</td><td>92</td></tr><tr><td>Additive gMLP,s(Z)= Z+ f(Z)</td><td>4.97</td><td>92</td></tr><tr><td>Multiplicative gMLP,s(Z) = Z  f(Z)</td><td>4.53</td><td>92</td></tr><tr><td>Multiplicative,Split gMLP,s(Z) = Z1  f(Z2),Z= Zi||Z2</td><td>4.35</td><td>102</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/731fe5bbb1001c5b9f64529e6e9ed073521439ba52c88469a01d1124d0dfaa34.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of the spatial filters in $\\mathrm { g M L P }$ learned on the MLM task. For each layer in the model we plot the row in $W$ associated with the token in the middle of the sequence. The $\\mathbf { X }$ -axis of each subplot has a length of 128 which equal the number of tokens in the sequence. The learned filters appear to be smooth and have several types: forward-looking (e.g., 1st in 2nd row), backward-looking (e.g., 5th in 2nd row) and bi-directional (e.g., 2nd last in the last row). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 Case Study: The Behavior of $\\mathbf { g } \\mathbf { M L P }$ as Model Size Increases ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Table 4, we investigate the scaling properties of Transformers and gMLPs in BERT as their model capacity grows. Specifically, we scale the depth of these models by a factor of $\\{ 0 . 5 , 1 , 2 , 4 \\} \\times$ and report the their pretraining MLM perplexities on the validation set as well as finetuning results on the dev sets of two tasks in GLUE [41]. Note each individual Transformer layer is effectively two consecutive blocks: one for self-attention and one for FFN. In the table below we use the notation of $1 2 + 1 2$ to refer to 12 of self-attention blocks plus 12 of FFN blocks in the Transformer baselines. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f9b9937d357c9507e5ed75ff7c559cbe2b327ae203d9f1866ee2864e0fbaa0c9.jpg",
+        "table_caption": [
+            "Table 4: Pretraining and dev-set finetuning results over increased model capacity. We use the relative positional encoding scheme for Transformers which performs the best in Table 3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>#L</td><td rowspan=1 colspan=2>Params (M) Perplexity</td><td rowspan=1 colspan=1>SST-2 MNLI-m</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>6+618</td><td rowspan=1 colspan=1>6759</td><td rowspan=1 colspan=1>4.915.25</td><td rowspan=1 colspan=1>90.4    81.591.2     77.7</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>12+1236</td><td rowspan=1 colspan=1>110102</td><td rowspan=1 colspan=1>4.264.35</td><td rowspan=1 colspan=1>91.3     83.392.3     80.9</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>24+2472</td><td rowspan=1 colspan=1>195187</td><td rowspan=1 colspan=1>3.833.79</td><td rowspan=1 colspan=1>92.1     85.293.5     82.8</td></tr><tr><td rowspan=1 colspan=1>TransformergMLP</td><td rowspan=1 colspan=1>48+48144</td><td rowspan=1 colspan=1>365357</td><td rowspan=1 colspan=1>3.473.43</td><td rowspan=1 colspan=1>92.8     86.395.1     84.6</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results above show that a deep enough gMLP is able to match and even outperform the perplexity of Transformers with comparable capacity.4 In addition, the perplexity-parameter relationships for both architecture families approximately follow a power law (left of Figure 5). This implies the empirical scaling laws originally observed for Transformer-based language models [42] might be broadly applicable across different model families. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/03ebb35b54fcbc8c3b3a364ab001ed17abb00c96397df6daed10ed232e24238b.jpg",
+        "image_caption": [
+            "Figure 5: Scaling properties with respect to perplexity and finetuning accuracies. The figures show that for pretraining, gMLPs are equally good at optimizing perplexity as Transformers. For finetuning, the two model families exhibit comparable scalability despite task-specific offsets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4 also leads to an interesting observation that the pretraining perplexities across different model families are not equal in terms of finetuning. While gMLPs outperform Transformers on SST-2, they are worse on MNLI. The results imply that the finetuning performance for NLP tasks is a function of not only the perplexity but also the inductive bias in the architecture. Figure 5 shows that despite the architecture-specific discrepancies between pretraining and finetuning, gMLPs and Transformers exhibit comparable scalability (slope) on both finetuning tasks. This means one can always offset the gap by enlarging the model capacity. In other words, the results indicate that model scalability with respect to downstream metrics can be independent from the presence of self-attention. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Ablation: The Usefulness of Tiny Attention in BERT’s Finetuning ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "So far we have found that self-attention is not a required component to achieve strong MLM perplexity or scalability. At the meantime, we also identified NLP finetuning tasks where gMLPs transfer less well than Transformers (Table 4). The fact that our MLP-like model is advantageous on SST-2 but worse on MNLI is particularly informative—the former is a single-sentence task whereas the latter involves sentence pairs (premise and hypothesis) [43]. We suspect the role of self-attention during finetuning is related to cross-sentence alignment. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To isolate the effect of self-attention, we experiment with a hybrid model where a tiny self-attention block is attached to the gating function of gMLP (Figure 6). Since gMLP itself is already capable in capturing spatial relationships, we hypothesize that this extra self-attention module does not have to be heavy, and that its presence is more relevant than its capacity. A typical tiny attention module in our experiments has only a single head with size 64, significantly smaller than a typical multi-head self-attention in Transformers with 12 heads and a total size of 768. In the following, we refer to the hybrid model, namely $\\mathrm { g M L P }$ with a tiny self-attention, as aMLP (“a” for attention). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Pseudo-code for the tiny attention module ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "def tiny_attn(x, d_out, d_attn=64): qkv $=$ proj(x, 3 ∗ d_attn, axis=\"channel\") q, k, v $=$ split(qkv, 3, axis=\"channel\") $\\kappa =$ einsum(\"bnd,bmd−>bnm\", q, k) a = softmax(w $^ *$ rsqrt(d_attn)) x = einsum(\"bnm,bmd−>bnd\", a, v) return proj(x, d_out, axis=\"channel\") ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/e228c75c14097e90bf3596da75b5d6c0ccbec039247e6155396cb13236ea093c.jpg",
+        "image_caption": [
+            "Figure 6: Hybrid spatial gating unit with a tiny self-attention module. We use the normalized input of the $\\mathrm { g M L P }$ block (endpoint after the input normalization and right before the channel expansion) as the input to the tiny self-attention. For SGU we have ${ d _ { \\mathrm { o u t } } = d _ { \\mathrm { f f n } } / 2 }$ due to the channel split. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure 7, we investigate the transferability of MLM models via the calibration plots between their pretraining perplexities and finetuning metrics. Models evaluated include $\\mathbf { B E R T _ { b a s e } }$ , gMLP and its hybrid version aMLP with a 64-d single-head self-attention (Figure 6). The data points were collected by varying the model depth by $\\{ 0 . 5 , 1 , 2 \\} \\times$ or data by $\\{ 1 , 2 , 4 , 8 \\} \\times$ . It can be seen that gMLPs transfer better to SST-2 than Transformers regardless of the presence of self-attention, While gMLP performs worse on MNLI, attaching a tiny bit of self-attention is sufficient to close the gap. In Appendix D we visualize the tiny self-attention modules in aMLP over MNLI examples, showing that they are primarily responsible for the alignment between sentence pairs. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/263412114ce248c9b9dccbc4dbdcbba36ac2aa648dce820d76c34069690a0cd1.jpg",
+        "image_caption": [
+            "Figure 7: Transferability from MLM pretraining perpexity to finetuning accuracies on GLUE. aMLP refers to $\\mathrm { g M L P }$ enhanced with a 64-d single-head self-attention, as illustrated in Figure 6. In contrast, each self-attention module in the BERT baseline contains 12 heads with a total size of 768. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8 we put together the scaling properties of the three models, showing that aMLP $\\mathrm { ( g M L P + }$ tiny attention) consistently outperforms Transformer on both finetuning tasks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/395d749d892cbad8add03840c7c07a237a2c72cef704c6aff4263e86c3de1d09.jpg",
+        "image_caption": [
+            "Figure 8: Comparing the scaling properties of Transformers, $\\mathrm { g M L P s }$ and aMLPs (with 64-d, singlehead attention). Results were obtained using the same setup in Section 4.2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Main Results for MLM in the BERT Setup ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Below we present pretraining and finetuning results in the full BERT setup. Different from ablation and case studies, here we use the full English C4 dataset and adopt a common MLM setup with batch size 256, max length 512 and 1M training steps. For fair comparison, we adjust the depth and width of $\\mathrm { g M L P s }$ to ensure comparable model capacity with the Transformer baselines. The model specifications are given in Table 5 and hyperparameters are detailed in Appendix A.2. For finetuning, we report the dev-set performance for SST-2 and MNLI in GLUE [41] and each result entry was obtained by taking the median of five independent runs. In addition, we report finetuning results on SQuAD [44, 45] to test the models’ ability in reasoning over a longer context. ",
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+    {
+        "type": "text",
+        "text": "Results are presented in Table 6. Consistent with our findings earlier in Section 4.1 and Section 4.2, gMLPs are competitive with Transformers in terms of perplexity, especially in the larger scale setup. There are several observations related to the finetuning results: ",
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+        "text": "First, on finetuning tasks where gMLPs underperform Transformers, the performance gap tends to narrow as the model capacity increases. For example, while $\\mathrm { g M L P }$ performs worse by $8 . 5 \\%$ on SQuAD-v2.0 in the base scale, the performance gap relative to the baseline decreases to $2 . 7 \\%$ at the larger scale. Notably, our ${ \\mathrm { g M L P } } _ { \\mathrm { l a r g e } }$ achieves $8 9 . 5 \\%$ F1 on SQuAD-v1.1 without any self-attention or dynamic spatial parameterization [28], which is well above the $8 8 . 5 \\%$ reported for $\\mathbf { B E R T _ { b a s e } }$ in Devlin et al. [2] and is only $1 . 4 \\%$ away from the original result for $\\mathbf { B E R T _ { l a r g e } }$ . We also include one additional data point by scaling up $\\mathrm { g M L P }$ even further. The resulting model, $\\mathrm { g } \\mathrm { \\bar { M } L P _ { \\mathrm { x l a r g e } } }$ , outperforms $\\mathbf { B E R T _ { l a r g e } }$ on SQuAD- $\\mathbf { \\nabla } \\cdot \\mathbf { v } 2 . 0 $ —a difficult task involving question-answer pairs—without any self-attention. While this is not a fair comparison due to different model sizes, it is an existence proof that MLP-like models can be competitive with Transformers on challenging NLP tasks. ",
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+        "table_caption": [
+            "Table 5: Model specifications in the full BERT setup. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Params (M)</td><td>FLOPs (B)</td><td>#L</td><td>dmodel</td><td>dffn</td></tr><tr><td rowspan=\"3\">BERTbase gMLPbase aMLPbase</td><td>110</td><td>100.8</td><td>12+12</td><td>768</td><td>3072</td></tr><tr><td>130</td><td>158.0</td><td>48</td><td>512</td><td>3072</td></tr><tr><td>109</td><td>128.9</td><td>36</td><td>512</td><td>3072</td></tr><tr><td rowspan=\"3\">BERTlarge gMLPlarge aMLPlarge</td><td>336</td><td>341.2</td><td>24+24</td><td>1024</td><td>4096</td></tr><tr><td>365</td><td>430.1</td><td>96</td><td>768</td><td>3072</td></tr><tr><td>316</td><td>370.3</td><td>72</td><td>768</td><td>3072</td></tr><tr><td> gMLPxlarge</td><td>941</td><td>1091.3</td><td>144</td><td>1024</td><td>4096</td></tr></table>",
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+        "table_caption": [
+            "Table 6: Pretraining perplexities and dev-set results for finetuning. “ours” indicates models trained using our setup. We report accuracies for SST-2 and MNLI, and F1 scores for SQuAD v1.1/2.0. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Perplexity</td><td rowspan=\"2\">SST-2</td><td rowspan=\"2\">MNLI (m/mm)</td><td colspan=\"2\">SQuAD</td><td rowspan=\"2\">Attn Size</td><td rowspan=\"2\">Params (M)</td></tr><tr><td>v1.1</td><td>v2.0</td></tr><tr><td>BERTbase [2]</td><td>1</td><td>92.7</td><td>84.4/-</td><td>88.5</td><td>76.3</td><td>768 (64 × 12)</td><td>110</td></tr><tr><td>BERTbase (ours)</td><td>4.17</td><td>93.8</td><td>85.6/85.7</td><td>90.2</td><td>78.6</td><td>768 (64 × 12)</td><td>110</td></tr><tr><td>gMLPbase</td><td>4.28</td><td>94.2</td><td>83.7/84.1</td><td>86.7</td><td>70.1</td><td>1</td><td>130</td></tr><tr><td>aMLPbase</td><td>3.95</td><td>93.4</td><td>85.9/85.8</td><td>90.7</td><td>80.9</td><td>64</td><td>109</td></tr><tr><td>BERTlarge [2]</td><td>1</td><td>93.7</td><td>86.6/-</td><td>90.9</td><td>81.8</td><td>1024 (64 × 16)</td><td>336</td></tr><tr><td>BERTlarge (ours)</td><td>3.35</td><td>94.3</td><td>87.0/87.4</td><td>92.0</td><td>81.0</td><td>1024 (64 × 16)</td><td>336</td></tr><tr><td>gMLPlarge</td><td>3.32</td><td>94.8</td><td>86.2/86.5</td><td>89.5</td><td>78.3</td><td>1</td><td>365</td></tr><tr><td>aMLPlarge</td><td>3.19</td><td>94.8</td><td>88.4/88.4</td><td>92.2</td><td>85.4</td><td>128</td><td>316</td></tr><tr><td>gMLPxlarge</td><td>2.89</td><td>95.6</td><td>87.7/87.7</td><td>90.9</td><td>82.1</td><td>1</td><td>941</td></tr></table>",
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+        "text": "Furthermore, we show that blending in a tiny single-head self-attention of size either 64 or 128 is sufficient to make gMLPs outperform Transformers of similar capacity, sometimes by a significant margin. For example, our hybrid model $\\mathrm { a M L P _ { l a r g e } }$ achieves $4 . 4 \\%$ higher F1 than Transformers on SQuAD-v2.0. The results suggest that the capacity in the multi-head self-attention of Transformers can be largely redundant, and that the majority of its functionalities can be captured by the spatial gating unit in gMLPs. The results also imply that the inductive biases in the spatial gating unit of gMLPs and the tiny attention are complementary to each other. While the benefits of architectural inductive bias may vanish over increased compute, tiny attention does improve the practical value of gMLPs in the regime that we investigate in this work. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the seminal work of Vaswani et al. [1], Transformers have been widely adopted across NLP and computer vision. This adoption has enabled many impressive results especially in NLP. To date, it is still unclear what empowers such success: is it the feedforward nature of Transformers or is it the multi-head self-attention layers in Transformers? ",
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+    {
+        "type": "text",
+        "text": "Our work suggests a simpler alternative to the multi-head self-attention layers in Transformers. We show that gMLPs, a simple variant of MLPs with gating, can be competitive with Transformers in terms of BERT’s pretraining perplexity and ViT’s accuracy. gMLPs are also comparable with Transformers in terms of the scalability over increased data and compute. As for BERT finetuning, we find gMLPs can achieve appealing results on challenging tasks such as SQuAD without self-attention, and can significantly outperform Transformers in certain cases. We also find the inductive bias in Transformer’s multi-head self-attention useful on downstream tasks that require cross-sentence alignment. However in those cases, making gMLP substantially larger closes the gap with Transformers. More practically, blending a small single-head self-attention into gMLP allows for an even better architecture without the need for increasing model size. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank Gabriel Bender, Neil Houlsby, Thang Luong, Niki Parmar, Hieu Pham, Jascha SohlDickstein, Noam Shazeer, Ilya Sutskever, Jakob Uszkoreit and Ashish Vaswani for their feedback. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+                                    "cross_page": true
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+                                    "score": 1.0,
+                                    "content": "similar regularization recipe as the one used in DeiT.2 To avoid extensive tuning, we adjust only the",
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+                                    "content": "strengths of stochastic depth [32] as we move from smaller to larger models in Table 1. All the other",
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+                                    "score": 1.0,
+                                    "content": "hyperparameters remain shared across our three models. See Appendix A.1 for details.",
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+                                            "content": "\\mathrm { g M L P }",
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+                                            "content": "models for vision.",
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+                                    "content": "{ \\mathrm { g M L P s } }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                                            "score": 1.0,
+                                            "content": "Table 2: ImageNet-1K results without extra data.",
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+                                            "html": "<table><tr><td rowspan=\"2\">Model</td><td>ImageNet Top-1 (%)*</td><td>Input Resolution</td><td>Params (M)</td><td>MAdds (B)</td></tr><tr><td colspan=\"4\">ConvNets</td></tr><tr><td>ResNet-152 [16]</td><td>78.3</td><td>224</td><td>60</td><td>11.3</td></tr><tr><td>RegNetY-8GF[39]</td><td>81.7</td><td>224</td><td>39</td><td>8.0</td></tr><tr><td>EfficientNet-B0 [17]</td><td>77.1</td><td>224</td><td>5</td><td>0.39</td></tr><tr><td>EfficientNet-B3[17]</td><td>81.6</td><td>300</td><td>12</td><td>1.8</td></tr><tr><td>EfficientNet-B7 [17]</td><td>84.3</td><td>600</td><td>66</td><td>37.0</td></tr><tr><td>NFNet-F0 [33]</td><td>83.6</td><td>192</td><td>72</td><td>12.4</td></tr><tr><td></td><td colspan=\"4\">Transformers</td></tr><tr><td>ViT-B/16 [7]</td><td>77.9</td><td>384</td><td>86</td><td>55.4</td></tr><tr><td>ViT-L/16 [7]</td><td>76.5</td><td>384</td><td>307</td><td>190.7</td></tr><tr><td>DeiT-Ti [8] (ViT+reg)</td><td>72.2</td><td>224</td><td>5</td><td>1.3</td></tr><tr><td>DeiT-S [8](ViT+reg)</td><td>79.8</td><td>224</td><td>22</td><td>4.6</td></tr><tr><td>DeiT-B [8] (ViT+reg)</td><td>81.8</td><td>224</td><td>86</td><td>17.5</td></tr><tr><td></td><td colspan=\"4\">MLP-like†</td></tr><tr><td>Mixer-B/16 [20]</td><td>76.4</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-B/16 (our setup)</td><td>77.3</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-L/16 [20]</td><td>71.8</td><td>224</td><td>207</td><td>44.8</td></tr><tr><td>ResMLP-12 [22]</td><td>76.6</td><td>224</td><td>15</td><td>3.0</td></tr><tr><td>ResMLP-24 [22]</td><td>79.4</td><td>224</td><td>30</td><td>6.0</td></tr><tr><td>ResMLP-36[22]</td><td>79.7</td><td>224</td><td>45</td><td>8.9</td></tr><tr><td>gMLP-Ti (ours)</td><td>72.3</td><td>224</td><td>6</td><td>1.4</td></tr><tr><td>gMLP-S (ours)</td><td>79.6</td><td>224</td><td>20</td><td>4.5</td></tr><tr><td>gMLP-B (ours)</td><td>81.6</td><td>224</td><td>73</td><td>15.8</td></tr></table>",
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+                                            "score": 1.0,
+                                            "content": "* Standard deviation across multiple independent runs is around 0.1.",
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+                                            "content": "† Tokenization & embedding process at the stem can be viewed as a convolution.",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "The depth and width are chosen so that the models are comparable with ViT/DeiT in capacity. Like",
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+                                    "content": "Transformers, we find gMLPs tend to drastically overfit the training data. We therefore apply a",
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+                                    "score": 1.0,
+                                    "content": "similar regularization recipe as the one used in DeiT.2 To avoid extensive tuning, we adjust only the",
+                                    "type": "text"
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+                            "index": 6
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+                                    "score": 1.0,
+                                    "content": "strengths of stochastic depth [32] as we move from smaller to larger models in Table 1. All the other",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "hyperparameters remain shared across our three models. See Appendix A.1 for details.",
+                                    "type": "text"
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+                    ],
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+                                            "content": "\\mathrm { g M L P }",
+                                            "type": "inline_equation"
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+                                            "score": 1.0,
+                                            "content": "models for vision.",
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+                                    "content": "Our ImageNet results are summarized in Table 1 and Figure 2.3 It is interesting to see that gMLPs are",
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+                                    "content": "fact, when the models are properly regularized, their accuracies seem better correlated with capacity",
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+                                    "content": "instead of the presence of self-attention. Moreover, the accuracy-parameter/FLOPs tradeoff of gMLPs",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "surpasses all concurrently proposed MLP-like architectures [20, 21, 22], which we attribute to the",
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+                                    "score": 0.57,
+                                    "content": "{ \\mathrm { g M L P s } }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "are competitive with vanilla Transformers, their performance is behind the best existing",
+                                    "type": "text"
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+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "ConvNet models (e.g., [33, 34]) or hybrid models (e.g., [35, 36, 37, 38, 10]).",
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+                            "index": 21
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+                                            "score": 1.0,
+                                            "content": "Table 2: ImageNet-1K results without extra data.",
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+                                            "html": "<table><tr><td rowspan=\"2\">Model</td><td>ImageNet Top-1 (%)*</td><td>Input Resolution</td><td>Params (M)</td><td>MAdds (B)</td></tr><tr><td colspan=\"4\">ConvNets</td></tr><tr><td>ResNet-152 [16]</td><td>78.3</td><td>224</td><td>60</td><td>11.3</td></tr><tr><td>RegNetY-8GF[39]</td><td>81.7</td><td>224</td><td>39</td><td>8.0</td></tr><tr><td>EfficientNet-B0 [17]</td><td>77.1</td><td>224</td><td>5</td><td>0.39</td></tr><tr><td>EfficientNet-B3[17]</td><td>81.6</td><td>300</td><td>12</td><td>1.8</td></tr><tr><td>EfficientNet-B7 [17]</td><td>84.3</td><td>600</td><td>66</td><td>37.0</td></tr><tr><td>NFNet-F0 [33]</td><td>83.6</td><td>192</td><td>72</td><td>12.4</td></tr><tr><td></td><td colspan=\"4\">Transformers</td></tr><tr><td>ViT-B/16 [7]</td><td>77.9</td><td>384</td><td>86</td><td>55.4</td></tr><tr><td>ViT-L/16 [7]</td><td>76.5</td><td>384</td><td>307</td><td>190.7</td></tr><tr><td>DeiT-Ti [8] (ViT+reg)</td><td>72.2</td><td>224</td><td>5</td><td>1.3</td></tr><tr><td>DeiT-S [8](ViT+reg)</td><td>79.8</td><td>224</td><td>22</td><td>4.6</td></tr><tr><td>DeiT-B [8] (ViT+reg)</td><td>81.8</td><td>224</td><td>86</td><td>17.5</td></tr><tr><td></td><td colspan=\"4\">MLP-like†</td></tr><tr><td>Mixer-B/16 [20]</td><td>76.4</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-B/16 (our setup)</td><td>77.3</td><td>224</td><td>59</td><td>12.7</td></tr><tr><td>Mixer-L/16 [20]</td><td>71.8</td><td>224</td><td>207</td><td>44.8</td></tr><tr><td>ResMLP-12 [22]</td><td>76.6</td><td>224</td><td>15</td><td>3.0</td></tr><tr><td>ResMLP-24 [22]</td><td>79.4</td><td>224</td><td>30</td><td>6.0</td></tr><tr><td>ResMLP-36[22]</td><td>79.7</td><td>224</td><td>45</td><td>8.9</td></tr><tr><td>gMLP-Ti (ours)</td><td>72.3</td><td>224</td><td>6</td><td>1.4</td></tr><tr><td>gMLP-S (ours)</td><td>79.6</td><td>224</td><td>20</td><td>4.5</td></tr><tr><td>gMLP-B (ours)</td><td>81.6</td><td>224</td><td>73</td><td>15.8</td></tr></table>",
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+                                            "score": 1.0,
+                                            "content": "* Standard deviation across multiple independent runs is around 0.1.",
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+                                            "content": "† Tokenization & embedding process at the stem can be viewed as a convolution.",
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+# VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING
+
+Oleg Ivanov   
+Samsung AI Center Moscow Moscow, Russia   
+tigvarts@gmail.com Michael Figurnov   
+National Research University Higher School of Economics ∗ Moscow, Russia   
+michael@figurnov.ru Dmitry Vetrov   
+Samsung-HSE Laboratory, National Research University Higher School of Economics Samsung AI Center Moscow Moscow, Russia   
+vetrovd@yandex.ru
+
+# ABSTRACT
+
+We propose a single neural probabilistic model based on variational autoencoder that can be conditioned on an arbitrary subset of observed features and then sample the remaining features in “one shot”. The features may be both real-valued and categorical. Training of the model is performed by stochastic variational Bayes. The experimental evaluation on synthetic data, as well as feature imputation and image inpainting problems, shows the effectiveness of the proposed approach and diversity of the generated samples.
+
+# 1 INTRODUCTION
+
+In past years, a number of generative probabilistic models based on neural networks have been proposed. The most popular approaches include variational autoencoder (Kingma & Welling, 2013) (VAE) and generative adversarial net (Goodfellow et al., 2014) (GANs). They learn a distribution over objects $p ( x )$ and allow sampling from this distribution.
+
+In many cases, we are interested in learning a conditional distribution $p ( x | y )$ . For instance, if $x$ is an image of a face, $y$ could be the characteristics describing the face (are glasses present or not; length of hair, etc.) Conditional variational autoencoder (Sohn et al., 2015) and conditional generative adversarial nets (Mirza & Osindero, 2014) are popular methods for this problem.
+
+In this paper, we consider the problem of learning all conditional distributions of the form $p ( x _ { I } | x _ { U \setminus I } )$ , where $U$ is the set of all features and $I$ is its arbitrary subset. This problem generalizes both learning the joint distribution $p ( x )$ and learning the conditional distribution $p ( x | y )$ . To tackle this problem, we propose a Variational Autoencoder with Arbitrary Conditioning (VAEAC) model. It is a latent variable model similar to VAE, but allows conditioning on an arbitrary subset of the features. The conditioning features affect the prior on the latent Gaussian variables which are used to generate unobserved features. The model is trained using stochastic gradient variational Bayes (Kingma & Welling, 2013).
+
+We consider two most natural applications of the proposed model. The first one is feature imputation where the goal is to restore the missing features given the observed ones. The imputed values may be valuable by themselves or may improve the performance of other machine learning algorithms which process the dataset. Another application is image inpainting in which the goal is to fill in an unobserved part of an image with an artificial content in a realistic way. This can be used for removing unnecessary objects from the images or, vice versa, for complementing the partially closed or corrupted object.
+
+The experimental evaluation shows that the proposed model successfully samples from the conditional distributions. The distribution over samples is close to the true conditional distribution. This property is very important when the true distribution has several modes. The model is shown to be effective in feature imputation problem which helps to increase the quality of subsequent discriminative models on different problems from UCI datasets collection (Lichman, 2013). We demonstrate that model can generate diverse and realistic image inpaintings on MNIST (LeCun et al., 1998), Omniglot (Lake et al., 2015) and CelebA (Liu et al., 2015) datasets, and works even better than the current state of the art inpainting techniques in terms of peak signal to noise ratio (PSNR).
+
+The paper is organized as follows. In section 2 we review the related works. In section 3 we briefly describe variational autoencoders and conditional variational autoencoders. In section 4 we define the problem, describe the VAEAC model and its training procedure. In section 5 we evaluate VAEAC. Section 6 concludes the paper. Appendix contains additional explanations, theoretical analysis, and experiments for VAEAC.
+
+# 2 RELATED WORK
+
+Universal Marginalizer (Douglas et al., 2017) is a model based on a feed-forward neural network which approximates marginals of unobserved features conditioned on observable values. A related idea of an autoregressive model of joint probability was previously proposed in Germain et al. (2015) and Uria et al. (2016). The description of the model and comparison with VAEAC are available in section 5.3.
+
+Yoon et al. (2018) propose a GANs-based model called GAIN which solves the same problem as VAEAC. In contrast to VAEAC, GAIN does not use unobserved data during training, which makes it easier to apply to the missing features imputation problem. Nevertheless, it turns into a disadvantage when the fully-observed training data is available but the missingness rate at the testing stage is high. For example, in inpainting setting GAIN cannot learn the conditional distribution over MNIST digits given one horizontal line of the image while VAEAC can (see appendix D.4). The comparison of VAEAC and GAIN on the missing feature imputation problem is given in section 5.1 and appendix D.2.
+
+Rezende et al. (2014) [Appendix F], Sohl-Dickstein et al. (2015), Goyal et al. (2017), and Bordes et al. (2017) propose to fill missing data with noise and run Markov chain with a learned transition operator. The stationary distribution of such chains approximates the true conditional distribution of the unobserved features. Bachman & Precup (2015) consider missing feature imputation in terms of Markov decision process and propose LSTM-based sequential decision making model to solve it. Nevertheless, these methods are computationally expensive at the test time and require fully-observed training data.
+
+Image inpainting is a classic computer vision problem. Most of the earlier methods rely on local and texture information or hand-crafted problem-specific features (Bertalmio et al., 2000). In past years multiple neural network based approaches have been proposed.
+
+Pathak et al. (2016), Yeh et al. (2016) and Yang et al. (2017) use different kinds and combinations of adversarial, reconstruction, texture and other losses. Li et al. (2017) focuses on face inpainting and uses two adversarial losses and one semantic parsing loss to train the generative model. In Yeh et al. (2017) GANs are first trained on the whole training dataset. The inpainting is an optimization procedure that finds the latent variables that explain the observed features best. Then, the obtained latents are passed through the generative model to restore the unobserved portion of the image. We can say that VAEAC is a similar model which uses prior network to find a proper latents instead of solving the optimization problem.
+
+All described methods aim to produce a single realistic inpainting, while VAEAC is capable of sampling diverse inpaintings. Additionally, Yeh et al. (2016), Yang et al. (2017) and Yeh et al. (2017) have high testtime computational complexity of inpainting, because they require an optimization problem to be solved. On the other hand, VAEAC is a “single-shot” method with a low computational cost.
+
+# 3 BACKGROUND
+
+# 3.1 VARIATIONAL AUTOENCODER
+
+Variational autoencoder (Kingma & Welling, 2013) (VAE) is a directed generative model with latent variables. The generative process in variational autoencoder is as follows: first, a latent variable $z$ is generated from the prior distribution $p ( z )$ , and then the data $x$ is generated from the generative distribution $p _ { \theta } ( x | z )$ , where $\theta$ are the generative model’s parameters. This process induces the distribution $p _ { \theta } ( x ) = \mathbb { E } _ { p ( z ) } p _ { \theta } ( x | z )$ . The distribution $p _ { \theta } ( x | z )$ is modeled by a neural network with parameters $\theta$ . $p ( z )$ is a standard Gaussian distribution.
+
+The parameters $\theta$ are tuned by maximizing the likelihood of the training data points $\{ x _ { i } \} _ { i = 1 } ^ { N }$ from the true data distribution $p _ { d } ( x )$ . In general, this optimization problem is challenging due to intractable posterior inference. However, a variational lower bound can be optimized efficiently using backpropagation and stochastic gradient descent:
+
+$$
+\begin{array} { r l } & { \log p _ { \theta } ( x ) = \mathbb { E } _ { q _ { \phi } ( z | x ) } \log \frac { p _ { \theta } ( x , z ) } { q _ { \phi } ( z | x ) } + D _ { \mathrm { K L } } \big ( q _ { \phi } ( z | x ) \| p ( z | x , \theta ) \big ) } \\ & { \qquad \quad \ge \mathbb { E } _ { q _ { \phi } ( z | x ) } \log p _ { \theta } ( x | z ) - D _ { \mathrm { K L } } \big ( q _ { \phi } ( z | x ) \| p ( z ) \big ) = L _ { V A E } \big ( x ; \theta , \phi \big ) } \end{array}
+$$
+
+Here $q _ { \phi } ( z | x )$ is a proposal distribution parameterized by neural network with parameters $\phi$ that approximates the posterior $p ( z | x , \theta )$ . Usually this distribution is Gaussian with a diagonal covariance matrix. The closer $q _ { \phi } ( z | x )$ to $p ( z | x , \theta )$ , the tighter variational lower bound $L _ { V A E } ( \theta , \phi )$ . To compute the gradient of the variational lower bound with respect to $\phi$ , reparameterization trick is used: $z = \bar { \mu } _ { \phi } ( x ) + \bar { \varepsilon } \sigma _ { \phi } ( x )$ where $\varepsilon \sim \mathcal { N } ( 0 , I )$ and $\mu _ { \phi }$ and $\sigma _ { \phi }$ are deterministic functions parameterized by neural networks. So the gradient can be estimated using Monte-Carlo method for the first term and computing the second term analytically:
+
+$$
+\frac { \partial L _ { V A E } ( x ; \theta , \phi ) } { \partial \phi } = \mathbb { E } _ { \varepsilon \sim \mathcal { N } ( 0 , I ) } \frac { \partial } { \partial \phi } \log p _ { \theta } ( x | \mu _ { \phi } ( x ) + \varepsilon \sigma _ { \phi } ( x ) ) - \frac { \partial } { \partial \phi } D _ { \mathrm { K L } } ( q _ { \phi } ( z | x ) \| p ( z ) ) .
+$$
+
+So $L _ { V A E } ( \theta , \phi )$ can be optimized using stochastic gradient ascent with respect to $\phi$ and $\theta$
+
+# 3.2 CONDITIONAL VARIATIONAL AUTOENCODER
+
+Conditional variational autoencoder (Sohn et al., 2015) (CVAE) approximates the conditional distribution $p _ { d } ( x | y )$ . It outperforms deterministic models when the distribution $p _ { d } ( x | y )$ is multi-modal (diverse $x \mathbf { s }$ are probable for the given $y$ ). For example, assume that $x$ is a real-valued image. Then, a deterministic regression model with mean squared error loss would predict the average blurry value for $x$ . On the other hand, CVAE learns the distribution of $x$ , from which one can sample diverse and realistic objects.
+
+Variational lower bound for CVAE can be derived similarly to VAE by conditioning all considered distributions on $y$ :
+
+$$
+L _ { C V A E } ( x , y ; \theta , \psi , \phi ) = \mathbb { E } _ { q _ { \phi } ( z | x , y ) } \log p _ { \theta } ( x | z , y ) - D _ { \mathrm { K L } } ( q _ { \phi } ( z | x , y ) | | p _ { \psi } ( z | y ) ) \leq \log p _ { \theta , \psi } ( x | y )
+$$
+
+Similarly to VAE, this objective is optimized using the reparameterization trick. Note that the prior distribution $p _ { \psi } ( z | y )$ is conditioned on $y$ and is modeled by a neural network with parameters $\psi$ . Thus, CVAE uses three trainable neural networks, while VAE only uses two.
+
+Also authors propose such modifications of CVAE as Gaussian stochastic neural network and hybrid model. These modifications can be applied to our model as well. Nevertheless, we don’t use them, because of their disadvantage which is described in appendix C.
+
+# 4 VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING
+
+# 4.1 PROBLEM STATEMENT
+
+Consider a distribution $p _ { d } ( x )$ over a $D$ -dimensional vector $x$ with real or categorical components. The components of the vector are called features.
+
+Let binary vector $b \in \{ 0 , 1 \} ^ { D }$ be the binary mask of unobserved features of the object. Then we describe the vector of unobserved features as $x _ { b } = \{ x _ { i : b _ { i } = 1 } \}$ . For example, $x _ { ( 0 , 1 , 1 , 0 , 1 ) } = ( x _ { 2 } , x _ { 3 } , x _ { 5 } )$ . Using this notation we denote $x _ { 1 - b }$ as a vector of observed features.
+
+Our goal is to build a model of the conditional distribution $p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b ) \approx p _ { d } ( x _ { b } | x _ { 1 - b } , b )$ for an arbitrary $b$ , where $\psi$ and $\theta$ are parameters that are used in our model at the testing stage.
+
+However, the true distribution $p _ { d } ( x _ { b } | x _ { 1 - b } , b )$ is intractable without strong assumptions about $p _ { d } ( x )$ . Therefore, our model $p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b )$ has to be more precise for some $b$ and less precise for others. To formalize our requirements about the accuracy of our model we introduce the distribution $p ( b )$ over different unobserved feature masks. The distribution $p ( b )$ is arbitrary and may be defined by the user depending on the problem. Generally it should have full support over $\{ 0 , 1 \} ^ { D }$ so that $p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b )$ can evaluate arbitrary conditioning. Nevertheless, it is not necessary if the model is used for specific kinds of conditioning (as we do in section 5.2).
+
+Using $p ( b )$ we can introduce the following log-likelihood objective function for the model:
+
+$$
+\operatorname* { m a x } _ { \psi , \theta } \mathbb { E } _ { p _ { d } ( \boldsymbol { x } ) } \mathbb { E } _ { p ( \boldsymbol { b } ) } \log p _ { \psi , \theta } \big ( x _ { b } | \boldsymbol { x } _ { 1 - \boldsymbol { b } } , \boldsymbol { b } \big )
+$$
+
+The special cases of the objective (4) are variational autoencoder $( b _ { i } = 1 \forall i \in \{ 1 , \ldots , D \} )$ and conditional variational autoencoder $^ { \textit { b } }$ is constant).
+
+# 4.2 MODEL DESCRIPTION
+
+The generative process of our model is similar to the generative process of CVAE: for each object firstly we generate $z \sim p _ { \psi } ( z | x _ { 1 - b } , b )$ using prior network, and then sample unobserved features $x _ { b } ~ \sim ~ p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b )$ using generative network. This process induces the following model distribution over unobserved features:
+
+$$
+p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b ) = \mathbb { E } _ { z \sim p _ { \psi } ( z | x _ { 1 - b } , b ) } p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b )
+$$
+
+We use $z ~ \in ~ \mathbb { R } ^ { d }$ , and Gaussian distribution $p _ { \psi }$ over $z$ , with parameters from a neural network with weights $\psi$ : $p _ { \psi } ( z | x _ { 1 - b } , b , \psi ) = \mathcal { N } ( z | \mu _ { \psi } ( x _ { 1 - b } , b ) , \sigma _ { \psi } ^ { 2 } ( x _ { 1 - b } , b ) I )$ . The real-valued components of distribution $p _ { \theta } ( x _ { b } | \boldsymbol { z } , x _ { 1 - b } , b )$ are defined likewise. Each categorical component $i$ of distribution $p _ { \theta } ( x _ { i } | \boldsymbol { z } , x _ { 1 - b } , b )$ is parameterized by a function $w _ { i , \theta } ( z , x _ { 1 - b } , b )$ , whose outputs are logits of probabilities for each category: $x _ { i } \sim \mathrm { C a t } [ \mathrm { S o f t m a x } ( w _ { i , \theta } ( z , x _ { 1 - b } , b ) ) ]$ . Therefore the components of the latent vector $z$ are conditionally independent given $x _ { 1 - b }$ and $b$ , and the components of $x _ { b }$ are conditionally independent given $z , x _ { 1 - b }$ and $b$ .
+
+The variables $x _ { b }$ and $x _ { 1 - b }$ have variable length that depends on $b$ . So in order to use architectures such as multi-layer perceptron and convolutional neural network we consider $x _ { 1 - b } = x \circ ( 1 - b )$ where $\circ$ is an element-wise product. So in implementation $x _ { 1 - b }$ has fixed length. The output of the generative network also has a fixed length, but we use only unobserved components to compute likelihood.
+
+The theoretical analysis of the model is available in appendix B.1.
+
+# 4.3 LEARNING VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING
+
+# 4.3.1 VARIATIONAL LOWER BOUND
+
+We can derive a lower bound for $\log p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b )$ as for variational autoencoder:
+
+$$
+\begin{array} { r l } & { \log p _ { \psi , \theta } ( x _ { b } | x _ { 1 - b } , b ) = \mathbb { E } _ { q _ { \phi } ( z | x , b ) } \log \frac { p _ { \psi , \theta } ( x _ { b } , z | x _ { 1 - b } , b ) } { q _ { \phi } ( z | x , b ) } + D _ { \mathrm { K L } } ( q _ { \phi } ( z | x , b ) | | p _ { \psi , \theta } ( z | x , b ) ) } \\ & { \qquad \geq \mathbb { E } _ { q _ { \phi } ( z | x , b ) } \log p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b ) - D _ { \mathrm { K L } } ( q _ { \phi } ( z | x , b ) | | p _ { \psi } ( z | x _ { 1 - b } , b ) ) = L _ { V A E A C } ( x , b ; \theta , \psi , \phi ) } \end{array}
+$$
+
+Therefore we have the following variational lower bound optimization problem:
+
+$$
+\operatorname* { m a x } _ { \theta , \psi , \phi } \mathbb { E } _ { p _ { d } ( x ) } \mathbb { E } _ { p ( b ) } L _ { V A E A C } ( x , b ; \theta , \psi , \phi )
+$$
+
+We use fully-factorized Gaussian proposal distribution $q _ { \phi }$ which allows us to perform reparameterization trick and compute KL divergence analytically in order to optimize (7).
+
+# 4.3.2 PRIOR IN LATENT SPACE
+
+During the optimization of objective (7) the parameters $\mu _ { \psi }$ and $\sigma _ { \psi }$ of the prior distribution of $z$ may tend to infinity, since there is no penalty for large values of those parameters. We usually observe the growth of $\left. z \right. _ { 2 }$ during training, though it is slow enough. To prevent potential numerical instabilities, we put a Normal-Gamma prior on the parameters of the prior distribution to prevent the divergence. Formally, we redefine $p _ { \psi } ( z | x _ { 1 - b } , b )$ as follows:
+
+$$
+p _ { \psi } ( z , \mu _ { \psi } , \sigma _ { \psi } | x _ { 1 - b } , b ) = \mathcal { N } ( z | \mu _ { \psi } , \sigma _ { \psi } ^ { 2 } ) \mathcal { N } ( \mu _ { \psi } | 0 , \sigma _ { \mu } ) \mathrm { G a m m a } ( \sigma _ { \psi } | 2 , \sigma _ { \sigma } )
+$$
+
+As a result, the regularizers $- \frac { \mu _ { \psi } ^ { 2 } } { 2 \sigma _ { \mu } ^ { 2 } }$ and $\sigma _ { \sigma } ( \log ( \sigma _ { \psi } ) - \sigma _ { \psi } )$ are added to the model log-likelihood. Hyperparameter $\sigma _ { \mu }$ is chosen to be large $( 1 0 ^ { 4 } )$ and $\sigma _ { \sigma }$ is taken to be a small positive number $( 1 0 ^ { - 4 } )$ . This distribution is close to uniform near zero, so it doesn’t affect the learning process significantly.
+
+# 4.3.3 MISSING FEATURES
+
+The optimization objective (7) requires all features of each object at the training stage: some of the features will be observed variables at the input of the model and other will be unobserved features used to evaluate the model. Nevertheless, in some problem settings the training data contains missing features too. We propose the following slight modification of the problem (7) in order to cover such problems as well.
+
+The missing values cannot be observed so $x _ { i } = \omega \Rightarrow b _ { i } = 1$ , where $\omega$ describes the missing value in the data. In order to meet this requirement, we redefine mask distribution as conditioned on $x$ : $p ( b )$ turns into $p ( b | x )$ in (4) and (7). In the reconstruction loss (5) we simply omit the missing features, i. e. marginalize them out:
+
+$$
+\log p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b ) = \sum _ { \substack { i : b _ { i } = 1 , x _ { i } \neq \omega } } \log p _ { \theta } ( x _ { i } | z , x _ { 1 - b } , b )
+$$
+
+The proposal network must be able to determine which features came from real object and which are just missing. So we use additional missing features mask which is fed to proposal network together with unobserved features mask $b$ and object $x$ .
+
+The proposed modifications are evaluated in section 5.1.
+
+Table 1: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations. Less is better.   
+
+<table><tr><td>Method/Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Zoo</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.964± 0.007</td><td>1.01 ± 0.01</td><td>0.334± 0.002</td><td>0.19±0.03</td><td>0.422± 0.006</td></tr><tr><td>MissForest</td><td>0.878 ± 0.009</td><td>1.02 ± 0.06</td><td>0.249 ± 0.006</td><td>0.16 ±0.02</td><td>0.422 ± 0.009</td></tr><tr><td>GAIN</td><td>0.97 ± 0.02</td><td>0.99 ± 0.03</td><td>0.271 ± 0.003</td><td>0.20± 0.02</td><td>0.427 ± 0.010</td></tr><tr><td>VAEAC</td><td>0.850 ± 0.007</td><td>0.94 ± 0.01</td><td>0.244 ± 0.002</td><td>0.16 ± 0.02</td><td>0.394± 0.006</td></tr></table>
+
+# 5 EXPERIMENTS
+
+In this section we validate the performance of VAEAC using several real-world datasets. In the first set of experiments we evaluate VAEAC missing features imputation performance using various UCI datasets (Lichman, 2013). We compare imputations from our model with imputations from such classical methods as MICE (Buuren & Groothuis-Oudshoorn, 2010) and MissForest (Stekhoven & Buhlmann, 2011) and recently ¨ proposed GANs-based method GAIN (Yoon et al., 2018). In the second set of experiments we use VAEAC to solve image inpainting problem. We show inpainitngs generated by VAEAC and compare our model with models from papers Pathak et al. (2016), Yeh et al. (2017) and Li et al. (2017) in terms of peak signal-to-noise ratio (PSNR) of obtained inpaintings on CelebA dataset (Liu et al., 2015) . And finally, we evaluate VAEAC against the competing method called Universal Marginalizer (Douglas et al., 2017). Additional experiments can be found in appendices C and D. The code is available at https://github.com/tigvarts/ vaeac.
+
+# 5.1 MISSING FEATURES IMPUTATION
+
+The datasets with missing features are widespread. Consider a dataset with $D$ -dimensional objects $x$ where each feature may be missing (which we denote by $x _ { i } ~ = ~ \omega$ ) and their target values $y$ . The majority of discriminative methods do not support missing values in the objects. The procedure of filling in the missing features values is called missing features imputation.
+
+In this section we evaluate the quality of imputations produced by VAEAC. For evaluation we use datasets from UCI repository (Lichman, 2013). Before training we drop randomly $50 \%$ of values both in train and test set. After that we impute missing features using MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest (Stekhoven & Buhlmann, 2011), GAIN (Yoon et al., 2018) and VAEAC trained on the observed data. The ¨ details of GAIN implementation are described in appendix A.4.
+
+Our model learns the distribution of the imputations, so it is able to sample from this distribution. We replace each object with missing features by $n = 1 0$ objects with sampled imputations, so the size of the dataset increases by $n$ times. This procedure is called missing features multiple imputation. MICE and GAIN are also capable of multiple imputation (we use $n = 1 0$ for them in experiments as well), but MissForest is not.
+
+For more details about the experimental setup see appendices A.1, A.2, and A.4.
+
+In table 1 we report NRMSE (i.e. RMSE normalized by the standard deviation of each feature and then averaged over all features) of imputations for continuous datasets and proportion of falsely classified (PFC) for categorical ones. For multiple imputation methods we average imputations of continuous variables and take most frequent imputation for categorical ones for each object.
+
+We also learn linear or logistic regression and report the regression or classification performance after applying imputations of different methods in table 2. For multiple imputation methods we average predictions for continuous targets and take most frequent prediction for categorical ones for each object in test set.
+
+Table 2: R2-score (for continuous targets) or accuracy (for categorical ones) of post-imputation regression or classification. Higher is better.   
+
+<table><tr><td>Method /Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Z00</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.13±0.02</td><td>0.41 ±0.02</td><td>0.92± 0.01</td><td>0.78± 0.05</td><td>0.75 ±0.02</td></tr><tr><td>MissForest</td><td>0.17 ± 0.01</td><td>0.42 ± 0.02</td><td>0.972 ± 0.003</td><td>0.71 ± 0.07</td><td>0.73 ± 0.02</td></tr><tr><td>GAIN</td><td>0.11 ± 0.01</td><td>0.39 ± 0.06</td><td>0.969 ± 0.005</td><td>0.67 ± 0.06</td><td>0.74 ± 0.03</td></tr><tr><td>VAEAC</td><td>0.17 ± 0.01</td><td>0.43 ± 0.01</td><td>0.983 ± 0.002</td><td>0.8 ± 0.1</td><td>0.74 ± 0.02</td></tr></table>
+
+As can be seen from the tables 1 and 2, VAEAC can learn joint data distribution and use it for missing feature imputation. The imputations are competitive with current state of the art imputation methods in terms of RMSE, PFC, post-imputation regression R2-score and classification accuracy. Nevertheless, we don’t claim that our method is state of the art in missing features imputation; for some datasets MICE or MissForest outperform it. The additional experiments can be found in appendix D.2.
+
+# 5.2 IMAGE INPAINTING
+
+The image inpainting problem has a number of different formulations. The formulation of our interest is as follows: some of the pixels of an image are unobserved and we want to restore them in a natural way. Unlike the majority of papers, we want to restore not just one most probable inpainting, but the distribution over all possible inpaintings from which we can sample. This distribution is extremely multi-modal because often there is a lot of different possible ways to inpaint the image.
+
+Unlike the previous subsection, here we have uncorrupted images without missing features in the training set, so $p ( b | x ) = p ( b )$ .
+
+As we show in section 2, state of the art results use different adversarial losses to achieve more sharp and realistic samples. VAEAC can be adapted to the image inpainting problem by using a combination of those adversarial losses as a part of reconstruction loss $p _ { \theta } ( x _ { b } | \boldsymbol { z } , x _ { 1 - b } , b )$ . Nevertheless, such construction is out of scope for this research, so we leave it for the future work. In the current work we show that the model can generate both diverse and realistic inpaintings.
+
+In figures 1, 2, 3 and 4 we visualize image inpaintings produced by VAEAC on binarized MNIST (LeCun et al., 1998), Omniglot (Lake et al., 2015) and CelebA (Liu et al., 2015). The details of learning procedure and description of datasets are available in appendixes A.1 and A.3.
+
+To the best of our knowledge, the most modern inpainting papers don’t consider the diverse inpainting problem, where the goal is to build diverse image inpaintings, so there is no straightforward way to compare with these models. Nevertheless, we compute peak signal-to-noise ratio (PSNR) for one random inpainting from VAEAC and the best PSNR among 10 random inpaintings from VAEAC. One inpainting might not be similar to the original image, so we also measure how good the inpainting which is most similar to the original image reconstructs it. We compare these two metrics computed for certain masks with the PSNRs for the same masks on CelebA from papers Yeh et al. (2017) and Li et al. (2017). The results are available in tables 3 and 4.
+
+We observe that for the majority of proposed masks our model outperforms the competing methods in terms of PSNR even with one sample, and for the rest (where the inpaintings are significantly diverse) the best PSNR over 10 inpaintings is larger than the same PSNR of the competing models. Even if PSNR does not reflect completely the visual quality of images and tends to encourage blurry VAE samples instead of realistic GANs samples, the results show that VAEAC is able to solve inpainting problem comparably to the state of the art methods. The disadvantage of VAEAC compared to Yeh et al. (2017) and Li et al. (2017) (but not Pathak et al. (2016)) is that it needs the distribution over masks at the training stage to be similar to the distribution over them at the test stage. However, it is not a very strict limitation for the practical usage.
+
+Table 3: PSNR of inpaintings for different masks for Context Encoder (Pathak et al., 2016), model from “Semantic Image Inpainting with Deep Generative Models” (Yeh et al., 2017) and VAEAC. Higher is better.   
+
+<table><tr><td>Method/Masks</td><td>Center</td><td>Pattern</td><td>Random</td><td>Half</td></tr><tr><td>Context Encoder 1</td><td>21.3</td><td>19.2</td><td>20.6</td><td>15.5</td></tr><tr><td>SIIDGM 1</td><td>19.4</td><td>17.4</td><td>22.8</td><td>13.7</td></tr><tr><td>VAEAC, 1 sample</td><td>22.1</td><td>21.4</td><td>29.3</td><td>14.9</td></tr><tr><td>VAEAC,10 samples</td><td>23.7</td><td>23.3</td><td>29.3</td><td>17.4</td></tr></table>
+
+Table 4: PSNR of inpaintings for different masks for Context Encoder (Pathak et al., 2016), model from “Generative Face Completion” (Li et al., 2017) and VAEAC. Higher is better.   
+
+<table><tr><td>Method/Masks</td><td>01</td><td>02</td><td>03</td><td>04</td><td>05</td><td>06</td></tr><tr><td>Context Encoder2</td><td>18.6</td><td>18.4</td><td>17.9</td><td>19.0</td><td>19.1</td><td>19.3</td></tr><tr><td>GFC ²</td><td>20.0</td><td>19.8</td><td>18.8</td><td>19.7</td><td>19.5</td><td>20.2</td></tr><tr><td>VAEAC,1 sample</td><td>20.8</td><td>21.0</td><td>19.5</td><td>20.3</td><td>20.3</td><td>21.0</td></tr><tr><td>VAEAC,10 samples</td><td>22.0</td><td>22.2</td><td>20.8</td><td>21.7</td><td>21.8</td><td>22.2</td></tr></table>
+
+# 5.3 UNIVERSAL MARGINALIZER
+
+Universal Marginalizer (Douglas et al., 2017) (UM) is a model which uses a single neural network to estimate the marginal distributions over the unobserved features. So it optimizes the following objective:
+
+$$
+\operatorname* { m a x } _ { \theta } \mathbb { E } _ { x \sim p _ { d } ( x ) } \mathbb { E } _ { b \sim p ( b ) } \sum _ { i = 1 } ^ { D } b _ { i } \log p _ { \theta } \big ( x _ { i } | x _ { 1 - b } , b \big )
+$$
+
+For given mask $b$ we fix a permutation of its unobserved components: $( i _ { 1 } , i _ { 2 } , \dots , i _ { | b | } )$ , where $| b |$ is a number of unobserved components. Using the learned model and the permutation we can generate objects from joint distribution and estimate their probability using chain rule.
+
+$$
+\log p _ { \theta } ( x _ { b } | x _ { 1 - b } , b ) = \sum _ { j = 1 } ^ { | b | } \log p _ { \theta } ( x _ { i _ { j } } | x _ { 1 - ( b - \sum _ { k = 1 } ^ { j - 1 } e _ { i _ { k } } ) } , b - \sum _ { k = 1 } ^ { j - 1 } e _ { i _ { k } } )
+$$
+
+For example, $p _ { \theta } ( x _ { 1 } , x _ { 4 } , x _ { 5 } | x _ { 2 } , x _ { 3 } ) = p _ { \theta } ( x _ { 4 } | x _ { 2 } , x _ { 3 } ) p _ { \theta } ( x _ { 1 } | x _ { 2 } , x _ { 3 } , x _ { 4 } ) p _ { \theta } ( x _ { 5 } | x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } ) .$
+
+Conditional sampling or conditional likelihood estimation for one object requires $| b |$ requests to UM to compute $p _ { \theta } ( x _ { i } | x _ { 1 - b } , b )$ . Each request is a forward pass through the neural network. In the case of conditional sampling those requests even cannot be paralleled because the input of the next request contains the output of the previous one.
+
+We propose a slight modification of the original UM training procedure which allows learning UM efficiently for any kind of masks including those considered in this paper. The details of the modification are described in appendix B.3.
+
+![](images/d8e3304c44f6641f3fcf46844d8ec076bf559af92021199814e73a3e8e86af57.jpg)  
+Figure 1: MNIST inpaintings.
+
+![](images/696a7457eff5fb7c33faf339e1df45fda1cdfba4264c7adccc10fee5a551d327.jpg)  
+Figure 2: Omniglot inpaintings.
+
+![](images/22952789d3ea86e1798ed0fcb93d6b5ec6a1f7740b49980530ccffa387686d74.jpg)  
+Figure 3: CelebA inpaintings.
+
+![](images/1a5cbe58c9eb61a7543d31d330e9a0fdd262a2099e64f42864b8627f3f719b25.jpg)  
+Figure 4: CelebA inpaintings with masks from (Yeh et al., 2017).
+
+Left: input. The gray pixels are unobserved. Middle: samples from VAEAC. Right: ground truth.
+
+Table 5: VAEAC and UM comparison on MNIST.   
+
+<table><tr><td>Method</td><td>VAEAC</td><td>UM</td></tr><tr><td>Negative log-likelihood</td><td>61</td><td>41</td></tr><tr><td>Training time (30 epochs)</td><td>5min 47s</td><td>3min 14s</td></tr><tr><td>Test time (1OO samples generation)</td><td>0.7ms</td><td>1s</td></tr></table>
+
+The results of using this modification of UM are provided in table 5. We can say that the relation between VAEAC and UM is similar to the relation between VAE and PixelCNN. The second one is much slower at the testing stage, but it easily takes into account local dependencies in data while the first one is faster but assumes conditional independence of the outputs. Nevertheless, there are a number of cases where UM cannot learn the distribution well while VAEAC can. For example, when the data is real-valued and marginal distributions have many local optima, there is no straightforward parametrization which allows UM to approximate them, and, therefore also the conditioned joint distribution. An example of such distribution and more illustrations for comparison of VAEAC and UM are available in appendix D.5.
+
+# 6 CONCLUSION
+
+In this paper we consider the problem of simultaneous learning of all conditional distributions for a vector. This problem has a number of different special cases with practical applications. We propose neural network based probabilistic model for distribution conditioning learning with Gaussian latent variables. This model is scalable and efficient in inference and learning. We propose several tricks to improve optimization and give recommendations about hyperparameters choice. The model is successfully applied to feature imputation and inpainting tasks. The experimental results show that the model is competitive with state of the art methods for both missing features imputation and image inpainting problems.
+
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+
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+
+# APPENDIX
+
+# A EXPERIMENTAL DETAILS
+
+A.1 NEURAL NETWORK ARCHITECTURES
+
+In all experiments we use optimization method Adam (Kingma & Ba, 2014), skip-connections between prior network and generative network inspired by (Mao et al., 2016), (Sønderby et al., 2016) and (Ronneberger et al., 2015), and convolutional neural networks based on ResNet blocks (He et al., 2016).
+
+Without skip-connections all information for decoder goes through the latent variables. In image inpainting we found skip-connections very useful in both terms of log-likelihood improvement and the image realism, because latent variables are responsible for the global information only while the local information passes through skip-connections. Therefore the border between image and inpainting becomes less conspicuous.
+
+The main idea of neural networks architecture is reflected in figure 5.
+
+![](images/5b67d91960752eb18692b5b61ef536a092947e8d387abb82437625c2d41f393a.jpg)  
+Figure 5: Neural network architecture for inpainting.
+
+The number of hidden layers, their widths and structure may be different.
+
+The neural networks we used for image inpainting have He-Uniform initialization of convolutional ResNet blocks, and the skip-connections are implemented using concatenation, not addition. The proposal network structure is exactly the same as the prior network except skip-connections.
+
+Also one could use much simpler fully-connected networks with one hidden layer as a proposal, prior and generative networks in VAEAC and still obtain nice inpaintings on MNIST.
+
+# A.2 MISSING FEATURES IMPUTATION
+
+We split the dataset into train and test set with size ratio 3:1. Before training we drop randomly $50 \%$ of values both in train and test set. We repeat each experiment 5 times with different train-test splits and dropped features and then average results and compute their standard deviation.
+
+As we show in appendix B.2, the better results can be achieved when the model learns the concatenation of objects features $x$ and targets $y$ . So we treat $y$ as an additional feature that is always unobserved during the testing time.
+
+To train our model we use distribution $p ( b _ { i } | x )$ in which $p ( b _ { i } | x _ { i } = \omega ) = 1$ and $p ( b _ { i } | x ) = 0 . 2$ otherwise. Also for VAEAC trainig we normalize real-valued features, fix $\sigma _ { \theta } = 1$ in the generative model of VAEAC in order to optimize RMSE, and use $2 5 \%$ of training data as validation set to select the best model among all epochs of training.
+
+For the test set, the classifier or regressor is applied to each of the $n$ imputed objects and the predictions are combined. For regression problems we report R2-score of combined predictions, so we use averaging as a combination method. For classification problem we report accuracy, and therefore choose the mode. We consider the workflow where the imputed values of $y$ are not fed to the classifier or regressor to make a fair comparison of feature imputation quality.
+
+Table 6: Generative Face Completion (Li et al., 2017) masks. Image size is 128x128.   
+
+<table><tr><td>Mask</td><td>Meaning</td><td>X1</td><td>x2</td><td>y1</td><td>y2</td></tr><tr><td>01</td><td>Left half of the face</td><td>33</td><td>70</td><td>52</td><td>115</td></tr><tr><td>02</td><td>Right half of the face</td><td>57</td><td>70</td><td>95</td><td>115</td></tr><tr><td>03</td><td>Two eyes</td><td>29</td><td>98</td><td>52</td><td>73</td></tr><tr><td>04</td><td>Left eye</td><td>29</td><td>66</td><td>52</td><td>73</td></tr><tr><td>05</td><td>Right eye</td><td>61</td><td>99</td><td>52</td><td>73</td></tr><tr><td>06</td><td>Lower half of the face</td><td>40</td><td>87</td><td>86</td><td>123</td></tr></table>
+
+NRMSE or PFC for dataset is computed as an average of NRMSE or PFC of all features of this dataset. NRMSE of a feature is just RMSE of imputations divided by the standard deviation of this feature. PFC of a feature is a proportion of imputations which are incorrect.
+
+# A.3 IMAGE INPAINTING DATASETS AND MASKS
+
+MNIST is a dataset of 60000 train and 10000 test grayscale images of digits from 0 to 9 of size $2 8 \mathbf { x } 2 8$ . We binarize all images in the dataset. For MNIST we consider Bernoulli log-likelihood as the reconstruction loss: $\begin{array} { r } { \log p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b ) = \sum _ { i : b _ { i } = 1 } \log \mathrm { B e r n o u l l i } ( x _ { i } | p _ { \theta , i } ( z , x _ { 1 - b } , b ) ) } \end{array}$ where $p _ { \theta , i } ( z , x _ { 1 - b } , b )$ is an output of the generative neural network. We use 16 latent variables. In the mask for this dataset the observed pixels form a three pixels wide horizontal line which position is distributed uniformly.
+
+Omniglot is a dataset of 19280 train and 13180 test black-and-white images of different alphabets symbols of size $1 0 5 \mathrm { x } 1 0 5$ . As in previous section, the brightness of each pixel is treated as a Bernoulli probability of it to be 1. The mask we use is a random rectangular which is described below. We use 64 latent variables. We train model for 50 epochs and choose best model according to IWAE log-likelihood estimation on the validation set after each epoch.
+
+CelebA is a dataset of 162770 train, 19867 validation and 19962 test color images of faces of celebrities of size $1 7 8 \mathrm { x } 2 1 8$ . Before learning we normalize the channels in dataset. We use logarithm of fully-factorized Gaussian distribution as reconstruction loss. The mask we use is a random rectangular which is describe below. We use 32 latent variables.
+
+Rectangular mask is the common shape of unobserved region in image inpainting. We use such mask for Omniglot and Celeba. We sample the corner points of rectangles uniprobably on the image, but reject those rectangles which area is less than a quarter of the image area.
+
+In Li et al. (2017) six different masks O1–O6 are used on the testing stage. We reconstruct the positions of masks from the illustrations in the paper and give their coordinates in table 6. The visualizations of the masks are available in figure 10.
+
+At the training stage we used a rectangle mask with uniprobable random corners. We reject masks with width or height less than 16pt. We use 64 latent variables and take the best model over 50 epochs based on the validation IWAE log-likelihood estimation. We can obtain slightly higher PSNR values than reported in table 4 if use only masks O1–O6 at the training stage.
+
+In Yeh et al. (2017) four types of masks are used. Center mask is just an unobserved $3 2 \mathrm { x } 3 2 $ square in the center of 64x64 image. Half mask mean that one of upper, lower, left or right half of the image is unobserved. All these types of a half are equiprobable. Random mask means that we use pixelwise-independent Bernoulli distribution with probability 0.8 to form a mask of unobserved pixels. Pattern mask is proposed in Pathak et al. (2016). As we deduced from the code 3, the generation process is follows: firstly we generate $6 0 0 \times 6 0 0$ one-channel image with uniform distribution over pixels, then bicubically interpolate it to image of size $1 0 0 0 0 \mathrm { x } 1 0 0 0 0$ , and then apply Heaviside step function $H ( x - 0 . 2 5 )$ (i. e. all points with value less than 0.25 are considered as unobserved). To sample a mask we sample a random position in this $1 0 0 0 0 \mathrm { x } 1 0 0 0 0$ binary image and crop $6 4 \mathrm { x } 6 4$ mask. If less than $20 \%$ or more than $30 \%$ of pixel are unobserved, than the mask is rejected and the position is sampled again. In comparison with this paper in section 5.2 we use the same distribution over masks at training and testing stages. We use VAEAC with 64 latent variables and take the best model over 50 epochs based on the validation IWAE log-likelihood estimation.
+
+# A.4 GAIN IMPLEMENTATION DETAILS
+
+For missing feature imputation we reimplemented GAIN in PyTorch based on the paper (Yoon et al., 2018) and the available TensorFlow source code for image inpainting 4.
+
+For categorical features we use one-hot encoding. We observe in experiments that it works better in terms of NRMSE and PFC than processing categorical features in GAIN as continuous ones and then rounding them to the nearest category.
+
+For categorical features we also use reconstruction loss $\begin{array} { r } { L _ { M } ( x _ { i } , x _ { i } ^ { \prime } ) = - \frac { 1 } { | X _ { i } | } \sum _ { j = 1 } ^ { | X _ { i } | } x _ { i , j } \log ( x _ { i , j } ^ { \prime } ) } \end{array}$ $\left| X _ { i } \right|$ the number of categories of the $i$ -th feature, and $x _ { i , j }$ is the $j$ -th component of one-hot encoding of the feature $x _ { i }$ . Such $L _ { M }$ enforces equal contribution of each categorical feature into the whole reconstruction loss.
+
+We use one more modification of $L _ { M } ( x , x ^ { \prime } )$ for binary and categorical features. Cross-entropy loss in $L _ { M }$ penalizes incorrect reconstructions of categorical and binary features much more than incorrect reconstructions for continuous ones. To avoid such imbalance we mixed L2 and cross-entropy reconstruction losses for binary and categorical features with weights 0.8 and 0.2 respectively:
+
+We observe in experiments that this modification also works better in terms of NRMSE and PFC than the original model.
+
+We use validation set which contains $5 \%$ of the observed features for the best model selection (hyperparameter is the number of iterations).
+
+In the original GAIN paper authors propose to use cross-validation for hyper-parameter $\alpha \in \{ 0 . 1 , 0 . 5 , 1 , 2 , 1 0 \}$ . We observe that using $\alpha ~ = ~ 1 0$ and a hint $h \ = \ b \circ \ m \ + \ 0 . 5 ( 1 \ - \ b )$ where vector $b$ is sampled from Bernoulli distribution with $p = 0 . 0 1$ provides better results in terms of NRMSE and PFC than the original model with every $\alpha \in \{ 0 . 1 , 0 . 5 , 1 , 2 , 1 0 \}$ . Such hint distribution makes model theoretically inconsistent but works well in practice (see table 7).
+
+Table 7 shows that our modifications provide consistently not worse or even better imputations than the original GAIN (in terms of NRMSE and PFC, on the considered datasets). So in this paper for the missing feature imputation problem we report the results of our modification of GAIN.
+
+Table 7: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations for different GAIN modifications. Less is better. “Our modification” includes the reconstruction loss $L _ { M } ^ { \prime }$ (12), Bernoulli distribution over $b$ in the hint generation procedure, and fixed $\alpha = 1 0$ . Other columns refers original GAIN without these modifications and with different values of $\alpha$ .   
+
+<table><tr><td>Dataset</td><td>Our modification</td><td>α=10</td><td>α=2</td><td>α=1</td><td>α= 0.5</td><td>α=0.1</td></tr><tr><td>Boston</td><td>0.78±0.03</td><td>0.87±0.02</td><td>1.0 ± 0.1</td><td>1.0 ± 0.1</td><td>1.02 ± 0.05</td><td>1.6±0.2</td></tr><tr><td>Breast</td><td>0.67 ± 0.01</td><td>0.80±0.05</td><td>1.00 ± 0.05</td><td>1.10 ± 0.07</td><td>1.19 ± 0.05</td><td>1.52 ± 0.06</td></tr><tr><td>Concrete</td><td>0.96 ± 0.01</td><td>0.98 ± 0.02</td><td>1.02 ± 0.02</td><td>1.13 ± 0.06</td><td>1.17 ± 0.04</td><td>1.3 ± 0.1</td></tr><tr><td>Diabetes</td><td>0.911 ± 0.009</td><td>0.93 ±0.03</td><td>1.05 ± 0.04</td><td>1.07 ± 0.07</td><td>1.21 ± 0.07</td><td>1.6 ± 0.1</td></tr><tr><td>Digits</td><td>0.79 ± 0.02</td><td>0.88 ± 0.01</td><td>1.05 ± 0.02</td><td>1.13 ± 0.02</td><td>1.24 ± 0.08</td><td>1.4± 0.2</td></tr><tr><td>Glass</td><td>1.06 ± 0.05</td><td>1.04 ± 0.05</td><td>1.19 ± 0.06</td><td>1.4 ± 0.2</td><td>1.6 ± 0.1</td><td>1.81 ± 0.10</td></tr><tr><td>Iris</td><td>0.72 ±0.04</td><td>0.73±0.06</td><td>0.83 ±0.08</td><td>0.97 ± 0.09</td><td>1.2 ± 0.2</td><td>1.3±0.2</td></tr><tr><td>Mushroom</td><td>0.271 ± 0.003</td><td>0.404 ± 0.004</td><td>0.52 ± 0.05</td><td>0.55 ± 0.01</td><td>0.56 ± 0.03</td><td>0.64± 0.06</td></tr><tr><td>Orthopedic</td><td>0.91 ± 0.03</td><td>0.91 ±0.08</td><td>1.1 ± 0.1</td><td>1.2 ± 0.1</td><td>1.34 ± 0.08</td><td>1.6 ± 0.2</td></tr><tr><td>Phishing</td><td>0.427 ± 0.010</td><td>0.52 ±0.02</td><td>0.54±0.02</td><td>0.543 ± 0.010</td><td>0.56 ± 0.01</td><td>0.57 ± 0.04</td></tr><tr><td>WallRobot</td><td>0.907 ± 0.005</td><td>0.924± 0.005</td><td>0.933 ± 0.008</td><td>0.95 ± 0.01</td><td>1.00 ± 0.02</td><td>1.26 ± 0.04</td></tr><tr><td>WhiteWine</td><td>0.97±0.02</td><td>1.02 ± 0.04</td><td>1.2 ± 0.1</td><td>1.3 ± 0.1</td><td>1.6 ± 0.1</td><td>1.86 ± 0.08</td></tr><tr><td>Yeast</td><td>0.99 ±0.03</td><td>1.3±0.2</td><td>1.6 ± 0.1</td><td>1.83 ± 0.09</td><td>1.9 ±0.1</td><td>2.4± 0.4</td></tr><tr><td>Zoo</td><td>0.20 ±0.02</td><td>0.24± 0.05</td><td>0.35 ± 0.06</td><td>0.36 ± 0.03</td><td>0.43 ± 0.04</td><td>0.433 ± 0.004</td></tr></table>
+
+# B THEORY
+
+# B.1 VAEAC UNIVERSALITY
+
+The theoretical guarantees that VAEAC can model arbitrary distribution are based on the same guarantees for Condtitional Variational Autoencoder (CVAE). We prove below that if CVAE can model each of the conditional distributions $p ( x _ { b } | x _ { 1 - b } )$ , then VAEAC can model all of them.
+
+We can imagine $2 ^ { D }$ CVAEs learned each for the certain mask. Because neural networks are universal approximators, VAEAC networks could model the union of CVAE networks, so that VAEAC network performs transformation defined by the same network of the corresponding to the given mask CVAE.
+
+$$
+p _ { \psi , V A E A C } ( z | x _ { 1 - b } , b ) = p _ { \psi , C V A E , 1 - b } ( z | x _ { 1 - b } ) \forall x , b
+$$
+
+$$
+p _ { \theta , V A E A C } ( x _ { b } | z , x _ { 1 - b } , b ) = p _ { \theta , C V A E , 1 - b } ( x _ { b } | z , x _ { 1 - b } ) \forall z , x , b
+$$
+
+So if CVAE models any distribution $p ( x | y )$ , VAEAC also do.
+
+The guarantees for CVAE in the case of continuous variables are based on the point that every smooth distribution can be approximated with a large enough mixture of Gaussians, which is a special case of CVAE’s generative model. These guarantees can be extended on the case of categorical-continuous variables also. Actually, there are distributions over categorical variables which CVAE with Gaussian prior and proposal distributions cannot learn. Nevertheless, this kind of limitation is not fundamental and is caused by poor proposal distribution family.
+
+# B.2 WHY VAEAC NEEDS TARGET VALUES FOR MISSING FEATURES IMPUTATION?
+
+Consider a dataset with $D$ -dimensional objects $x$ where each feature may be missing (which we denote by $x _ { i } = \omega$ ) and their target values $y$ . In this section we show that the better results are achieved when our model learns the concatenation of objects features $x$ and targets $y$ . The example that shows the necessity of it is following. Consider a dataset where $x _ { 1 } = 1$ , $x _ { 2 } \sim \mathcal { N } ( \bar { x } _ { 2 } | y , 1 )$ , $p _ { d } ( y = \mathrm { { 0 } ) = { { p } ( y = 5 ) = 0 . 5 } }$ . In this case $p _ { d } ( x _ { 2 } | x _ { 1 } = 1 ) = 0 . 5 \mathcal { N } ( x _ { 2 } | 0 , 1 ) + 0 . 5 \mathcal { N } ( x _ { 2 } | 5 , 1 )$ . We can see that generating data from $p _ { d } ( x _ { 2 } | x _ { 1 } )$ may only confuse the classifier, because with probability 0.5 it generates $x _ { 2 } \sim \bar { \mathcal { N } } ( 0 , 1 )$ for $y = 5$ and $x _ { 2 } \sim \mathcal { N } ( 5 , 1 )$ for $y = 0$ . On the other hand, $p _ { d } ( x _ { 2 } | x _ { 1 } , y ) = \mathcal { N } ( x _ { 2 } | y , 1 )$ . Filling gaps using $p _ { d } ( x _ { 2 } | x _ { 1 } , y )$ may only improve classifier or regressor by giving it some information from the joint distribution $p _ { d } ( x , y )$ and thus simplifying the dependence to be learned at the training time. So we treat $y$ as an additional feature that is always unobserved during the testing time.
+
+# B.3 UNIVERSAL MARGINALIZER: TRAINING PROCEDURE MODIFICATION
+
+The problem authors did not address in the original paper is the relation between the distribution of unobserved components $p ( b )$ at the testing stage and the distribution of masks in the requests to UM ${ \hat { p } } ( b )$ . The distribution over masks $p ( b )$ induces the distribution ${ \hat { p } } ( b )$ , and in the most cases $p ( b ) \neq { \hat { p } } ( b )$ . The distribution ${ \hat { p } } ( b )$ also depends on the permutations $( i _ { 1 } , i _ { 2 } , \dots , i _ { | b | } )$ that we use to generate objects.
+
+We observed in experiments, that UM must be trained using unobserved mask distribution ${ \hat { p } } ( b )$ . For example, if all masks from $p ( b )$ have a fixed number of unobserved components (e. g., $\begin{array} { l } { { \frac { D } { 2 } } } \end{array}$ ), then UM will never see an example of mask with $\begin{array} { r } { { 1 , 2 , \ldots , \frac { D } { 2 } - 1 } } \end{array}$ unobserved components, which is necessary to generate a sample conditioned on $\textstyle { \frac { D } { 2 } }$ components. That leads to drastically low likelihood estimate for the test set and unrealistic samples.
+
+We developed an easy generative process for ${ \hat { p } } ( b )$ for arbitrary $p ( b )$ if the permutation of unobserved components $( i _ { 1 } , i _ { 2 } , \dots , i _ { | b | } )$ is chosen randomly and equiprobably: firstly we generate $b _ { 0 } \sim p ( b )$ , $u \sim U [ 0 , 1 ]$ , then $b _ { 1 } \sim ( \mathrm { B e r n o u l l i } ( u ) ) ^ { D }$ and $b = b _ { 0 } \circ b _ { 1 }$ . More complicated generative process exists for a sorted permutation where $i _ { j - 1 } < i _ { j } \forall j : 2 \le j \le | b |$ .
+
+In experiments we use uniform distribution over the permutations.
+
+# C GAUSSIAN STOCHASTIC NEURAL NETWORK
+
+Gaussian stochastic neural network (13) and hybrid model (14) are originally proposed in the paper on Conditional VAE (Sohn et al., 2015). The motivation authors mention in the paper is as follows. During training the proposal distribution $q _ { \phi } ( z | x , y )$ is used to generate the latent variables $z$ , while during the testing stage the prior $p _ { \psi } ( z | y )$ is used. KL divergence tries to close the gap between two distributions but, according to authors, it is not enough. To overcome the issue authors propose to use a hybrid model (14), a weighted mixture of variational lower bound (3) and a single-sample Monte-Carlo estimation of log-likelihood (13). The model corresponding to the second term is called Gaussian Stochastic Neural Network (13), because it is a feed-forward neural network with a single Gaussian stochastic layer in the middle. Also GSNN is a special case of CVAE where $q _ { \phi } ( z | x , y ) = p _ { \psi } ( z | y )$ .
+
+$$
+\begin{array} { r } { L _ { G S N N } ( x , y ; \theta , \psi ) = \mathbb { E } _ { p _ { \psi } ( z | y ) } \log p _ { \theta } ( x | z , y ) \qquad } \\ { L ( x , y ; \theta , \psi , \phi ) = \alpha L _ { C V A E } ( x , y ; \theta , \psi , \phi ) + ( 1 - \alpha ) L _ { G S N N } ( x , y ; \theta , \psi ) , \quad \alpha \in [ 0 , 1 ] } \end{array}
+$$
+
+Authors report that hybrid model and GSNN outperform CVAE in terms of segmentation accuracy on the majority of datasets.
+
+We can also add that this technique seems to soften the “holes problem” (Makhzani et al., 2016). In Makhzani et al. (2016) authors observe that vectors $z$ from prior distribution may be different enough from all vectors $z$ from the proposal distribution at the training stage, so the generator network may be confused at the testing stage. Due to this problem CVAE can have good reconstructions of $y$ given $z \sim q _ { \phi } ( z | x , y )$ , while samples of $y$ given $z \sim p _ { \psi } ( z | x )$ are not realistic.
+
+The same trick is applicable to our model as well:
+
+$$
+\begin{array} { r l } & { L _ { G S N N } ( x , b ; \theta , \psi ) = \mathbb { E } _ { p _ { \psi } ( z | x _ { 1 - b } , b ) } \log p _ { \theta } ( x _ { b } | z , x _ { 1 - b } , b ) } \\ & { L ( x , b ; \theta , \psi , \phi ) = \alpha L _ { V A E A C } ( x , b ; \theta , \psi , \phi ) + ( 1 - \alpha ) L _ { G S N N } ( x , b ; \theta , \psi ) , \quad \alpha \in [ 0 , 1 ] } \end{array}
+$$
+
+In order to reflect the difference between sampling $z$ from prior and proposal distributions, authors of CVAE use two methods of log-likelihood estimation:
+
+$$
+\log p _ { \theta , \psi } ( x | y ) \approx \log \frac { 1 } { S } \sum _ { i = 1 } ^ { S } p _ { \theta } ( x | z _ { i } , y ) , ~ z _ { i } \sim p _ { \psi } ( z | y )
+$$
+
+$$
+\log p _ { \theta , \psi } ( x | y ) \approx \log \frac { 1 } { S } \sum _ { i = 1 } ^ { S } \frac { p _ { \theta } ( x | z _ { i } , y ) p _ { \psi } ( z _ { i } | y ) } { q _ { \phi } ( z _ { i } | x , y ) } , ~ z _ { i } \sim q _ { \phi } ( z | x , y )
+$$
+
+The first estimator is called Monte-Carlo estimator and the second one is called Importance Sampling estimator (also known as IWAE). They are asymptotically equivalent, but in practice the Monte-Carlo estimator requires much more samples to obtain the same accuracy of estimation. Small $S$ leads to underestimation of the log-likelihood for both Monte-Carlo and Importance Sampling (Burda et al., 2015), but for Monte-Carlo the underestimation is expressed much stronger.
+
+We perform an additional study of GSNN and hybrid model and show that they have drawbacks when the target distribution $p ( x | y )$ is has multiple different local maximums.
+
+# C.1 THEORETICAL STUDY
+
+In this section we show why GSNN cannot learn distributions with several different modes and leads to a blurry image samples.
+
+For the simplicity of the notation we consider hybrid model for a standard VAE:
+
+$$
+L ( x ; \phi , \psi , \theta ) = \alpha \mathbb { E } _ { z \sim q _ { \phi } ( z \mid x ) } \log \frac { p _ { \theta } ( x \mid z ) p _ { \psi } ( z ) } { q _ { \phi } ( z \mid x ) } + ( 1 - \alpha ) \mathbb { E } _ { z \sim p _ { \psi } ( z ) } \log p _ { \theta } ( x \mid z )
+$$
+
+The hybrid model (16) for VAEAC can be obtained from (19) by replacing $x$ with $x _ { b }$ and conditioning all distributions on $x _ { 1 - b }$ and $b$ . The validity of the further equations and conclusions remains for VAEAC after this replacement.
+
+Consider now a categorical latent variable $z$ which can take one of $K$ values. Let $x$ be a random variable witfor $p _ { d } ( x )$ to be modele some values following true data distribut. So the true distribution has n: d $\begin{array} { r } { p _ { d } ( x = x _ { i } ) = \frac { 1 } { K } } \end{array}$ $i \in \{ 1 , 2 , \ldots , K \}$ $x _ { 1 } , x _ { 2 } , \dotsc , x _ { K }$ $K$ able modes. Suppose the generator network $N N _ { \theta }$ which models mapping from $z$ to some vector of parameters $\begin{array} { r l r } { v _ { z } } & { { } = } & { \bar { N } N _ { \theta } ( z ) } \end{array}$ . Thus, we define generative distribution as some function of these parameters: $p _ { \theta } ( x | z ) \ : = \ : f ( x , v _ { z } )$ . Therefore, the parameters $\theta$ are just the set of $v _ { 1 } , v _ { 2 } , \dotsc , v _ { K }$ .
+
+For the simplicity of the model we assume $\begin{array} { r } { p _ { \psi } ( z ) = \frac { 1 } { K } } \end{array}$ . Taking into account $\begin{array} { r } { p _ { \psi } ( z ) = \frac { 1 } { K } } \end{array}$ , we obtain optimal $\begin{array} { r } { q ( z = i | x ) = \frac { f ( x , v _ { i } ) } { \sum _ { j = 1 } ^ { K } f ( x , v _ { j } ) } } \end{array}$ Using (19) and the above formulas for $q _ { \phi } , p _ { \psi }$ and $p _ { \theta }$ we obtain the following optimization problem:
+
+$$
+\operatorname* { m a x } _ { v _ { 1 } , v _ { 2 } , \ldots , v _ { K } } \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \left[ \alpha \sum _ { j = 1 } ^ { K } \frac { f ( x _ { i } , v _ { j } ) } { \sum _ { k = 1 } ^ { K } f ( x _ { i } , v _ { k } ) } \log \frac { f ( x _ { i } , v _ { j } ) \frac { 1 } { K } } { \frac { f ( x _ { i } , v _ { j } ) } { \sum _ { k = 1 } ^ { K } f ( x _ { i } , v _ { k } ) } } + ( 1 - \alpha ) \sum _ { j = 1 } ^ { K } \frac { 1 } { K } \log f ( x _ { i } , v _ { j } ) \right]
+$$
+
+Table 8: Negative log-likelihood estimation of a hybrid model on the synthetic data. IS- $S$ refers to Importance Sampling log-likelihood estimation with $S$ samples for each object (18). MC- $S$ refers to Monte-Carlo log-likelihood estimation with $S$ samples for each object (17).
+
+<table><tr><td>VAEAC weight</td><td>IS-10</td><td>MC-10</td></tr><tr><td>a=1</td><td>0.22</td><td>85</td></tr><tr><td>α = 0.99</td><td>0.35</td><td>11</td></tr><tr><td>α = 0.9</td><td>0.62</td><td>1.7</td></tr></table>
+
+It is easy to show that (20) is equivalent to the following optimization problem:
+
+$$
+\operatorname* { m a x } _ { v _ { 1 } , v _ { 2 } , \ldots , v _ { K } } \sum _ { i = 1 } ^ { K } \left[ \alpha \log \frac { \sum _ { j = 1 } ^ { K } f ( x _ { i } , v _ { j } ) } { K } + ( 1 - \alpha ) \sum _ { j = 1 } ^ { K } \frac { 1 } { K } \log f ( x _ { i } , v _ { j } ) \right]
+$$
+
+It is clear from (21) that when $\alpha = 1$ the log-likelihood of the initial model is optimized. On the other hand, when influe $\alpha = 0$ the optimal point is  generative process, a $v _ { 1 } = v _ { 2 } = \cdots = v _ { K } = \operatorname { a r g m a x } _ { v } \sum _ { i = 1 } ^ { K } \log f ( x _ { i } , v ) .$ , i. e. mizes $z$ doesn’telihood $z$ $v$   
+estimation of the generative model $f ( x , v )$ for the given dataset of $x$ ’s. For Bernoulli and Gaussian generative distributions $f$ such $v$ is just average of all modes $x _ { 1 } , x _ { 2 } , \dotsc , x _ { K }$ . That explains why further we observe blurry images when using GSNN model.
+
+The same conclusion holds for for continuous latent variables instead of categorical. Given $K$ different modes in true data distribution, VAE uses proposal network to separate prior distribution into $K$ components (i. e. regions in the latent space), so that each region corresponds to one mode. On the other hand, in GSNN $z$ is sampled independently on the mode which is to be reconstructed from it, so for each $z$ the generator have to produce parameters suitable for all modes.
+
+From this point of view, there is no difference between VAE and VAEAC. If the true conditional distribution has several different modes, then VAEAC can fit them all, while GSNN learns their average. If true conditional distribution has one mode, GSNN and VAEAC are equal, and GSNN may even learn faster because it has less parameters.
+
+Hybrid model is a trade-off between VAEAC and GSNN: the closer $\alpha$ to zero, the more blurry and closer to the average is the distribution of the model. The exact dependence of the model distribution on $\alpha$ can be derived analytically for the simple data distributions or evaluated experimentally. We perform such experimental evaluation in the next sections.
+
+# C.2 SYNTHETIC DATA
+
+In this section we show that VAEAC is capable of learning a complex multimodal distribution of synthetic   
+data while GSNN and hybrid model are not. Let $x \in \mathbb { R } ^ { \bar { 2 } }$ and $p ( \bar { b } _ { 1 } = 1 ) = p ( b _ { 2 } = 1 ) = 0 . 5$ . $p _ { d } ( x ) =$   
+$\begin{array} { r l } { \frac { 1 } { 8 } \sum _ { i = 1 } ^ { 8 } \mathcal { N } ( x | \mu _ { i } , \frac { 1 } { 1 0 } I ) } & { { } } \end{array}$ where s samp $\mu _ { i } \sim \mathcal N ( \mu _ { i } | 0 , I )$ . The distribution e use multi-layer p $p ( x )$ is plotted in figure 6. The datasetptron with four ReLU layers of size $p _ { d } ( x )$   
+400-200-100-50, 25-dimensional Gaussian latent variables.
+
+For different mixture coefficients $\alpha$ we visualize samples from the learned distributions $p _ { \psi , \theta } ( x _ { 1 } , x _ { 2 } )$ , $p _ { \psi , \theta } ( x _ { 1 } | x _ { 2 } )$ , and $p _ { \psi , \theta } ( x _ { 2 } | x _ { 1 } )$ . The observed features for the conditional distributions are generated from the marginal distributions $p ( x _ { 2 } )$ and $p ( x _ { 1 } )$ respectively.
+
+We see in table 8 and in figure 7, that even with very small weight GSNN prevents model from learning distributions with several local optimas. GSNN also increases Monte-Carlo log-likelihood estimation with a few samples and decreases much more precise Importance Sampling log-likelihood estimation. When $\alpha = 0 . 9$ the whole distribution structure is lost.
+
+![](images/7407e71d027399c997ec577ddb914586b76471425f58bff6404a5cd035d567fc.jpg)  
+Figure 6: Probability density function of synthetic data distribution.
+
+![](images/a4bd4e815fbae18fb3e5afb3429dcbb43b07bb985c3a0166b18e1666e16f7080.jpg)  
+Figure 7: VAEAC for synthetic data.
+
+![](images/54463305da3093916463d301746c0407e7d8d2daa9a2e48291c407dc4d828109.jpg)  
+Figure 8: MNIST inpaintings. Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth.
+
+We see that using $\alpha \neq 1$ ruins multimodality of the restored distribution, so we highly recommend to use $\alpha = 1$ or at least $\alpha \approx 1$ .
+
+Table 9: Average negative log-likelihood of inpaintings for 1000 objects. IS- $S$ refers to Importance Sampling log-likelihood estimation with $S$ samples for each object (18). MC- $S$ refers to Monte-Carlo log-likelihood estimation with $S$ samples for each object (17). Naive Bayes is a baseline method which assumes pixels and colors independence.   
+
+<table><tr><td>Method</td><td>MNIST</td><td>Omniglot</td><td>CelebA</td></tr><tr><td>VAEAC IS-102</td><td>61±1</td><td>275±17</td><td>34035 ± 1609</td></tr><tr><td>VAEAC MC-104</td><td>94±4</td><td>1452 ± 109</td><td>41513 ± 2163</td></tr><tr><td>VAEAC MC-102</td><td>156 ±1</td><td>2203 ± 150</td><td>53904 ± 3121</td></tr><tr><td>GSNN MC-104</td><td>141 ±7</td><td>1199 ± 62</td><td>53427 ± 2208</td></tr><tr><td>GSNN MC-10²</td><td>141 ±1</td><td>1200 ± 62</td><td>53486 ± 2210</td></tr><tr><td>Naive Bayes</td><td>205</td><td>2490</td><td>269480</td></tr></table>
+
+![](images/3126337f9f6dd982a9196015b6247be9dab5083bb45c2e9eb0945400c4c56f31.jpg)  
+Figure 9: Convergence of VAE and VAEAC on MNIST dataset.
+
+C.3 COMPARISON ON THE IMAGE INPAINTING PROBLEM
+
+In figure 8 we can see that the inpaintings produced by GSNN are smooth, blurry and not diverse compared with VAEAC.
+
+Table 9 shows that VAEAC learns distribution over inpaintings better than GSNN in terms of test loglikelihood. Nevertheless, Monte-Carlo estimations with a small number of samples sometimes are better for GSNN, which means less local modes in the learned distribution and more blurriness in the samples.
+
+# D ADDITIONAL EXPERIMENTS
+
+# D.1 CONVERGENCE SPEED
+
+In figure 9 one can see that VAEAC has similar convergence speed to VAE in terms of iterations on MNIST dataset. In our experiments we observed the same behaviour for other datasets. Each iteration of VAEAC is about 1.5 times slower than VAE due to usage of three networks instead of two.
+
+Table 10: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations. Less is better.   
+
+<table><tr><td>Dataset</td><td>MICE</td><td>MissForest</td><td>GAIN</td><td>VAEAC</td><td>GSNN</td><td>NN</td></tr><tr><td>Boston</td><td>0.69 ± 0.02</td><td>0.58±0.02</td><td>0.78 ± 0.03</td><td>0.71 ± 0.02</td><td>0.70 ± 0.01</td><td>0.69 ± 0.01</td></tr><tr><td>Breast</td><td>0.58 ±0.02</td><td>0.515 ± 0.008</td><td>0.67 ±0.01</td><td>0.55±0.02</td><td>0.55 ± 0.02</td><td>0.52 ±0.02</td></tr><tr><td>Concrete</td><td>0.850 ± 0.007</td><td>0.78 ± 0.01</td><td>0.96 ±0.01</td><td>0.84 ±0.02</td><td>0.85 ± 0.01</td><td>2±3</td></tr><tr><td>Diabetes</td><td>0.80 ±0.01</td><td>0.84±0.02</td><td>0.911 ± 0.009</td><td>0.90 ±0.03</td><td>0.91 ± 0.03</td><td>0.90 ±0.02</td></tr><tr><td>Digits</td><td>0.69 ±0.02</td><td>0.61± 0.02</td><td>0.79±0.02</td><td>0.69 ±0.02</td><td>0.69 ± 0.02</td><td>0.67 ±0.02</td></tr><tr><td>Glass</td><td>0.91 ±0.02</td><td>0.83± 0.04</td><td>1.06 ±0.05</td><td>0.91 ±0.04</td><td>0.91 ± 0.05</td><td>0.87 ± 0.04</td></tr><tr><td>Iris</td><td>0.59 ± 0.02</td><td>0.62 ± 0.04</td><td>0.72 ± 0.04</td><td>0.64± 0.04</td><td>0.62 ± 0.04</td><td>0.61± 0.02</td></tr><tr><td>Mushroom</td><td>0.334 ± 0.002</td><td>0.249 ±0.006</td><td>0.271 ±0.003</td><td>0.241 ± 0.002</td><td>0.2412 ± 0.0009</td><td>0.239 ± 0.001</td></tr><tr><td>Orthopedic</td><td>0.76 ±0.02</td><td>0.79±0.03</td><td>0.91±0.03</td><td>0.80±0.03</td><td>0.81 ±0.03</td><td>0.81 ±0.02</td></tr><tr><td>Phishing</td><td>0.422 ± 0.006</td><td>0.422 ±0.009</td><td>0.427 ± 0.010</td><td>0.397 ± 0.010</td><td>0.392 ±0.009</td><td>0.41 ± 0.01</td></tr><tr><td>WallRobot</td><td>0.885 ± 0.003</td><td>0.640 ± 0.003</td><td>0.907 ± 0.005</td><td>0.78 ±0.01</td><td>0.776 ±0.007</td><td>0.757± 0.005</td></tr><tr><td>WhiteWine</td><td>0.964 ± 0.007</td><td>0.878 ±0.009</td><td>0.97 ± 0.02</td><td>0.850 ± 0.005</td><td>0.848 ± 0.007</td><td>0.85 ± 0.01</td></tr><tr><td>Yeast</td><td>0.98±0.02</td><td>1.00 ± 0.02</td><td>0.99 ±0.03</td><td>0.95 ± 0.01</td><td>0.958 ± 0.007</td><td>0.97 ± 0.03</td></tr><tr><td>Zoo</td><td>0.19 ± 0.03</td><td>0.16 ±0.02</td><td>0.20 ±0.02</td><td>0.16±0.02</td><td>0.17 ±0.02</td><td>0.16 ±0.01</td></tr></table>
+
+Table 11: R2-score (for continuous targets) or accuracy (for categorical ones) of post-imputation regression or classification. Higher is better.   
+
+<table><tr><td>Dataset</td><td>MICE</td><td>MissForest</td><td>GAIN</td><td>VAEAC</td><td>GSNN</td><td>NN</td></tr><tr><td>Boston</td><td>0.57 ± 0.08</td><td>0.6 ± 0.1</td><td>0.50 ± 0.10</td><td>0.5 ± 0.1</td><td>0.5± 0.1</td><td>0.50±0.09</td></tr><tr><td>Breast</td><td>0.96 ± 0.02</td><td>0.95 ± 0.02</td><td>0.94± 0.01</td><td>0.95 ± 0.02</td><td>0.96 ± 0.02</td><td>0.95 ± 0.02</td></tr><tr><td>Concrete</td><td>0.35 ± 0.05</td><td>0.33 ± 0.04</td><td>0.28±0.06</td><td>0.30 ± 0.08</td><td>0.32 ± 0.05</td><td>0±1</td></tr><tr><td>Diabetes</td><td>0.37 ± 0.06</td><td>0.34±0.06</td><td>0.34±0.03</td><td>0.34± 0.04</td><td>0.33 ±0.04</td><td>0.27±0.06</td></tr><tr><td>Digits</td><td>0.86±0.02</td><td>0.887 ±0.008</td><td>0.83 ±0.03</td><td>0.892 ± 0.010</td><td>0.895 ± 0.010</td><td>0.912 ± 0.010</td></tr><tr><td>Glass</td><td>0.44 ±0.08</td><td>0.53 ±0.05</td><td>0.37±0.05</td><td>0.49 ±0.09</td><td>0.47 ±0.09</td><td>0.48 ±0.09</td></tr><tr><td>Iris</td><td>0.81± 0.02</td><td>0.84±0.02</td><td>0.66 ±0.06</td><td>0.84±0.05</td><td>0.82±0.06</td><td>0.73±0.09</td></tr><tr><td>Mushroom</td><td>0.92 ±0.01</td><td>0.972 ± 0.003</td><td>0.969 ± 0.005</td><td>0.987 ± 0.001</td><td>0.986 ± 0.002</td><td>0.989 ± 0.003</td></tr><tr><td>Orthopedic</td><td>0.71 ± 0.02</td><td>0.72 ± 0.03</td><td>0.60±0.03</td><td>0.71 ±0.02</td><td>0.70± 0.04</td><td>0.61±0.04</td></tr><tr><td>Phishing</td><td>0.75 ± 0.02</td><td>0.73±0.03</td><td>0.74± 0.03</td><td>0.75 ± 0.01</td><td>0.74±0.04</td><td>0.73±0.02</td></tr><tr><td>WallRobot</td><td>0.55 ±0.01</td><td>0.697 ± 0.005</td><td>0.56 ±0.01</td><td>0.62±0.02</td><td>0.62 ± 0.01</td><td>0.64±0.02</td></tr><tr><td>WhiteWine</td><td>0.13 ± 0.02</td><td>0.17 ± 0.01</td><td>0.11± 0.01</td><td>0.18 ±0.02</td><td>0.17 ± 0.01</td><td>0.15 ± 0.03</td></tr><tr><td>Yeast</td><td>0.42 ±0.02</td><td>0.41 ±0.02</td><td>0.39 ±0.06</td><td>0.42 ± 0.01</td><td>0.425 ± 0.010</td><td>0.33±0.03</td></tr><tr><td>Zoo</td><td>0.78 ± 0.06</td><td>0.71 ±0.08</td><td>0.67±0.06</td><td>0.77 ± 0.09</td><td>0.8±0.1</td><td>0.83 ± 0.08</td></tr></table>
+
+# D.2 MISSING FEATURES IMPUTATION
+
+We evaluate the quality of imputations on different datasets (mostly from UCI (Lichman, 2013)). The evaluation is performed for VAEAC, GSNN (15) and NN (neural network; can be considered as a special case of GSNN where $p _ { \theta } ( z | x _ { 1 - b } , b )$ is delta-function; produces single imputation). We compare these methods with MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest (Stekhoven & Buhlmann, 2011), and GAIN ¨ (Yoon et al., 2018).
+
+We see that for some datasets MICE and MissForest outperform VAEAC, GSNN and NN. The reason is that for some datasets random forest is more natural structure than neural network.
+
+The results also show that VAEAC, GSNN and NN show similar imputation performance in terms of NRMSE, PFC, post-imputation R2-score and accuracy. Given the result from appendix C we can take this as a weak evidence that the distribution of imputations has only one local maximum for datasets from (Lichman, 2013).
+
+![](images/21041c7b3c7fc1baa4707c17e3944ec8fc55aadb337ff273f13ec39db52954c7.jpg)  
+Figure 10: CelebA inpaintings with masks from (Li et al., 2017). ft: input. The gray pixels are unobserved. Middle: samples from VAEAC. Right: ground truth.
+
+# D.3 FACE INPAINTINGS
+
+In figure 10 we provide samples of VAEAC on the CelebA dataset for the masks from (Li et al., 2017).
+
+# D.4 GAIN FOR IMAGE INPAINTING
+
+GAIN (Yoon et al., 2018) doesnt use unobserved data during training, which makes it easier to apply to the missing features imputation problem. Nevertheless, it turns into a disadvantage when the fully-observed training data is available but the missingness rate at the testing stage is high.
+
+We consider the horizontal line mask for MNIST which is described in appendix A.3. We use the released GAIN code 5 with a different mask generator. The inpaintings from VAEAC which uses the unobserved pixels during training are available in figure 1. The inpaintings from GAIN which ignores unobserved pixels are provided in figure 11. As can be seen in figure 11, GAIN fails to learn conditional distribution for given mask distribution ${ \dot { p } } ( b )$ .
+
+Nevertheless, we don’t claim that GAIN is not suitable for image inpainting. As it was shown in the supplementary of (Yoon et al., 2018) and in the corresponding code, GAIN is able to learn conditional distributions when $p ( b )$ is pixel-wise independent Bernoulli distribution with probability 0.5.
+
+![](images/77978a748fdfece774bbdba6a454e4caa9d264c220caf33b0d01dfee4003f59b.jpg)  
+Figure 11: MNIST inpaintings from GAIN.
+
+Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth.
+
+![](images/3addba9b5d9e7527dc59c64b67c88b55ee2d758e97d2f9f04dcf482aada3e5b6.jpg)  
+Figure 12: MNIST inpaintings. Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth.
+
+# D.5 UNIVERSAL MARGINALIZER: ILLUSTRATIONS
+
+In figure 12 we provide samples of Universal Marginalizer (UM) and VAEAC for the same inputs.
+
+Consider the case when UM marginal distributions are parametrized with Gaussians. The most simple example of a distribution, which UM cannot learn but VAEAC can, is given in figure 13.
+
+![](images/34b2efedea65ec8dd1cd6662835f9b38b5a648676328c4f707b65cec76fc2de2.jpg)  
+Figure 13: Distribution learning: VAEAC vs UM.
\ No newline at end of file
diff --git a/parse/train/SyxtJh0qYm/SyxtJh0qYm_content_list.json b/parse/train/SyxtJh0qYm/SyxtJh0qYm_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..96a50e647b53de90f44b9d1317132b38d772c571
--- /dev/null
+++ b/parse/train/SyxtJh0qYm/SyxtJh0qYm_content_list.json
@@ -0,0 +1,2990 @@
+[
+    {
+        "type": "text",
+        "text": "VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            117,
+            558,
+            162
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Oleg Ivanov   \nSamsung AI Center Moscow Moscow, Russia   \ntigvarts@gmail.com Michael Figurnov   \nNational Research University Higher School of Economics ∗ Moscow, Russia   \nmichael@figurnov.ru Dmitry Vetrov   \nSamsung-HSE Laboratory, National Research University Higher School of Economics Samsung AI Center Moscow Moscow, Russia   \nvetrovd@yandex.ru ",
+        "bbox": [
+            156,
+            186,
+            348,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            387,
+            186,
+            589,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            622,
+            188,
+            816,
+            284
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            320,
+            544,
+            335
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a single neural probabilistic model based on variational autoencoder that can be conditioned on an arbitrary subset of observed features and then sample the remaining features in “one shot”. The features may be both real-valued and categorical. Training of the model is performed by stochastic variational Bayes. The experimental evaluation on synthetic data, as well as feature imputation and image inpainting problems, shows the effectiveness of the proposed approach and diversity of the generated samples. ",
+        "bbox": [
+            207,
+            349,
+            792,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            457,
+            310,
+            473
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In past years, a number of generative probabilistic models based on neural networks have been proposed. The most popular approaches include variational autoencoder (Kingma & Welling, 2013) (VAE) and generative adversarial net (Goodfellow et al., 2014) (GANs). They learn a distribution over objects $p ( x )$ and allow sampling from this distribution. ",
+        "bbox": [
+            148,
+            488,
+            849,
+            544
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In many cases, we are interested in learning a conditional distribution $p ( x | y )$ . For instance, if $x$ is an image of a face, $y$ could be the characteristics describing the face (are glasses present or not; length of hair, etc.) Conditional variational autoencoder (Sohn et al., 2015) and conditional generative adversarial nets (Mirza & Osindero, 2014) are popular methods for this problem. ",
+        "bbox": [
+            148,
+            550,
+            849,
+            606
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we consider the problem of learning all conditional distributions of the form $p ( x _ { I } | x _ { U \\setminus I } )$ , where $U$ is the set of all features and $I$ is its arbitrary subset. This problem generalizes both learning the joint distribution $p ( x )$ and learning the conditional distribution $p ( x | y )$ . To tackle this problem, we propose a Variational Autoencoder with Arbitrary Conditioning (VAEAC) model. It is a latent variable model similar to VAE, but allows conditioning on an arbitrary subset of the features. The conditioning features affect the prior on the latent Gaussian variables which are used to generate unobserved features. The model is trained using stochastic gradient variational Bayes (Kingma & Welling, 2013). ",
+        "bbox": [
+            148,
+            613,
+            851,
+            712
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider two most natural applications of the proposed model. The first one is feature imputation where the goal is to restore the missing features given the observed ones. The imputed values may be valuable by themselves or may improve the performance of other machine learning algorithms which process the dataset. Another application is image inpainting in which the goal is to fill in an unobserved part of an image with an artificial content in a realistic way. This can be used for removing unnecessary objects from the images or, vice versa, for complementing the partially closed or corrupted object. ",
+        "bbox": [
+            148,
+            718,
+            851,
+            801
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The experimental evaluation shows that the proposed model successfully samples from the conditional distributions. The distribution over samples is close to the true conditional distribution. This property is very important when the true distribution has several modes. The model is shown to be effective in feature imputation problem which helps to increase the quality of subsequent discriminative models on different problems from UCI datasets collection (Lichman, 2013). We demonstrate that model can generate diverse and realistic image inpaintings on MNIST (LeCun et al., 1998), Omniglot (Lake et al., 2015) and CelebA (Liu et al., 2015) datasets, and works even better than the current state of the art inpainting techniques in terms of peak signal to noise ratio (PSNR). ",
+        "bbox": [
+            148,
+            119,
+            851,
+            232
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The paper is organized as follows. In section 2 we review the related works. In section 3 we briefly describe variational autoencoders and conditional variational autoencoders. In section 4 we define the problem, describe the VAEAC model and its training procedure. In section 5 we evaluate VAEAC. Section 6 concludes the paper. Appendix contains additional explanations, theoretical analysis, and experiments for VAEAC. ",
+        "bbox": [
+            148,
+            238,
+            849,
+            295
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            320,
+            318,
+            335
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Universal Marginalizer (Douglas et al., 2017) is a model based on a feed-forward neural network which approximates marginals of unobserved features conditioned on observable values. A related idea of an autoregressive model of joint probability was previously proposed in Germain et al. (2015) and Uria et al. (2016). The description of the model and comparison with VAEAC are available in section 5.3. ",
+        "bbox": [
+            148,
+            354,
+            851,
+            410
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Yoon et al. (2018) propose a GANs-based model called GAIN which solves the same problem as VAEAC. In contrast to VAEAC, GAIN does not use unobserved data during training, which makes it easier to apply to the missing features imputation problem. Nevertheless, it turns into a disadvantage when the fully-observed training data is available but the missingness rate at the testing stage is high. For example, in inpainting setting GAIN cannot learn the conditional distribution over MNIST digits given one horizontal line of the image while VAEAC can (see appendix D.4). The comparison of VAEAC and GAIN on the missing feature imputation problem is given in section 5.1 and appendix D.2. ",
+        "bbox": [
+            148,
+            417,
+            851,
+            515
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Rezende et al. (2014) [Appendix F], Sohl-Dickstein et al. (2015), Goyal et al. (2017), and Bordes et al. (2017) propose to fill missing data with noise and run Markov chain with a learned transition operator. The stationary distribution of such chains approximates the true conditional distribution of the unobserved features. Bachman & Precup (2015) consider missing feature imputation in terms of Markov decision process and propose LSTM-based sequential decision making model to solve it. Nevertheless, these methods are computationally expensive at the test time and require fully-observed training data. ",
+        "bbox": [
+            148,
+            522,
+            849,
+            606
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Image inpainting is a classic computer vision problem. Most of the earlier methods rely on local and texture information or hand-crafted problem-specific features (Bertalmio et al., 2000). In past years multiple neural network based approaches have been proposed. ",
+        "bbox": [
+            148,
+            613,
+            849,
+            655
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Pathak et al. (2016), Yeh et al. (2016) and Yang et al. (2017) use different kinds and combinations of adversarial, reconstruction, texture and other losses. Li et al. (2017) focuses on face inpainting and uses two adversarial losses and one semantic parsing loss to train the generative model. In Yeh et al. (2017) GANs are first trained on the whole training dataset. The inpainting is an optimization procedure that finds the latent variables that explain the observed features best. Then, the obtained latents are passed through the generative model to restore the unobserved portion of the image. We can say that VAEAC is a similar model which uses prior network to find a proper latents instead of solving the optimization problem. ",
+        "bbox": [
+            148,
+            661,
+            849,
+            760
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "All described methods aim to produce a single realistic inpainting, while VAEAC is capable of sampling diverse inpaintings. Additionally, Yeh et al. (2016), Yang et al. (2017) and Yeh et al. (2017) have high testtime computational complexity of inpainting, because they require an optimization problem to be solved. On the other hand, VAEAC is a “single-shot” method with a low computational cost. ",
+        "bbox": [
+            148,
+            766,
+            849,
+            821
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            118,
+            300,
+            135
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 VARIATIONAL AUTOENCODER ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            150,
+            397,
+            165
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Variational autoencoder (Kingma & Welling, 2013) (VAE) is a directed generative model with latent variables. The generative process in variational autoencoder is as follows: first, a latent variable $z$ is generated from the prior distribution $p ( z )$ , and then the data $x$ is generated from the generative distribution $p _ { \\theta } ( x | z )$ , where $\\theta$ are the generative model’s parameters. This process induces the distribution $p _ { \\theta } ( x ) = \\mathbb { E } _ { p ( z ) } p _ { \\theta } ( x | z )$ . The distribution $p _ { \\theta } ( x | z )$ is modeled by a neural network with parameters $\\theta$ . $p ( z )$ is a standard Gaussian distribution. ",
+        "bbox": [
+            147,
+            176,
+            852,
+            262
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The parameters $\\theta$ are tuned by maximizing the likelihood of the training data points $\\{ x _ { i } \\} _ { i = 1 } ^ { N }$ from the true data distribution $p _ { d } ( x )$ . In general, this optimization problem is challenging due to intractable posterior inference. However, a variational lower bound can be optimized efficiently using backpropagation and stochastic gradient descent: ",
+        "bbox": [
+            148,
+            267,
+            849,
+            325
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fed675d9452648daab0a6ca45636d1aac44900502c485323a4dc8122dfe3f9fe.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p _ { \\theta } ( x ) = \\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\log \\frac { p _ { \\theta } ( x , z ) } { q _ { \\phi } ( z | x ) } + D _ { \\mathrm { K L } } \\big ( q _ { \\phi } ( z | x ) \\| p ( z | x , \\theta ) \\big ) } \\\\ & { \\qquad \\quad \\ge \\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\log p _ { \\theta } ( x | z ) - D _ { \\mathrm { K L } } \\big ( q _ { \\phi } ( z | x ) \\| p ( z ) \\big ) = L _ { V A E } \\big ( x ; \\theta , \\phi \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            165,
+            335,
+            815,
+            392
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here $q _ { \\phi } ( z | x )$ is a proposal distribution parameterized by neural network with parameters $\\phi$ that approximates the posterior $p ( z | x , \\theta )$ . Usually this distribution is Gaussian with a diagonal covariance matrix. The closer $q _ { \\phi } ( z | x )$ to $p ( z | x , \\theta )$ , the tighter variational lower bound $L _ { V A E } ( \\theta , \\phi )$ . To compute the gradient of the variational lower bound with respect to $\\phi$ , reparameterization trick is used: $z = \\bar { \\mu } _ { \\phi } ( x ) + \\bar { \\varepsilon } \\sigma _ { \\phi } ( x )$ where $\\varepsilon \\sim \\mathcal { N } ( 0 , I )$ and $\\mu _ { \\phi }$ and $\\sigma _ { \\phi }$ are deterministic functions parameterized by neural networks. So the gradient can be estimated using Monte-Carlo method for the first term and computing the second term analytically: ",
+        "bbox": [
+            148,
+            396,
+            852,
+            481
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/392ee19314ad1169b0b4fd7d080fa9dd49328ced339e9d76af78ca28090587f9.jpg",
+        "text": "$$\n\\frac { \\partial L _ { V A E } ( x ; \\theta , \\phi ) } { \\partial \\phi } = \\mathbb { E } _ { \\varepsilon \\sim \\mathcal { N } ( 0 , I ) } \\frac { \\partial } { \\partial \\phi } \\log p _ { \\theta } ( x | \\mu _ { \\phi } ( x ) + \\varepsilon \\sigma _ { \\phi } ( x ) ) - \\frac { \\partial } { \\partial \\phi } D _ { \\mathrm { K L } } ( q _ { \\phi } ( z | x ) \\| p ( z ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            488,
+            787,
+            521
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "So $L _ { V A E } ( \\theta , \\phi )$ can be optimized using stochastic gradient ascent with respect to $\\phi$ and $\\theta$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 CONDITIONAL VARIATIONAL AUTOENCODER ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            560,
+            503,
+            575
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Conditional variational autoencoder (Sohn et al., 2015) (CVAE) approximates the conditional distribution $p _ { d } ( x | y )$ . It outperforms deterministic models when the distribution $p _ { d } ( x | y )$ is multi-modal (diverse $x \\mathbf { s }$ are probable for the given $y$ ). For example, assume that $x$ is a real-valued image. Then, a deterministic regression model with mean squared error loss would predict the average blurry value for $x$ . On the other hand, CVAE learns the distribution of $x$ , from which one can sample diverse and realistic objects. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Variational lower bound for CVAE can be derived similarly to VAE by conditioning all considered distributions on $y$ : ",
+        "bbox": [
+            142,
+            662,
+            849,
+            693
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a364d873a354bd42b6ac3c1a473c1e2729b1b87d4dce0f0bfde7ab681819e08.jpg",
+        "text": "$$\nL _ { C V A E } ( x , y ; \\theta , \\psi , \\phi ) = \\mathbb { E } _ { q _ { \\phi } ( z | x , y ) } \\log p _ { \\theta } ( x | z , y ) - D _ { \\mathrm { K L } } ( q _ { \\phi } ( z | x , y ) | | p _ { \\psi } ( z | y ) ) \\leq \\log p _ { \\theta , \\psi } ( x | y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            699,
+            812,
+            718
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similarly to VAE, this objective is optimized using the reparameterization trick. Note that the prior distribution $p _ { \\psi } ( z | y )$ is conditioned on $y$ and is modeled by a neural network with parameters $\\psi$ . Thus, CVAE uses three trainable neural networks, while VAE only uses two. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Also authors propose such modifications of CVAE as Gaussian stochastic neural network and hybrid model. These modifications can be applied to our model as well. Nevertheless, we don’t use them, because of their disadvantage which is described in appendix C. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING",
+        "text_level": 1,
+        "bbox": [
+            147,
+            118,
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+            136
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 PROBLEM STATEMENT ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            152,
+            346,
+            167
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Consider a distribution $p _ { d } ( x )$ over a $D$ -dimensional vector $x$ with real or categorical components. The components of the vector are called features. ",
+        "bbox": [
+            147,
+            180,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let binary vector $b \\in \\{ 0 , 1 \\} ^ { D }$ be the binary mask of unobserved features of the object. Then we describe the vector of unobserved features as $x _ { b } = \\{ x _ { i : b _ { i } = 1 } \\}$ . For example, $x _ { ( 0 , 1 , 1 , 0 , 1 ) } = ( x _ { 2 } , x _ { 3 } , x _ { 5 } )$ . Using this notation we denote $x _ { 1 - b }$ as a vector of observed features. ",
+        "bbox": [
+            147,
+            215,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to build a model of the conditional distribution $p _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b ) \\approx p _ { d } ( x _ { b } | x _ { 1 - b } , b )$ for an arbitrary $b$ , where $\\psi$ and $\\theta$ are parameters that are used in our model at the testing stage. ",
+        "bbox": [
+            148,
+            265,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, the true distribution $p _ { d } ( x _ { b } | x _ { 1 - b } , b )$ is intractable without strong assumptions about $p _ { d } ( x )$ . Therefore, our model $p _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b )$ has to be more precise for some $b$ and less precise for others. To formalize our requirements about the accuracy of our model we introduce the distribution $p ( b )$ over different unobserved feature masks. The distribution $p ( b )$ is arbitrary and may be defined by the user depending on the problem. Generally it should have full support over $\\{ 0 , 1 \\} ^ { D }$ so that $p _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b )$ can evaluate arbitrary conditioning. Nevertheless, it is not necessary if the model is used for specific kinds of conditioning (as we do in section 5.2). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using $p ( b )$ we can introduce the following log-likelihood objective function for the model: ",
+        "bbox": [
+            147,
+            404,
+            741,
+            420
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6eacc0f28bc2e238ba172a9f915eb5066998bc34e8c21e5ae8d8543613577821.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\psi , \\theta } \\mathbb { E } _ { p _ { d } ( \\boldsymbol { x } ) } \\mathbb { E } _ { p ( \\boldsymbol { b } ) } \\log p _ { \\psi , \\theta } \\big ( x _ { b } | \\boldsymbol { x } _ { 1 - \\boldsymbol { b } } , \\boldsymbol { b } \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            430,
+            625,
+            454
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The special cases of the objective (4) are variational autoencoder $( b _ { i } = 1 \\forall i \\in \\{ 1 , \\ldots , D \\} )$ and conditional variational autoencoder $^ { \\textit { b } }$ is constant). ",
+        "bbox": [
+            147,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 MODEL DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            517,
+            343,
+            531
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The generative process of our model is similar to the generative process of CVAE: for each object firstly we generate $z \\sim p _ { \\psi } ( z | x _ { 1 - b } , b )$ using prior network, and then sample unobserved features $x _ { b } ~ \\sim ~ p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b )$ using generative network. This process induces the following model distribution over unobserved features: ",
+        "bbox": [
+            147,
+            545,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1010c9b588784d89851c33cee48ac5e4d9b7fc2302b1356bbdce195c382c31ce.jpg",
+        "text": "$$\np _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b ) = \\mathbb { E } _ { z \\sim p _ { \\psi } ( z | x _ { 1 - b } , b ) } p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            612,
+            673,
+            631
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use $z ~ \\in ~ \\mathbb { R } ^ { d }$ , and Gaussian distribution $p _ { \\psi }$ over $z$ , with parameters from a neural network with weights $\\psi$ : $p _ { \\psi } ( z | x _ { 1 - b } , b , \\psi ) = \\mathcal { N } ( z | \\mu _ { \\psi } ( x _ { 1 - b } , b ) , \\sigma _ { \\psi } ^ { 2 } ( x _ { 1 - b } , b ) I )$ . The real-valued components of distribution $p _ { \\theta } ( x _ { b } | \\boldsymbol { z } , x _ { 1 - b } , b )$ are defined likewise. Each categorical component $i$ of distribution $p _ { \\theta } ( x _ { i } | \\boldsymbol { z } , x _ { 1 - b } , b )$ is parameterized by a function $w _ { i , \\theta } ( z , x _ { 1 - b } , b )$ , whose outputs are logits of probabilities for each category: $x _ { i } \\sim \\mathrm { C a t } [ \\mathrm { S o f t m a x } ( w _ { i , \\theta } ( z , x _ { 1 - b } , b ) ) ]$ . Therefore the components of the latent vector $z$ are conditionally independent given $x _ { 1 - b }$ and $b$ , and the components of $x _ { b }$ are conditionally independent given $z , x _ { 1 - b }$ and $b$ . ",
+        "bbox": [
+            147,
+            650,
+            851,
+            739
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The variables $x _ { b }$ and $x _ { 1 - b }$ have variable length that depends on $b$ . So in order to use architectures such as multi-layer perceptron and convolutional neural network we consider $x _ { 1 - b } = x \\circ ( 1 - b )$ where $\\circ$ is an element-wise product. So in implementation $x _ { 1 - b }$ has fixed length. The output of the generative network also has a fixed length, but we use only unobserved components to compute likelihood. ",
+        "bbox": [
+            148,
+            744,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The theoretical analysis of the model is available in appendix B.1. ",
+        "bbox": [
+            147,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 LEARNING VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING",
+        "text_level": 1,
+        "bbox": [
+            145,
+            119,
+            717,
+            136
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3.1 VARIATIONAL LOWER BOUND ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            146,
+            413,
+            161
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can derive a lower bound for $\\log p _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b )$ as for variational autoencoder: ",
+        "bbox": [
+            147,
+            171,
+            700,
+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23cab9595cb2df09a6fe520b5c3c4ac610eb31ffc336c62a0f70f0e3846d00aa.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p _ { \\psi , \\theta } ( x _ { b } | x _ { 1 - b } , b ) = \\mathbb { E } _ { q _ { \\phi } ( z | x , b ) } \\log \\frac { p _ { \\psi , \\theta } ( x _ { b } , z | x _ { 1 - b } , b ) } { q _ { \\phi } ( z | x , b ) } + D _ { \\mathrm { K L } } ( q _ { \\phi } ( z | x , b ) | | p _ { \\psi , \\theta } ( z | x , b ) ) } \\\\ & { \\qquad \\geq \\mathbb { E } _ { q _ { \\phi } ( z | x , b ) } \\log p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b ) - D _ { \\mathrm { K L } } ( q _ { \\phi } ( z | x , b ) | | p _ { \\psi } ( z | x _ { 1 - b } , b ) ) = L _ { V A E A C } ( x , b ; \\theta , \\psi , \\phi ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            163,
+            200,
+            815,
+            256
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Therefore we have the following variational lower bound optimization problem: ",
+        "bbox": [
+            147,
+            261,
+            671,
+            276
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0adebae96daeaa3c98aa6ef63b82c1e9fa0b29000c3c6a3af4e0beff92f397c1.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta , \\psi , \\phi } \\mathbb { E } _ { p _ { d } ( x ) } \\mathbb { E } _ { p ( b ) } L _ { V A E A C } ( x , b ; \\theta , \\psi , \\phi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            285,
+            633,
+            309
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use fully-factorized Gaussian proposal distribution $q _ { \\phi }$ which allows us to perform reparameterization trick and compute KL divergence analytically in order to optimize (7). ",
+        "bbox": [
+            148,
+            318,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3.2 PRIOR IN LATENT SPACE ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            363,
+            377,
+            377
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "During the optimization of objective (7) the parameters $\\mu _ { \\psi }$ and $\\sigma _ { \\psi }$ of the prior distribution of $z$ may tend to infinity, since there is no penalty for large values of those parameters. We usually observe the growth of $\\left. z \\right. _ { 2 }$ during training, though it is slow enough. To prevent potential numerical instabilities, we put a Normal-Gamma prior on the parameters of the prior distribution to prevent the divergence. Formally, we redefine $p _ { \\psi } ( z | x _ { 1 - b } , b )$ as follows: ",
+        "bbox": [
+            147,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/42bd1f2e7f266115ecbfdf42ffc85525498becdfe4a210dad151bb0713eff63a.jpg",
+        "text": "$$\np _ { \\psi } ( z , \\mu _ { \\psi } , \\sigma _ { \\psi } | x _ { 1 - b } , b ) = \\mathcal { N } ( z | \\mu _ { \\psi } , \\sigma _ { \\psi } ^ { 2 } ) \\mathcal { N } ( \\mu _ { \\psi } | 0 , \\sigma _ { \\mu } ) \\mathrm { G a m m a } ( \\sigma _ { \\psi } | 2 , \\sigma _ { \\sigma } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            468,
+            735,
+            488
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As a result, the regularizers $- \\frac { \\mu _ { \\psi } ^ { 2 } } { 2 \\sigma _ { \\mu } ^ { 2 } }$ and $\\sigma _ { \\sigma } ( \\log ( \\sigma _ { \\psi } ) - \\sigma _ { \\psi } )$ are added to the model log-likelihood. Hyperparameter $\\sigma _ { \\mu }$ is chosen to be large $( 1 0 ^ { 4 } )$ and $\\sigma _ { \\sigma }$ is taken to be a small positive number $( 1 0 ^ { - 4 } )$ . This distribution is close to uniform near zero, so it doesn’t affect the learning process significantly. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3.3 MISSING FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            568,
+            341,
+            583
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The optimization objective (7) requires all features of each object at the training stage: some of the features will be observed variables at the input of the model and other will be unobserved features used to evaluate the model. Nevertheless, in some problem settings the training data contains missing features too. We propose the following slight modification of the problem (7) in order to cover such problems as well. ",
+        "bbox": [
+            148,
+            593,
+            851,
+            650
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The missing values cannot be observed so $x _ { i } = \\omega \\Rightarrow b _ { i } = 1$ , where $\\omega$ describes the missing value in the data. In order to meet this requirement, we redefine mask distribution as conditioned on $x$ : $p ( b )$ turns into $p ( b | x )$ in (4) and (7). In the reconstruction loss (5) we simply omit the missing features, i. e. marginalize them out: ",
+        "bbox": [
+            147,
+            656,
+            851,
+            712
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd9e44e756fd89b81c9ee7ee19237edf2b2753f3ffd55278ac0c898fa8a3729d.jpg",
+        "text": "$$\n\\log p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b ) = \\sum _ { \\substack { i : b _ { i } = 1 , x _ { i } \\neq \\omega } } \\log p _ { \\theta } ( x _ { i } | z , x _ { 1 - b } , b )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            709,
+            684,
+            746
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proposal network must be able to determine which features came from real object and which are just missing. So we use additional missing features mask which is fed to proposal network together with unobserved features mask $b$ and object $x$ . ",
+        "bbox": [
+            148,
+            758,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proposed modifications are evaluated in section 5.1. ",
+        "bbox": [
+            148,
+            808,
+            516,
+            823
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a4721c4c0839da51aa1143ab97f04e2031a1758359b3d6ea12919686c3c0741.jpg",
+        "table_caption": [
+            "Table 1: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations. Less is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method/Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Zoo</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.964± 0.007</td><td>1.01 ± 0.01</td><td>0.334± 0.002</td><td>0.19±0.03</td><td>0.422± 0.006</td></tr><tr><td>MissForest</td><td>0.878 ± 0.009</td><td>1.02 ± 0.06</td><td>0.249 ± 0.006</td><td>0.16 ±0.02</td><td>0.422 ± 0.009</td></tr><tr><td>GAIN</td><td>0.97 ± 0.02</td><td>0.99 ± 0.03</td><td>0.271 ± 0.003</td><td>0.20± 0.02</td><td>0.427 ± 0.010</td></tr><tr><td>VAEAC</td><td>0.850 ± 0.007</td><td>0.94 ± 0.01</td><td>0.244 ± 0.002</td><td>0.16 ± 0.02</td><td>0.394± 0.006</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            247,
+            300,
+            263
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we validate the performance of VAEAC using several real-world datasets. In the first set of experiments we evaluate VAEAC missing features imputation performance using various UCI datasets (Lichman, 2013). We compare imputations from our model with imputations from such classical methods as MICE (Buuren & Groothuis-Oudshoorn, 2010) and MissForest (Stekhoven & Buhlmann, 2011) and recently ¨ proposed GANs-based method GAIN (Yoon et al., 2018). In the second set of experiments we use VAEAC to solve image inpainting problem. We show inpainitngs generated by VAEAC and compare our model with models from papers Pathak et al. (2016), Yeh et al. (2017) and Li et al. (2017) in terms of peak signal-to-noise ratio (PSNR) of obtained inpaintings on CelebA dataset (Liu et al., 2015) . And finally, we evaluate VAEAC against the competing method called Universal Marginalizer (Douglas et al., 2017). Additional experiments can be found in appendices C and D. The code is available at https://github.com/tigvarts/ vaeac. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 MISSING FEATURES IMPUTATION ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            463,
+            418,
+            478
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The datasets with missing features are widespread. Consider a dataset with $D$ -dimensional objects $x$ where each feature may be missing (which we denote by $x _ { i } ~ = ~ \\omega$ ) and their target values $y$ . The majority of discriminative methods do not support missing values in the objects. The procedure of filling in the missing features values is called missing features imputation. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we evaluate the quality of imputations produced by VAEAC. For evaluation we use datasets from UCI repository (Lichman, 2013). Before training we drop randomly $50 \\%$ of values both in train and test set. After that we impute missing features using MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest (Stekhoven & Buhlmann, 2011), GAIN (Yoon et al., 2018) and VAEAC trained on the observed data. The ¨ details of GAIN implementation are described in appendix A.4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our model learns the distribution of the imputations, so it is able to sample from this distribution. We replace each object with missing features by $n = 1 0$ objects with sampled imputations, so the size of the dataset increases by $n$ times. This procedure is called missing features multiple imputation. MICE and GAIN are also capable of multiple imputation (we use $n = 1 0$ for them in experiments as well), but MissForest is not. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For more details about the experimental setup see appendices A.1, A.2, and A.4. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In table 1 we report NRMSE (i.e. RMSE normalized by the standard deviation of each feature and then averaged over all features) of imputations for continuous datasets and proportion of falsely classified (PFC) for categorical ones. For multiple imputation methods we average imputations of continuous variables and take most frequent imputation for categorical ones for each object. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also learn linear or logistic regression and report the regression or classification performance after applying imputations of different methods in table 2. For multiple imputation methods we average predictions for continuous targets and take most frequent prediction for categorical ones for each object in test set. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f713f50af0782fe9152224004ad7b2409e899b7731c683b2a4bdd8d12d03d1da.jpg",
+        "table_caption": [
+            "Table 2: R2-score (for continuous targets) or accuracy (for categorical ones) of post-imputation regression or classification. Higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method /Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Z00</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.13±0.02</td><td>0.41 ±0.02</td><td>0.92± 0.01</td><td>0.78± 0.05</td><td>0.75 ±0.02</td></tr><tr><td>MissForest</td><td>0.17 ± 0.01</td><td>0.42 ± 0.02</td><td>0.972 ± 0.003</td><td>0.71 ± 0.07</td><td>0.73 ± 0.02</td></tr><tr><td>GAIN</td><td>0.11 ± 0.01</td><td>0.39 ± 0.06</td><td>0.969 ± 0.005</td><td>0.67 ± 0.06</td><td>0.74 ± 0.03</td></tr><tr><td>VAEAC</td><td>0.17 ± 0.01</td><td>0.43 ± 0.01</td><td>0.983 ± 0.002</td><td>0.8 ± 0.1</td><td>0.74 ± 0.02</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "As can be seen from the tables 1 and 2, VAEAC can learn joint data distribution and use it for missing feature imputation. The imputations are competitive with current state of the art imputation methods in terms of RMSE, PFC, post-imputation regression R2-score and classification accuracy. Nevertheless, we don’t claim that our method is state of the art in missing features imputation; for some datasets MICE or MissForest outperform it. The additional experiments can be found in appendix D.2. ",
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+    {
+        "type": "text",
+        "text": "5.2 IMAGE INPAINTING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The image inpainting problem has a number of different formulations. The formulation of our interest is as follows: some of the pixels of an image are unobserved and we want to restore them in a natural way. Unlike the majority of papers, we want to restore not just one most probable inpainting, but the distribution over all possible inpaintings from which we can sample. This distribution is extremely multi-modal because often there is a lot of different possible ways to inpaint the image. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Unlike the previous subsection, here we have uncorrupted images without missing features in the training set, so $p ( b | x ) = p ( b )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As we show in section 2, state of the art results use different adversarial losses to achieve more sharp and realistic samples. VAEAC can be adapted to the image inpainting problem by using a combination of those adversarial losses as a part of reconstruction loss $p _ { \\theta } ( x _ { b } | \\boldsymbol { z } , x _ { 1 - b } , b )$ . Nevertheless, such construction is out of scope for this research, so we leave it for the future work. In the current work we show that the model can generate both diverse and realistic inpaintings. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In figures 1, 2, 3 and 4 we visualize image inpaintings produced by VAEAC on binarized MNIST (LeCun et al., 1998), Omniglot (Lake et al., 2015) and CelebA (Liu et al., 2015). The details of learning procedure and description of datasets are available in appendixes A.1 and A.3. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the most modern inpainting papers don’t consider the diverse inpainting problem, where the goal is to build diverse image inpaintings, so there is no straightforward way to compare with these models. Nevertheless, we compute peak signal-to-noise ratio (PSNR) for one random inpainting from VAEAC and the best PSNR among 10 random inpaintings from VAEAC. One inpainting might not be similar to the original image, so we also measure how good the inpainting which is most similar to the original image reconstructs it. We compare these two metrics computed for certain masks with the PSNRs for the same masks on CelebA from papers Yeh et al. (2017) and Li et al. (2017). The results are available in tables 3 and 4. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We observe that for the majority of proposed masks our model outperforms the competing methods in terms of PSNR even with one sample, and for the rest (where the inpaintings are significantly diverse) the best PSNR over 10 inpaintings is larger than the same PSNR of the competing models. Even if PSNR does not reflect completely the visual quality of images and tends to encourage blurry VAE samples instead of realistic GANs samples, the results show that VAEAC is able to solve inpainting problem comparably to the state of the art methods. The disadvantage of VAEAC compared to Yeh et al. (2017) and Li et al. (2017) (but not Pathak et al. (2016)) is that it needs the distribution over masks at the training stage to be similar to the distribution over them at the test stage. However, it is not a very strict limitation for the practical usage. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5026e89f974837461ecf99590c5cc0d8aa5c4f5fb91a1c67e40c826d645b2e9c.jpg",
+        "table_caption": [
+            "Table 3: PSNR of inpaintings for different masks for Context Encoder (Pathak et al., 2016), model from “Semantic Image Inpainting with Deep Generative Models” (Yeh et al., 2017) and VAEAC. Higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method/Masks</td><td>Center</td><td>Pattern</td><td>Random</td><td>Half</td></tr><tr><td>Context Encoder 1</td><td>21.3</td><td>19.2</td><td>20.6</td><td>15.5</td></tr><tr><td>SIIDGM 1</td><td>19.4</td><td>17.4</td><td>22.8</td><td>13.7</td></tr><tr><td>VAEAC, 1 sample</td><td>22.1</td><td>21.4</td><td>29.3</td><td>14.9</td></tr><tr><td>VAEAC,10 samples</td><td>23.7</td><td>23.3</td><td>29.3</td><td>17.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/651996553ca128a2271022b3f6a402952fe4db1b266c3fc41a688b8cbaf6c735.jpg",
+        "table_caption": [
+            "Table 4: PSNR of inpaintings for different masks for Context Encoder (Pathak et al., 2016), model from “Generative Face Completion” (Li et al., 2017) and VAEAC. Higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method/Masks</td><td>01</td><td>02</td><td>03</td><td>04</td><td>05</td><td>06</td></tr><tr><td>Context Encoder2</td><td>18.6</td><td>18.4</td><td>17.9</td><td>19.0</td><td>19.1</td><td>19.3</td></tr><tr><td>GFC ²</td><td>20.0</td><td>19.8</td><td>18.8</td><td>19.7</td><td>19.5</td><td>20.2</td></tr><tr><td>VAEAC,1 sample</td><td>20.8</td><td>21.0</td><td>19.5</td><td>20.3</td><td>20.3</td><td>21.0</td></tr><tr><td>VAEAC,10 samples</td><td>22.0</td><td>22.2</td><td>20.8</td><td>21.7</td><td>21.8</td><td>22.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.3 UNIVERSAL MARGINALIZER ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Universal Marginalizer (Douglas et al., 2017) (UM) is a model which uses a single neural network to estimate the marginal distributions over the unobserved features. So it optimizes the following objective: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acbf51b909c72b7e6a56e480067a9402f1912d3b957be1fdb0f7435430250ce2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\mathbb { E } _ { x \\sim p _ { d } ( x ) } \\mathbb { E } _ { b \\sim p ( b ) } \\sum _ { i = 1 } ^ { D } b _ { i } \\log p _ { \\theta } \\big ( x _ { i } | x _ { 1 - b } , b \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For given mask $b$ we fix a permutation of its unobserved components: $( i _ { 1 } , i _ { 2 } , \\dots , i _ { | b | } )$ , where $| b |$ is a number of unobserved components. Using the learned model and the permutation we can generate objects from joint distribution and estimate their probability using chain rule. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/29a588d63f7f9b56a9a4a873e8bb8c472a9609e06bfe1de20bf3ead78d6521b2.jpg",
+        "text": "$$\n\\log p _ { \\theta } ( x _ { b } | x _ { 1 - b } , b ) = \\sum _ { j = 1 } ^ { | b | } \\log p _ { \\theta } ( x _ { i _ { j } } | x _ { 1 - ( b - \\sum _ { k = 1 } ^ { j - 1 } e _ { i _ { k } } ) } , b - \\sum _ { k = 1 } ^ { j - 1 } e _ { i _ { k } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For example, $p _ { \\theta } ( x _ { 1 } , x _ { 4 } , x _ { 5 } | x _ { 2 } , x _ { 3 } ) = p _ { \\theta } ( x _ { 4 } | x _ { 2 } , x _ { 3 } ) p _ { \\theta } ( x _ { 1 } | x _ { 2 } , x _ { 3 } , x _ { 4 } ) p _ { \\theta } ( x _ { 5 } | x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } ) .$ ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Conditional sampling or conditional likelihood estimation for one object requires $| b |$ requests to UM to compute $p _ { \\theta } ( x _ { i } | x _ { 1 - b } , b )$ . Each request is a forward pass through the neural network. In the case of conditional sampling those requests even cannot be paralleled because the input of the next request contains the output of the previous one. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We propose a slight modification of the original UM training procedure which allows learning UM efficiently for any kind of masks including those considered in this paper. The details of the modification are described in appendix B.3. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d8e3304c44f6641f3fcf46844d8ec076bf559af92021199814e73a3e8e86af57.jpg",
+        "image_caption": [
+            "Figure 1: MNIST inpaintings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/696a7457eff5fb7c33faf339e1df45fda1cdfba4264c7adccc10fee5a551d327.jpg",
+        "image_caption": [
+            "Figure 2: Omniglot inpaintings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/22952789d3ea86e1798ed0fcb93d6b5ec6a1f7740b49980530ccffa387686d74.jpg",
+        "image_caption": [
+            "Figure 3: CelebA inpaintings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a5cbe58c9eb61a7543d31d330e9a0fdd262a2099e64f42864b8627f3f719b25.jpg",
+        "image_caption": [
+            "Figure 4: CelebA inpaintings with masks from (Yeh et al., 2017). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Left: input. The gray pixels are unobserved. Middle: samples from VAEAC. Right: ground truth. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/cbb12bac85d7349bfd6184e4aa519378b2ab3130d2d4cea064e8fd8f042f3272.jpg",
+        "table_caption": [
+            "Table 5: VAEAC and UM comparison on MNIST. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>VAEAC</td><td>UM</td></tr><tr><td>Negative log-likelihood</td><td>61</td><td>41</td></tr><tr><td>Training time (30 epochs)</td><td>5min 47s</td><td>3min 14s</td></tr><tr><td>Test time (1OO samples generation)</td><td>0.7ms</td><td>1s</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The results of using this modification of UM are provided in table 5. We can say that the relation between VAEAC and UM is similar to the relation between VAE and PixelCNN. The second one is much slower at the testing stage, but it easily takes into account local dependencies in data while the first one is faster but assumes conditional independence of the outputs. Nevertheless, there are a number of cases where UM cannot learn the distribution well while VAEAC can. For example, when the data is real-valued and marginal distributions have many local optima, there is no straightforward parametrization which allows UM to approximate them, and, therefore also the conditioned joint distribution. An example of such distribution and more illustrations for comparison of VAEAC and UM are available in appendix D.5. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            363,
+            292,
+            380
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper we consider the problem of simultaneous learning of all conditional distributions for a vector. This problem has a number of different special cases with practical applications. We propose neural network based probabilistic model for distribution conditioning learning with Gaussian latent variables. This model is scalable and efficient in inference and learning. We propose several tricks to improve optimization and give recommendations about hyperparameters choice. The model is successfully applied to feature imputation and inpainting tasks. The experimental results show that the model is competitive with state of the art methods for both missing features imputation and image inpainting problems. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "type": "text",
+        "text": "Jinsung Yoon, James Jordon, and Mihaela van der Schaar. GAIN: Missing data imputation using generative adversarial nets. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 5689–5698, Stockholmsmssan, Stockholm Sweden, 10–15 Jul 2018. PMLR. URL http://proceedings.mlr. press/v80/yoon18a.html. ",
+        "bbox": [
+            148,
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            638,
+            393,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 NEURAL NETWORK ARCHITECTURES ",
+        "bbox": [
+            150,
+            670,
+            452,
+            684
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In all experiments we use optimization method Adam (Kingma & Ba, 2014), skip-connections between prior network and generative network inspired by (Mao et al., 2016), (Sønderby et al., 2016) and (Ronneberger et al., 2015), and convolutional neural networks based on ResNet blocks (He et al., 2016). ",
+        "bbox": [
+            148,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Without skip-connections all information for decoder goes through the latent variables. In image inpainting we found skip-connections very useful in both terms of log-likelihood improvement and the image realism, because latent variables are responsible for the global information only while the local information passes through skip-connections. Therefore the border between image and inpainting becomes less conspicuous. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The main idea of neural networks architecture is reflected in figure 5. ",
+        "bbox": [
+            147,
+            808,
+            598,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/5b67d91960752eb18692b5b61ef536a092947e8d387abb82437625c2d41f393a.jpg",
+        "image_caption": [
+            "Figure 5: Neural network architecture for inpainting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            161,
+            137,
+            833,
+            352
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The number of hidden layers, their widths and structure may be different. ",
+        "bbox": [
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+            426,
+            627,
+            441
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The neural networks we used for image inpainting have He-Uniform initialization of convolutional ResNet blocks, and the skip-connections are implemented using concatenation, not addition. The proposal network structure is exactly the same as the prior network except skip-connections. ",
+        "bbox": [
+            148,
+            448,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Also one could use much simpler fully-connected networks with one hidden layer as a proposal, prior and generative networks in VAEAC and still obtain nice inpaintings on MNIST. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 MISSING FEATURES IMPUTATION ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            559,
+            421,
+            573
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We split the dataset into train and test set with size ratio 3:1. Before training we drop randomly $50 \\%$ of values both in train and test set. We repeat each experiment 5 times with different train-test splits and dropped features and then average results and compute their standard deviation. ",
+        "bbox": [
+            147,
+            590,
+            856,
+            633
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As we show in appendix B.2, the better results can be achieved when the model learns the concatenation of objects features $x$ and targets $y$ . So we treat $y$ as an additional feature that is always unobserved during the testing time. ",
+        "bbox": [
+            147,
+            640,
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+            683
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To train our model we use distribution $p ( b _ { i } | x )$ in which $p ( b _ { i } | x _ { i } = \\omega ) = 1$ and $p ( b _ { i } | x ) = 0 . 2$ otherwise. Also for VAEAC trainig we normalize real-valued features, fix $\\sigma _ { \\theta } = 1$ in the generative model of VAEAC in order to optimize RMSE, and use $2 5 \\%$ of training data as validation set to select the best model among all epochs of training. ",
+        "bbox": [
+            148,
+            689,
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+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the test set, the classifier or regressor is applied to each of the $n$ imputed objects and the predictions are combined. For regression problems we report R2-score of combined predictions, so we use averaging as a combination method. For classification problem we report accuracy, and therefore choose the mode. We consider the workflow where the imputed values of $y$ are not fed to the classifier or regressor to make a fair comparison of feature imputation quality. ",
+        "bbox": [
+            148,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/04d4687dcf9699d512eac4a44124a37465506a687d19b17f4d1f42a5bbb655e0.jpg",
+        "table_caption": [
+            "Table 6: Generative Face Completion (Li et al., 2017) masks. Image size is 128x128. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Mask</td><td>Meaning</td><td>X1</td><td>x2</td><td>y1</td><td>y2</td></tr><tr><td>01</td><td>Left half of the face</td><td>33</td><td>70</td><td>52</td><td>115</td></tr><tr><td>02</td><td>Right half of the face</td><td>57</td><td>70</td><td>95</td><td>115</td></tr><tr><td>03</td><td>Two eyes</td><td>29</td><td>98</td><td>52</td><td>73</td></tr><tr><td>04</td><td>Left eye</td><td>29</td><td>66</td><td>52</td><td>73</td></tr><tr><td>05</td><td>Right eye</td><td>61</td><td>99</td><td>52</td><td>73</td></tr><tr><td>06</td><td>Lower half of the face</td><td>40</td><td>87</td><td>86</td><td>123</td></tr></table>",
+        "bbox": [
+            312,
+            142,
+            686,
+            244
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "NRMSE or PFC for dataset is computed as an average of NRMSE or PFC of all features of this dataset. NRMSE of a feature is just RMSE of imputations divided by the standard deviation of this feature. PFC of a feature is a proportion of imputations which are incorrect. ",
+        "bbox": [
+            148,
+            270,
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+            313
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 IMAGE INPAINTING DATASETS AND MASKS ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            329,
+            493,
+            344
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "MNIST is a dataset of 60000 train and 10000 test grayscale images of digits from 0 to 9 of size $2 8 \\mathbf { x } 2 8$ . We binarize all images in the dataset. For MNIST we consider Bernoulli log-likelihood as the reconstruction loss: $\\begin{array} { r } { \\log p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b ) = \\sum _ { i : b _ { i } = 1 } \\log \\mathrm { B e r n o u l l i } ( x _ { i } | p _ { \\theta , i } ( z , x _ { 1 - b } , b ) ) } \\end{array}$ where $p _ { \\theta , i } ( z , x _ { 1 - b } , b )$ is an output of the generative neural network. We use 16 latent variables. In the mask for this dataset the observed pixels form a three pixels wide horizontal line which position is distributed uniformly. ",
+        "bbox": [
+            147,
+            356,
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+            426
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Omniglot is a dataset of 19280 train and 13180 test black-and-white images of different alphabets symbols of size $1 0 5 \\mathrm { x } 1 0 5$ . As in previous section, the brightness of each pixel is treated as a Bernoulli probability of it to be 1. The mask we use is a random rectangular which is described below. We use 64 latent variables. We train model for 50 epochs and choose best model according to IWAE log-likelihood estimation on the validation set after each epoch. ",
+        "bbox": [
+            147,
+            441,
+            849,
+            512
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "CelebA is a dataset of 162770 train, 19867 validation and 19962 test color images of faces of celebrities of size $1 7 8 \\mathrm { x } 2 1 8$ . Before learning we normalize the channels in dataset. We use logarithm of fully-factorized Gaussian distribution as reconstruction loss. The mask we use is a random rectangular which is describe below. We use 32 latent variables. ",
+        "bbox": [
+            148,
+            527,
+            849,
+            584
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Rectangular mask is the common shape of unobserved region in image inpainting. We use such mask for Omniglot and Celeba. We sample the corner points of rectangles uniprobably on the image, but reject those rectangles which area is less than a quarter of the image area. ",
+        "bbox": [
+            148,
+            599,
+            851,
+            642
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Li et al. (2017) six different masks O1–O6 are used on the testing stage. We reconstruct the positions of masks from the illustrations in the paper and give their coordinates in table 6. The visualizations of the masks are available in figure 10. ",
+        "bbox": [
+            148,
+            659,
+            849,
+            700
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "At the training stage we used a rectangle mask with uniprobable random corners. We reject masks with width or height less than 16pt. We use 64 latent variables and take the best model over 50 epochs based on the validation IWAE log-likelihood estimation. We can obtain slightly higher PSNR values than reported in table 4 if use only masks O1–O6 at the training stage. ",
+        "bbox": [
+            148,
+            707,
+            849,
+            763
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Yeh et al. (2017) four types of masks are used. Center mask is just an unobserved $3 2 \\mathrm { x } 3 2 $ square in the center of 64x64 image. Half mask mean that one of upper, lower, left or right half of the image is unobserved. All these types of a half are equiprobable. Random mask means that we use pixelwise-independent Bernoulli distribution with probability 0.8 to form a mask of unobserved pixels. Pattern mask is proposed in Pathak et al. (2016). As we deduced from the code 3, the generation process is follows: firstly we generate $6 0 0 \\times 6 0 0$ one-channel image with uniform distribution over pixels, then bicubically interpolate it to image of size $1 0 0 0 0 \\mathrm { x } 1 0 0 0 0$ , and then apply Heaviside step function $H ( x - 0 . 2 5 )$ (i. e. all points with value less than 0.25 are considered as unobserved). To sample a mask we sample a random position in this $1 0 0 0 0 \\mathrm { x } 1 0 0 0 0$ binary image and crop $6 4 \\mathrm { x } 6 4$ mask. If less than $20 \\%$ or more than $30 \\%$ of pixel are unobserved, than the mask is rejected and the position is sampled again. In comparison with this paper in section 5.2 we use the same distribution over masks at training and testing stages. We use VAEAC with 64 latent variables and take the best model over 50 epochs based on the validation IWAE log-likelihood estimation. ",
+        "bbox": [
+            148,
+            780,
+            851,
+            823
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            119,
+            851,
+            246
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 GAIN IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            271,
+            431,
+            285
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For missing feature imputation we reimplemented GAIN in PyTorch based on the paper (Yoon et al., 2018) and the available TensorFlow source code for image inpainting 4. ",
+        "bbox": [
+            147,
+            300,
+            844,
+            330
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For categorical features we use one-hot encoding. We observe in experiments that it works better in terms of NRMSE and PFC than processing categorical features in GAIN as continuous ones and then rounding them to the nearest category. ",
+        "bbox": [
+            148,
+            335,
+            851,
+            378
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For categorical features we also use reconstruction loss $\\begin{array} { r } { L _ { M } ( x _ { i } , x _ { i } ^ { \\prime } ) = - \\frac { 1 } { | X _ { i } | } \\sum _ { j = 1 } ^ { | X _ { i } | } x _ { i , j } \\log ( x _ { i , j } ^ { \\prime } ) } \\end{array}$ $\\left| X _ { i } \\right|$ the number of categories of the $i$ -th feature, and $x _ { i , j }$ is the $j$ -th component of one-hot encoding of the feature $x _ { i }$ . Such $L _ { M }$ enforces equal contribution of each categorical feature into the whole reconstruction loss. ",
+        "bbox": [
+            148,
+            386,
+            849,
+            434
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use one more modification of $L _ { M } ( x , x ^ { \\prime } )$ for binary and categorical features. Cross-entropy loss in $L _ { M }$ penalizes incorrect reconstructions of categorical and binary features much more than incorrect reconstructions for continuous ones. To avoid such imbalance we mixed L2 and cross-entropy reconstruction losses for binary and categorical features with weights 0.8 and 0.2 respectively: ",
+        "bbox": [
+            148,
+            439,
+            848,
+            496
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We observe in experiments that this modification also works better in terms of NRMSE and PFC than the original model. ",
+        "bbox": [
+            151,
+            566,
+            851,
+            595
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use validation set which contains $5 \\%$ of the observed features for the best model selection (hyperparameter is the number of iterations). ",
+        "bbox": [
+            147,
+            602,
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+            630
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the original GAIN paper authors propose to use cross-validation for hyper-parameter $\\alpha \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 1 0 \\}$ . We observe that using $\\alpha ~ = ~ 1 0$ and a hint $h \\ = \\ b \\circ \\ m \\ + \\ 0 . 5 ( 1 \\ - \\ b )$ where vector $b$ is sampled from Bernoulli distribution with $p = 0 . 0 1$ provides better results in terms of NRMSE and PFC than the original model with every $\\alpha \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 1 0 \\}$ . Such hint distribution makes model theoretically inconsistent but works well in practice (see table 7). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 7 shows that our modifications provide consistently not worse or even better imputations than the original GAIN (in terms of NRMSE and PFC, on the considered datasets). So in this paper for the missing feature imputation problem we report the results of our modification of GAIN. ",
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+        "type": "table",
+        "img_path": "images/bd6b6b98454a1eb53c64484cc4462e2c757911704352b0336a4bd11fa6d9a4c0.jpg",
+        "table_caption": [
+            "Table 7: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations for different GAIN modifications. Less is better. “Our modification” includes the reconstruction loss $L _ { M } ^ { \\prime }$ (12), Bernoulli distribution over $b$ in the hint generation procedure, and fixed $\\alpha = 1 0$ . Other columns refers original GAIN without these modifications and with different values of $\\alpha$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Our modification</td><td>α=10</td><td>α=2</td><td>α=1</td><td>α= 0.5</td><td>α=0.1</td></tr><tr><td>Boston</td><td>0.78±0.03</td><td>0.87±0.02</td><td>1.0 ± 0.1</td><td>1.0 ± 0.1</td><td>1.02 ± 0.05</td><td>1.6±0.2</td></tr><tr><td>Breast</td><td>0.67 ± 0.01</td><td>0.80±0.05</td><td>1.00 ± 0.05</td><td>1.10 ± 0.07</td><td>1.19 ± 0.05</td><td>1.52 ± 0.06</td></tr><tr><td>Concrete</td><td>0.96 ± 0.01</td><td>0.98 ± 0.02</td><td>1.02 ± 0.02</td><td>1.13 ± 0.06</td><td>1.17 ± 0.04</td><td>1.3 ± 0.1</td></tr><tr><td>Diabetes</td><td>0.911 ± 0.009</td><td>0.93 ±0.03</td><td>1.05 ± 0.04</td><td>1.07 ± 0.07</td><td>1.21 ± 0.07</td><td>1.6 ± 0.1</td></tr><tr><td>Digits</td><td>0.79 ± 0.02</td><td>0.88 ± 0.01</td><td>1.05 ± 0.02</td><td>1.13 ± 0.02</td><td>1.24 ± 0.08</td><td>1.4± 0.2</td></tr><tr><td>Glass</td><td>1.06 ± 0.05</td><td>1.04 ± 0.05</td><td>1.19 ± 0.06</td><td>1.4 ± 0.2</td><td>1.6 ± 0.1</td><td>1.81 ± 0.10</td></tr><tr><td>Iris</td><td>0.72 ±0.04</td><td>0.73±0.06</td><td>0.83 ±0.08</td><td>0.97 ± 0.09</td><td>1.2 ± 0.2</td><td>1.3±0.2</td></tr><tr><td>Mushroom</td><td>0.271 ± 0.003</td><td>0.404 ± 0.004</td><td>0.52 ± 0.05</td><td>0.55 ± 0.01</td><td>0.56 ± 0.03</td><td>0.64± 0.06</td></tr><tr><td>Orthopedic</td><td>0.91 ± 0.03</td><td>0.91 ±0.08</td><td>1.1 ± 0.1</td><td>1.2 ± 0.1</td><td>1.34 ± 0.08</td><td>1.6 ± 0.2</td></tr><tr><td>Phishing</td><td>0.427 ± 0.010</td><td>0.52 ±0.02</td><td>0.54±0.02</td><td>0.543 ± 0.010</td><td>0.56 ± 0.01</td><td>0.57 ± 0.04</td></tr><tr><td>WallRobot</td><td>0.907 ± 0.005</td><td>0.924± 0.005</td><td>0.933 ± 0.008</td><td>0.95 ± 0.01</td><td>1.00 ± 0.02</td><td>1.26 ± 0.04</td></tr><tr><td>WhiteWine</td><td>0.97±0.02</td><td>1.02 ± 0.04</td><td>1.2 ± 0.1</td><td>1.3 ± 0.1</td><td>1.6 ± 0.1</td><td>1.86 ± 0.08</td></tr><tr><td>Yeast</td><td>0.99 ±0.03</td><td>1.3±0.2</td><td>1.6 ± 0.1</td><td>1.83 ± 0.09</td><td>1.9 ±0.1</td><td>2.4± 0.4</td></tr><tr><td>Zoo</td><td>0.20 ±0.02</td><td>0.24± 0.05</td><td>0.35 ± 0.06</td><td>0.36 ± 0.03</td><td>0.43 ± 0.04</td><td>0.433 ± 0.004</td></tr></table>",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B THEORY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.1 VAEAC UNIVERSALITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The theoretical guarantees that VAEAC can model arbitrary distribution are based on the same guarantees for Condtitional Variational Autoencoder (CVAE). We prove below that if CVAE can model each of the conditional distributions $p ( x _ { b } | x _ { 1 - b } )$ , then VAEAC can model all of them. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We can imagine $2 ^ { D }$ CVAEs learned each for the certain mask. Because neural networks are universal approximators, VAEAC networks could model the union of CVAE networks, so that VAEAC network performs transformation defined by the same network of the corresponding to the given mask CVAE. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebd07ceb7235754b921ee91da23d856f5d6aa8b4c7359835c4a8bda147c1fb4c.jpg",
+        "text": "$$\np _ { \\psi , V A E A C } ( z | x _ { 1 - b } , b ) = p _ { \\psi , C V A E , 1 - b } ( z | x _ { 1 - b } ) \\forall x , b\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6dbaafed2c4133461a5e6db047a62653e52e4a0973ef592faa2ba8d413f8f487.jpg",
+        "text": "$$\np _ { \\theta , V A E A C } ( x _ { b } | z , x _ { 1 - b } , b ) = p _ { \\theta , C V A E , 1 - b } ( x _ { b } | z , x _ { 1 - b } ) \\forall z , x , b\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "So if CVAE models any distribution $p ( x | y )$ , VAEAC also do. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The guarantees for CVAE in the case of continuous variables are based on the point that every smooth distribution can be approximated with a large enough mixture of Gaussians, which is a special case of CVAE’s generative model. These guarantees can be extended on the case of categorical-continuous variables also. Actually, there are distributions over categorical variables which CVAE with Gaussian prior and proposal distributions cannot learn. Nevertheless, this kind of limitation is not fundamental and is caused by poor proposal distribution family. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.2 WHY VAEAC NEEDS TARGET VALUES FOR MISSING FEATURES IMPUTATION? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Consider a dataset with $D$ -dimensional objects $x$ where each feature may be missing (which we denote by $x _ { i } = \\omega$ ) and their target values $y$ . In this section we show that the better results are achieved when our model learns the concatenation of objects features $x$ and targets $y$ . The example that shows the necessity of it is following. Consider a dataset where $x _ { 1 } = 1$ , $x _ { 2 } \\sim \\mathcal { N } ( \\bar { x } _ { 2 } | y , 1 )$ , $p _ { d } ( y = \\mathrm { { 0 } ) = { { p } ( y = 5 ) = 0 . 5 } }$ . In this case $p _ { d } ( x _ { 2 } | x _ { 1 } = 1 ) = 0 . 5 \\mathcal { N } ( x _ { 2 } | 0 , 1 ) + 0 . 5 \\mathcal { N } ( x _ { 2 } | 5 , 1 )$ . We can see that generating data from $p _ { d } ( x _ { 2 } | x _ { 1 } )$ may only confuse the classifier, because with probability 0.5 it generates $x _ { 2 } \\sim \\bar { \\mathcal { N } } ( 0 , 1 )$ for $y = 5$ and $x _ { 2 } \\sim \\mathcal { N } ( 5 , 1 )$ for $y = 0$ . On the other hand, $p _ { d } ( x _ { 2 } | x _ { 1 } , y ) = \\mathcal { N } ( x _ { 2 } | y , 1 )$ . Filling gaps using $p _ { d } ( x _ { 2 } | x _ { 1 } , y )$ may only improve classifier or regressor by giving it some information from the joint distribution $p _ { d } ( x , y )$ and thus simplifying the dependence to be learned at the training time. So we treat $y$ as an additional feature that is always unobserved during the testing time. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "B.3 UNIVERSAL MARGINALIZER: TRAINING PROCEDURE MODIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The problem authors did not address in the original paper is the relation between the distribution of unobserved components $p ( b )$ at the testing stage and the distribution of masks in the requests to UM ${ \\hat { p } } ( b )$ . The distribution over masks $p ( b )$ induces the distribution ${ \\hat { p } } ( b )$ , and in the most cases $p ( b ) \\neq { \\hat { p } } ( b )$ . The distribution ${ \\hat { p } } ( b )$ also depends on the permutations $( i _ { 1 } , i _ { 2 } , \\dots , i _ { | b | } )$ that we use to generate objects. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We observed in experiments, that UM must be trained using unobserved mask distribution ${ \\hat { p } } ( b )$ . For example, if all masks from $p ( b )$ have a fixed number of unobserved components (e. g., $\\begin{array} { l } { { \\frac { D } { 2 } } } \\end{array}$ ), then UM will never see an example of mask with $\\begin{array} { r } { { 1 , 2 , \\ldots , \\frac { D } { 2 } - 1 } } \\end{array}$ unobserved components, which is necessary to generate a sample conditioned on $\\textstyle { \\frac { D } { 2 } }$ components. That leads to drastically low likelihood estimate for the test set and unrealistic samples. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We developed an easy generative process for ${ \\hat { p } } ( b )$ for arbitrary $p ( b )$ if the permutation of unobserved components $( i _ { 1 } , i _ { 2 } , \\dots , i _ { | b | } )$ is chosen randomly and equiprobably: firstly we generate $b _ { 0 } \\sim p ( b )$ , $u \\sim U [ 0 , 1 ]$ , then $b _ { 1 } \\sim ( \\mathrm { B e r n o u l l i } ( u ) ) ^ { D }$ and $b = b _ { 0 } \\circ b _ { 1 }$ . More complicated generative process exists for a sorted permutation where $i _ { j - 1 } < i _ { j } \\forall j : 2 \\le j \\le | b |$ . ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In experiments we use uniform distribution over the permutations. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C GAUSSIAN STOCHASTIC NEURAL NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Gaussian stochastic neural network (13) and hybrid model (14) are originally proposed in the paper on Conditional VAE (Sohn et al., 2015). The motivation authors mention in the paper is as follows. During training the proposal distribution $q _ { \\phi } ( z | x , y )$ is used to generate the latent variables $z$ , while during the testing stage the prior $p _ { \\psi } ( z | y )$ is used. KL divergence tries to close the gap between two distributions but, according to authors, it is not enough. To overcome the issue authors propose to use a hybrid model (14), a weighted mixture of variational lower bound (3) and a single-sample Monte-Carlo estimation of log-likelihood (13). The model corresponding to the second term is called Gaussian Stochastic Neural Network (13), because it is a feed-forward neural network with a single Gaussian stochastic layer in the middle. Also GSNN is a special case of CVAE where $q _ { \\phi } ( z | x , y ) = p _ { \\psi } ( z | y )$ . ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6eda0f59e7f92ff41e13fc03a857ebaa2e07dbc1dc3cfeb2358c4de8661097dd.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { G S N N } ( x , y ; \\theta , \\psi ) = \\mathbb { E } _ { p _ { \\psi } ( z | y ) } \\log p _ { \\theta } ( x | z , y ) \\qquad } \\\\ { L ( x , y ; \\theta , \\psi , \\phi ) = \\alpha L _ { C V A E } ( x , y ; \\theta , \\psi , \\phi ) + ( 1 - \\alpha ) L _ { G S N N } ( x , y ; \\theta , \\psi ) , \\quad \\alpha \\in [ 0 , 1 ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Authors report that hybrid model and GSNN outperform CVAE in terms of segmentation accuracy on the majority of datasets. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We can also add that this technique seems to soften the “holes problem” (Makhzani et al., 2016). In Makhzani et al. (2016) authors observe that vectors $z$ from prior distribution may be different enough from all vectors $z$ from the proposal distribution at the training stage, so the generator network may be confused at the testing stage. Due to this problem CVAE can have good reconstructions of $y$ given $z \\sim q _ { \\phi } ( z | x , y )$ , while samples of $y$ given $z \\sim p _ { \\psi } ( z | x )$ are not realistic. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The same trick is applicable to our model as well: ",
+        "bbox": [
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+            119,
+            475,
+            136
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c33502d1bb26f985e83664b8c446d3988a4e59347df95247f5e7bef837ca1711.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { G S N N } ( x , b ; \\theta , \\psi ) = \\mathbb { E } _ { p _ { \\psi } ( z | x _ { 1 - b } , b ) } \\log p _ { \\theta } ( x _ { b } | z , x _ { 1 - b } , b ) } \\\\ & { L ( x , b ; \\theta , \\psi , \\phi ) = \\alpha L _ { V A E A C } ( x , b ; \\theta , \\psi , \\phi ) + ( 1 - \\alpha ) L _ { G S N N } ( x , b ; \\theta , \\psi ) , \\quad \\alpha \\in [ 0 , 1 ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            142,
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+            181
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In order to reflect the difference between sampling $z$ from prior and proposal distributions, authors of CVAE use two methods of log-likelihood estimation: ",
+        "bbox": [
+            143,
+            194,
+            856,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/583a13fe422be7a69bc3cb86f05f54ddd22119c286cd1b5e0c02498f08abc599.jpg",
+        "text": "$$\n\\log p _ { \\theta , \\psi } ( x | y ) \\approx \\log \\frac { 1 } { S } \\sum _ { i = 1 } ^ { S } p _ { \\theta } ( x | z _ { i } , y ) , ~ z _ { i } \\sim p _ { \\psi } ( z | y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            231,
+            684,
+            275
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ccb2cb6db773a0184ddddcd940622b1174f4ffcf1db0c6a0ee7b8d59e0ea61b.jpg",
+        "text": "$$\n\\log p _ { \\theta , \\psi } ( x | y ) \\approx \\log \\frac { 1 } { S } \\sum _ { i = 1 } ^ { S } \\frac { p _ { \\theta } ( x | z _ { i } , y ) p _ { \\psi } ( z _ { i } | y ) } { q _ { \\phi } ( z _ { i } | x , y ) } , ~ z _ { i } \\sim q _ { \\phi } ( z | x , y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The first estimator is called Monte-Carlo estimator and the second one is called Importance Sampling estimator (also known as IWAE). They are asymptotically equivalent, but in practice the Monte-Carlo estimator requires much more samples to obtain the same accuracy of estimation. Small $S$ leads to underestimation of the log-likelihood for both Monte-Carlo and Importance Sampling (Burda et al., 2015), but for Monte-Carlo the underestimation is expressed much stronger. ",
+        "bbox": [
+            147,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We perform an additional study of GSNN and hybrid model and show that they have drawbacks when the target distribution $p ( x | y )$ is has multiple different local maximums. ",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "C.1 THEORETICAL STUDY ",
+        "text_level": 1,
+        "bbox": [
+            148,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In this section we show why GSNN cannot learn distributions with several different modes and leads to a blurry image samples. ",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "For the simplicity of the notation we consider hybrid model for a standard VAE: ",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1559493b96d28a0f47430d3d517775906770a2762939c39f8bd304ea066eecd.jpg",
+        "text": "$$\nL ( x ; \\phi , \\psi , \\theta ) = \\alpha \\mathbb { E } _ { z \\sim q _ { \\phi } ( z \\mid x ) } \\log \\frac { p _ { \\theta } ( x \\mid z ) p _ { \\psi } ( z ) } { q _ { \\phi } ( z \\mid x ) } + ( 1 - \\alpha ) \\mathbb { E } _ { z \\sim p _ { \\psi } ( z ) } \\log p _ { \\theta } ( x \\mid z )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The hybrid model (16) for VAEAC can be obtained from (19) by replacing $x$ with $x _ { b }$ and conditioning all distributions on $x _ { 1 - b }$ and $b$ . The validity of the further equations and conclusions remains for VAEAC after this replacement. ",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Consider now a categorical latent variable $z$ which can take one of $K$ values. Let $x$ be a random variable witfor $p _ { d } ( x )$ to be modele some values following true data distribut. So the true distribution has n: d $\\begin{array} { r } { p _ { d } ( x = x _ { i } ) = \\frac { 1 } { K } } \\end{array}$ $i \\in \\{ 1 , 2 , \\ldots , K \\}$ $x _ { 1 } , x _ { 2 } , \\dotsc , x _ { K }$ $K$ able modes. Suppose the generator network $N N _ { \\theta }$ which models mapping from $z$ to some vector of parameters $\\begin{array} { r l r } { v _ { z } } & { { } = } & { \\bar { N } N _ { \\theta } ( z ) } \\end{array}$ . Thus, we define generative distribution as some function of these parameters: $p _ { \\theta } ( x | z ) \\ : = \\ : f ( x , v _ { z } )$ . Therefore, the parameters $\\theta$ are just the set of $v _ { 1 } , v _ { 2 } , \\dotsc , v _ { K }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the simplicity of the model we assume $\\begin{array} { r } { p _ { \\psi } ( z ) = \\frac { 1 } { K } } \\end{array}$ . Taking into account $\\begin{array} { r } { p _ { \\psi } ( z ) = \\frac { 1 } { K } } \\end{array}$ , we obtain optimal $\\begin{array} { r } { q ( z = i | x ) = \\frac { f ( x , v _ { i } ) } { \\sum _ { j = 1 } ^ { K } f ( x , v _ { j } ) } } \\end{array}$ Using (19) and the above formulas for $q _ { \\phi } , p _ { \\psi }$ and $p _ { \\theta }$ we obtain the following optimization problem: ",
+        "bbox": [
+            147,
+            715,
+            851,
+            768
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07b4e4827ba4de27da9506a427292dfd27100f30c89238d1b1a7b66b439dfe73.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { v _ { 1 } , v _ { 2 } , \\ldots , v _ { K } } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\left[ \\alpha \\sum _ { j = 1 } ^ { K } \\frac { f ( x _ { i } , v _ { j } ) } { \\sum _ { k = 1 } ^ { K } f ( x _ { i } , v _ { k } ) } \\log \\frac { f ( x _ { i } , v _ { j } ) \\frac { 1 } { K } } { \\frac { f ( x _ { i } , v _ { j } ) } { \\sum _ { k = 1 } ^ { K } f ( x _ { i } , v _ { k } ) } } + ( 1 - \\alpha ) \\sum _ { j = 1 } ^ { K } \\frac { 1 } { K } \\log f ( x _ { i } , v _ { j } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            776,
+            792,
+            827
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Negative log-likelihood estimation of a hybrid model on the synthetic data. IS- $S$ refers to Importance Sampling log-likelihood estimation with $S$ samples for each object (18). MC- $S$ refers to Monte-Carlo log-likelihood estimation with $S$ samples for each object (17). ",
+        "bbox": [
+            147,
+            117,
+            851,
+            160
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/65cc93c875903136313a61e9ac8afdb60ebc425608483cce53104e3064921176.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>VAEAC weight</td><td>IS-10</td><td>MC-10</td></tr><tr><td>a=1</td><td>0.22</td><td>85</td></tr><tr><td>α = 0.99</td><td>0.35</td><td>11</td></tr><tr><td>α = 0.9</td><td>0.62</td><td>1.7</td></tr></table>",
+        "bbox": [
+            375,
+            170,
+            622,
+            229
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "It is easy to show that (20) is equivalent to the following optimization problem: ",
+        "bbox": [
+            145,
+            252,
+            665,
+            268
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fba50a965f88da607ef696227097c6d0af74dd0f5711b3ed679ed890002f6b06.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { v _ { 1 } , v _ { 2 } , \\ldots , v _ { K } } \\sum _ { i = 1 } ^ { K } \\left[ \\alpha \\log \\frac { \\sum _ { j = 1 } ^ { K } f ( x _ { i } , v _ { j } ) } { K } + ( 1 - \\alpha ) \\sum _ { j = 1 } ^ { K } \\frac { 1 } { K } \\log f ( x _ { i } , v _ { j } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            272,
+            735,
+            323
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "It is clear from (21) that when $\\alpha = 1$ the log-likelihood of the initial model is optimized. On the other hand, when influe $\\alpha = 0$ the optimal point is  generative process, a $v _ { 1 } = v _ { 2 } = \\cdots = v _ { K } = \\operatorname { a r g m a x } _ { v } \\sum _ { i = 1 } ^ { K } \\log f ( x _ { i } , v ) .$ , i. e. mizes $z$ doesn’telihood $z$ $v$   \nestimation of the generative model $f ( x , v )$ for the given dataset of $x$ ’s. For Bernoulli and Gaussian generative distributions $f$ such $v$ is just average of all modes $x _ { 1 } , x _ { 2 } , \\dotsc , x _ { K }$ . That explains why further we observe blurry images when using GSNN model. ",
+        "bbox": [
+            147,
+            333,
+            851,
+            420
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The same conclusion holds for for continuous latent variables instead of categorical. Given $K$ different modes in true data distribution, VAE uses proposal network to separate prior distribution into $K$ components (i. e. regions in the latent space), so that each region corresponds to one mode. On the other hand, in GSNN $z$ is sampled independently on the mode which is to be reconstructed from it, so for each $z$ the generator have to produce parameters suitable for all modes. ",
+        "bbox": [
+            148,
+            426,
+            851,
+            497
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "From this point of view, there is no difference between VAE and VAEAC. If the true conditional distribution has several different modes, then VAEAC can fit them all, while GSNN learns their average. If true conditional distribution has one mode, GSNN and VAEAC are equal, and GSNN may even learn faster because it has less parameters. ",
+        "bbox": [
+            148,
+            503,
+            849,
+            560
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Hybrid model is a trade-off between VAEAC and GSNN: the closer $\\alpha$ to zero, the more blurry and closer to the average is the distribution of the model. The exact dependence of the model distribution on $\\alpha$ can be derived analytically for the simple data distributions or evaluated experimentally. We perform such experimental evaluation in the next sections. ",
+        "bbox": [
+            148,
+            565,
+            849,
+            622
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "C.2 SYNTHETIC DATA ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            638,
+            315,
+            654
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In this section we show that VAEAC is capable of learning a complex multimodal distribution of synthetic   \ndata while GSNN and hybrid model are not. Let $x \\in \\mathbb { R } ^ { \\bar { 2 } }$ and $p ( \\bar { b } _ { 1 } = 1 ) = p ( b _ { 2 } = 1 ) = 0 . 5$ . $p _ { d } ( x ) =$   \n$\\begin{array} { r l } { \\frac { 1 } { 8 } \\sum _ { i = 1 } ^ { 8 } \\mathcal { N } ( x | \\mu _ { i } , \\frac { 1 } { 1 0 } I ) } & { { } } \\end{array}$ where s samp $\\mu _ { i } \\sim \\mathcal N ( \\mu _ { i } | 0 , I )$ . The distribution e use multi-layer p $p ( x )$ is plotted in figure 6. The datasetptron with four ReLU layers of size $p _ { d } ( x )$   \n400-200-100-50, 25-dimensional Gaussian latent variables. ",
+        "bbox": [
+            148,
+            665,
+            851,
+            738
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "For different mixture coefficients $\\alpha$ we visualize samples from the learned distributions $p _ { \\psi , \\theta } ( x _ { 1 } , x _ { 2 } )$ , $p _ { \\psi , \\theta } ( x _ { 1 } | x _ { 2 } )$ , and $p _ { \\psi , \\theta } ( x _ { 2 } | x _ { 1 } )$ . The observed features for the conditional distributions are generated from the marginal distributions $p ( x _ { 2 } )$ and $p ( x _ { 1 } )$ respectively. ",
+        "bbox": [
+            148,
+            744,
+            851,
+            787
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We see in table 8 and in figure 7, that even with very small weight GSNN prevents model from learning distributions with several local optimas. GSNN also increases Monte-Carlo log-likelihood estimation with a few samples and decreases much more precise Importance Sampling log-likelihood estimation. When $\\alpha = 0 . 9$ the whole distribution structure is lost. ",
+        "bbox": [
+            147,
+            794,
+            849,
+            823
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7407e71d027399c997ec577ddb914586b76471425f58bff6404a5cd035d567fc.jpg",
+        "image_caption": [
+            "Figure 6: Probability density function of synthetic data distribution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            217,
+            185,
+            348,
+            267
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/a4bd4e815fbae18fb3e5afb3429dcbb43b07bb985c3a0166b18e1666e16f7080.jpg",
+        "image_caption": [
+            "Figure 7: VAEAC for synthetic data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            539,
+            119,
+            838,
+            347
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/54463305da3093916463d301746c0407e7d8d2daa9a2e48291c407dc4d828109.jpg",
+        "image_caption": [
+            "Figure 8: MNIST inpaintings. Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            429,
+            834,
+            659
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            758,
+            849,
+            787
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "We see that using $\\alpha \\neq 1$ ruins multimodality of the restored distribution, so we highly recommend to use $\\alpha = 1$ or at least $\\alpha \\approx 1$ . ",
+        "bbox": [
+            147,
+            794,
+            846,
+            823
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/a38ab45727edbf8bab4aa92d10957fbb17674c382278cea78615c2779533612a.jpg",
+        "table_caption": [
+            "Table 9: Average negative log-likelihood of inpaintings for 1000 objects. IS- $S$ refers to Importance Sampling log-likelihood estimation with $S$ samples for each object (18). MC- $S$ refers to Monte-Carlo log-likelihood estimation with $S$ samples for each object (17). Naive Bayes is a baseline method which assumes pixels and colors independence. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>MNIST</td><td>Omniglot</td><td>CelebA</td></tr><tr><td>VAEAC IS-102</td><td>61±1</td><td>275±17</td><td>34035 ± 1609</td></tr><tr><td>VAEAC MC-104</td><td>94±4</td><td>1452 ± 109</td><td>41513 ± 2163</td></tr><tr><td>VAEAC MC-102</td><td>156 ±1</td><td>2203 ± 150</td><td>53904 ± 3121</td></tr><tr><td>GSNN MC-104</td><td>141 ±7</td><td>1199 ± 62</td><td>53427 ± 2208</td></tr><tr><td>GSNN MC-10²</td><td>141 ±1</td><td>1200 ± 62</td><td>53486 ± 2210</td></tr><tr><td>Naive Bayes</td><td>205</td><td>2490</td><td>269480</td></tr></table>",
+        "bbox": [
+            289,
+            184,
+            709,
+            287
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/3126337f9f6dd982a9196015b6247be9dab5083bb45c2e9eb0945400c4c56f31.jpg",
+        "image_caption": [
+            "Figure 9: Convergence of VAE and VAEAC on MNIST dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            329,
+            632,
+            479
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "C.3 COMPARISON ON THE IMAGE INPAINTING PROBLEM ",
+        "bbox": [
+            147,
+            563,
+            553,
+            578
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "In figure 8 we can see that the inpaintings produced by GSNN are smooth, blurry and not diverse compared with VAEAC. ",
+        "bbox": [
+            147,
+            594,
+            849,
+            623
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Table 9 shows that VAEAC learns distribution over inpaintings better than GSNN in terms of test loglikelihood. Nevertheless, Monte-Carlo estimations with a small number of samples sometimes are better for GSNN, which means less local modes in the learned distribution and more blurriness in the samples. ",
+        "bbox": [
+            148,
+            630,
+            851,
+            672
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "D ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            708,
+            418,
+            724
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "D.1 CONVERGENCE SPEED ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            747,
+            349,
+            762
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "In figure 9 one can see that VAEAC has similar convergence speed to VAE in terms of iterations on MNIST dataset. In our experiments we observed the same behaviour for other datasets. Each iteration of VAEAC is about 1.5 times slower than VAE due to usage of three networks instead of two. ",
+        "bbox": [
+            148,
+            780,
+            851,
+            821
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "table",
+        "img_path": "images/a28b110ac71efa360491f823a261c429125aef3652390412ba5272709177d2d9.jpg",
+        "table_caption": [
+            "Table 10: NRMSE (for continuous datasets) or PFC (for categorical ones) of imputations. Less is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>MICE</td><td>MissForest</td><td>GAIN</td><td>VAEAC</td><td>GSNN</td><td>NN</td></tr><tr><td>Boston</td><td>0.69 ± 0.02</td><td>0.58±0.02</td><td>0.78 ± 0.03</td><td>0.71 ± 0.02</td><td>0.70 ± 0.01</td><td>0.69 ± 0.01</td></tr><tr><td>Breast</td><td>0.58 ±0.02</td><td>0.515 ± 0.008</td><td>0.67 ±0.01</td><td>0.55±0.02</td><td>0.55 ± 0.02</td><td>0.52 ±0.02</td></tr><tr><td>Concrete</td><td>0.850 ± 0.007</td><td>0.78 ± 0.01</td><td>0.96 ±0.01</td><td>0.84 ±0.02</td><td>0.85 ± 0.01</td><td>2±3</td></tr><tr><td>Diabetes</td><td>0.80 ±0.01</td><td>0.84±0.02</td><td>0.911 ± 0.009</td><td>0.90 ±0.03</td><td>0.91 ± 0.03</td><td>0.90 ±0.02</td></tr><tr><td>Digits</td><td>0.69 ±0.02</td><td>0.61± 0.02</td><td>0.79±0.02</td><td>0.69 ±0.02</td><td>0.69 ± 0.02</td><td>0.67 ±0.02</td></tr><tr><td>Glass</td><td>0.91 ±0.02</td><td>0.83± 0.04</td><td>1.06 ±0.05</td><td>0.91 ±0.04</td><td>0.91 ± 0.05</td><td>0.87 ± 0.04</td></tr><tr><td>Iris</td><td>0.59 ± 0.02</td><td>0.62 ± 0.04</td><td>0.72 ± 0.04</td><td>0.64± 0.04</td><td>0.62 ± 0.04</td><td>0.61± 0.02</td></tr><tr><td>Mushroom</td><td>0.334 ± 0.002</td><td>0.249 ±0.006</td><td>0.271 ±0.003</td><td>0.241 ± 0.002</td><td>0.2412 ± 0.0009</td><td>0.239 ± 0.001</td></tr><tr><td>Orthopedic</td><td>0.76 ±0.02</td><td>0.79±0.03</td><td>0.91±0.03</td><td>0.80±0.03</td><td>0.81 ±0.03</td><td>0.81 ±0.02</td></tr><tr><td>Phishing</td><td>0.422 ± 0.006</td><td>0.422 ±0.009</td><td>0.427 ± 0.010</td><td>0.397 ± 0.010</td><td>0.392 ±0.009</td><td>0.41 ± 0.01</td></tr><tr><td>WallRobot</td><td>0.885 ± 0.003</td><td>0.640 ± 0.003</td><td>0.907 ± 0.005</td><td>0.78 ±0.01</td><td>0.776 ±0.007</td><td>0.757± 0.005</td></tr><tr><td>WhiteWine</td><td>0.964 ± 0.007</td><td>0.878 ±0.009</td><td>0.97 ± 0.02</td><td>0.850 ± 0.005</td><td>0.848 ± 0.007</td><td>0.85 ± 0.01</td></tr><tr><td>Yeast</td><td>0.98±0.02</td><td>1.00 ± 0.02</td><td>0.99 ±0.03</td><td>0.95 ± 0.01</td><td>0.958 ± 0.007</td><td>0.97 ± 0.03</td></tr><tr><td>Zoo</td><td>0.19 ± 0.03</td><td>0.16 ±0.02</td><td>0.20 ±0.02</td><td>0.16±0.02</td><td>0.17 ±0.02</td><td>0.16 ±0.01</td></tr></table>",
+        "bbox": [
+            155,
+            141,
+            848,
+            343
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "table",
+        "img_path": "images/7d3a3cbee5bc4c33e77c2d6020bcaa263208c0156dec7db727d7d8e3202cc1d5.jpg",
+        "table_caption": [
+            "Table 11: R2-score (for continuous targets) or accuracy (for categorical ones) of post-imputation regression or classification. Higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>MICE</td><td>MissForest</td><td>GAIN</td><td>VAEAC</td><td>GSNN</td><td>NN</td></tr><tr><td>Boston</td><td>0.57 ± 0.08</td><td>0.6 ± 0.1</td><td>0.50 ± 0.10</td><td>0.5 ± 0.1</td><td>0.5± 0.1</td><td>0.50±0.09</td></tr><tr><td>Breast</td><td>0.96 ± 0.02</td><td>0.95 ± 0.02</td><td>0.94± 0.01</td><td>0.95 ± 0.02</td><td>0.96 ± 0.02</td><td>0.95 ± 0.02</td></tr><tr><td>Concrete</td><td>0.35 ± 0.05</td><td>0.33 ± 0.04</td><td>0.28±0.06</td><td>0.30 ± 0.08</td><td>0.32 ± 0.05</td><td>0±1</td></tr><tr><td>Diabetes</td><td>0.37 ± 0.06</td><td>0.34±0.06</td><td>0.34±0.03</td><td>0.34± 0.04</td><td>0.33 ±0.04</td><td>0.27±0.06</td></tr><tr><td>Digits</td><td>0.86±0.02</td><td>0.887 ±0.008</td><td>0.83 ±0.03</td><td>0.892 ± 0.010</td><td>0.895 ± 0.010</td><td>0.912 ± 0.010</td></tr><tr><td>Glass</td><td>0.44 ±0.08</td><td>0.53 ±0.05</td><td>0.37±0.05</td><td>0.49 ±0.09</td><td>0.47 ±0.09</td><td>0.48 ±0.09</td></tr><tr><td>Iris</td><td>0.81± 0.02</td><td>0.84±0.02</td><td>0.66 ±0.06</td><td>0.84±0.05</td><td>0.82±0.06</td><td>0.73±0.09</td></tr><tr><td>Mushroom</td><td>0.92 ±0.01</td><td>0.972 ± 0.003</td><td>0.969 ± 0.005</td><td>0.987 ± 0.001</td><td>0.986 ± 0.002</td><td>0.989 ± 0.003</td></tr><tr><td>Orthopedic</td><td>0.71 ± 0.02</td><td>0.72 ± 0.03</td><td>0.60±0.03</td><td>0.71 ±0.02</td><td>0.70± 0.04</td><td>0.61±0.04</td></tr><tr><td>Phishing</td><td>0.75 ± 0.02</td><td>0.73±0.03</td><td>0.74± 0.03</td><td>0.75 ± 0.01</td><td>0.74±0.04</td><td>0.73±0.02</td></tr><tr><td>WallRobot</td><td>0.55 ±0.01</td><td>0.697 ± 0.005</td><td>0.56 ±0.01</td><td>0.62±0.02</td><td>0.62 ± 0.01</td><td>0.64±0.02</td></tr><tr><td>WhiteWine</td><td>0.13 ± 0.02</td><td>0.17 ± 0.01</td><td>0.11± 0.01</td><td>0.18 ±0.02</td><td>0.17 ± 0.01</td><td>0.15 ± 0.03</td></tr><tr><td>Yeast</td><td>0.42 ±0.02</td><td>0.41 ±0.02</td><td>0.39 ±0.06</td><td>0.42 ± 0.01</td><td>0.425 ± 0.010</td><td>0.33±0.03</td></tr><tr><td>Zoo</td><td>0.78 ± 0.06</td><td>0.71 ±0.08</td><td>0.67±0.06</td><td>0.77 ± 0.09</td><td>0.8±0.1</td><td>0.83 ± 0.08</td></tr></table>",
+        "bbox": [
+            151,
+            396,
+            848,
+            601
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "D.2 MISSING FEATURES IMPUTATION ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            627,
+            423,
+            642
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the quality of imputations on different datasets (mostly from UCI (Lichman, 2013)). The evaluation is performed for VAEAC, GSNN (15) and NN (neural network; can be considered as a special case of GSNN where $p _ { \\theta } ( z | x _ { 1 - b } , b )$ is delta-function; produces single imputation). We compare these methods with MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest (Stekhoven & Buhlmann, 2011), and GAIN ¨ (Yoon et al., 2018). ",
+        "bbox": [
+            148,
+            654,
+            851,
+            724
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We see that for some datasets MICE and MissForest outperform VAEAC, GSNN and NN. The reason is that for some datasets random forest is more natural structure than neural network. ",
+        "bbox": [
+            145,
+            731,
+            851,
+            760
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "The results also show that VAEAC, GSNN and NN show similar imputation performance in terms of NRMSE, PFC, post-imputation R2-score and accuracy. Given the result from appendix C we can take this as a weak evidence that the distribution of imputations has only one local maximum for datasets from (Lichman, 2013). ",
+        "bbox": [
+            148,
+            766,
+            851,
+            823
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/21041c7b3c7fc1baa4707c17e3944ec8fc55aadb337ff273f13ec39db52954c7.jpg",
+        "image_caption": [
+            "Figure 10: CelebA inpaintings with masks from (Li et al., 2017). ft: input. The gray pixels are unobserved. Middle: samples from VAEAC. Right: ground truth. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            119,
+            759,
+            458
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "D.3 FACE INPAINTINGS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            537,
+            323,
+            553
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "In figure 10 we provide samples of VAEAC on the CelebA dataset for the masks from (Li et al., 2017). ",
+        "bbox": [
+            150,
+            564,
+            816,
+            579
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "D.4 GAIN FOR IMAGE INPAINTING ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            598,
+            406,
+            613
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "GAIN (Yoon et al., 2018) doesnt use unobserved data during training, which makes it easier to apply to the missing features imputation problem. Nevertheless, it turns into a disadvantage when the fully-observed training data is available but the missingness rate at the testing stage is high. ",
+        "bbox": [
+            148,
+            626,
+            849,
+            667
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "We consider the horizontal line mask for MNIST which is described in appendix A.3. We use the released GAIN code 5 with a different mask generator. The inpaintings from VAEAC which uses the unobserved pixels during training are available in figure 1. The inpaintings from GAIN which ignores unobserved pixels are provided in figure 11. As can be seen in figure 11, GAIN fails to learn conditional distribution for given mask distribution ${ \\dot { p } } ( b )$ . ",
+        "bbox": [
+            148,
+            674,
+            849,
+            744
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Nevertheless, we don’t claim that GAIN is not suitable for image inpainting. As it was shown in the supplementary of (Yoon et al., 2018) and in the corresponding code, GAIN is able to learn conditional distributions when $p ( b )$ is pixel-wise independent Bernoulli distribution with probability 0.5. ",
+        "bbox": [
+            148,
+            751,
+            849,
+            794
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/77978a748fdfece774bbdba6a454e4caa9d264c220caf33b0d01dfee4003f59b.jpg",
+        "image_caption": [
+            "Figure 11: MNIST inpaintings from GAIN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            348,
+            122,
+            650,
+            290
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth. ",
+        "bbox": [
+            166,
+            320,
+            823,
+            335
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/3addba9b5d9e7527dc59c64b67c88b55ee2d758e97d2f9f04dcf482aada3e5b6.jpg",
+        "image_caption": [
+            "Figure 12: MNIST inpaintings. Left: input. The gray pixels are unobserved. Middle: samples from the model. Right: ground truth. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            353,
+            834,
+            580
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "D.5 UNIVERSAL MARGINALIZER: ILLUSTRATIONS ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            654,
+            511,
+            669
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "In figure 12 we provide samples of Universal Marginalizer (UM) and VAEAC for the same inputs. ",
+        "bbox": [
+            148,
+            679,
+            792,
+            695
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Consider the case when UM marginal distributions are parametrized with Gaussians. The most simple example of a distribution, which UM cannot learn but VAEAC can, is given in figure 13. ",
+        "bbox": [
+            143,
+            700,
+            852,
+            731
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/34b2efedea65ec8dd1cd6662835f9b38b5a648676328c4f707b65cec76fc2de2.jpg",
+        "image_caption": [
+            "Figure 13: Distribution learning: VAEAC vs UM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            158,
+            375,
+            839,
+            530
+        ],
+        "page_idx": 24
+    }
+]
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diff --git a/parse/train/SyxtJh0qYm/SyxtJh0qYm_middle.json b/parse/train/SyxtJh0qYm/SyxtJh0qYm_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..678da957d3fcc10d21511ab39dba5c7bc86ef903
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+                                    "score": 1.0,
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+                                    "content": "approximates marginals of unobserved features conditioned on observable values. A related idea of an",
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+                                    "score": 1.0,
+                                    "content": "autoregressive model of joint probability was previously proposed in Germain et al. (2015) and Uria et al.",
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+                                    "content": "training data is available but the missingness rate at the testing stage is high. For example, in inpainting",
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+                                            "html": "<table><tr><td>Method/Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Zoo</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.964± 0.007</td><td>1.01 ± 0.01</td><td>0.334± 0.002</td><td>0.19±0.03</td><td>0.422± 0.006</td></tr><tr><td>MissForest</td><td>0.878 ± 0.009</td><td>1.02 ± 0.06</td><td>0.249 ± 0.006</td><td>0.16 ±0.02</td><td>0.422 ± 0.009</td></tr><tr><td>GAIN</td><td>0.97 ± 0.02</td><td>0.99 ± 0.03</td><td>0.271 ± 0.003</td><td>0.20± 0.02</td><td>0.427 ± 0.010</td></tr><tr><td>VAEAC</td><td>0.850 ± 0.007</td><td>0.94 ± 0.01</td><td>0.244 ± 0.002</td><td>0.16 ± 0.02</td><td>0.394± 0.006</td></tr></table>",
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+                                    "content": "(Lichman, 2013). We compare imputations from our model with imputations from such classical methods as",
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+                                    "content": "proposed GANs-based method GAIN (Yoon et al., 2018). In the second set of experiments we use VAEAC",
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+                                    "content": "to solve image inpainting problem. We show inpainitngs generated by VAEAC and compare our model with",
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+                                    "content": "models from papers Pathak et al. (2016), Yeh et al. (2017) and Li et al. (2017) in terms of peak signal-to-noise",
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+                                    "score": 1.0,
+                                    "content": "ratio (PSNR) of obtained inpaintings on CelebA dataset (Liu et al., 2015) . And finally, we evaluate VAEAC",
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+                                    "content": "against the competing method called Universal Marginalizer (Douglas et al., 2017). Additional experiments",
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+                                    "content": "can be found in appendices C and D. The code is available at https://github.com/tigvarts/",
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+                                            "html": "<table><tr><td>Method/Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Zoo</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.964± 0.007</td><td>1.01 ± 0.01</td><td>0.334± 0.002</td><td>0.19±0.03</td><td>0.422± 0.006</td></tr><tr><td>MissForest</td><td>0.878 ± 0.009</td><td>1.02 ± 0.06</td><td>0.249 ± 0.006</td><td>0.16 ±0.02</td><td>0.422 ± 0.009</td></tr><tr><td>GAIN</td><td>0.97 ± 0.02</td><td>0.99 ± 0.03</td><td>0.271 ± 0.003</td><td>0.20± 0.02</td><td>0.427 ± 0.010</td></tr><tr><td>VAEAC</td><td>0.850 ± 0.007</td><td>0.94 ± 0.01</td><td>0.244 ± 0.002</td><td>0.16 ± 0.02</td><td>0.394± 0.006</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "5 EXPERIMENTS",
+                                    "type": "text"
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+                            "index": 4
+                        }
+                    ],
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+                                    "content": "(Lichman, 2013). We compare imputations from our model with imputations from such classical methods as",
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+                                    "content": "MICE (Buuren & Groothuis-Oudshoorn, 2010) and MissForest (Stekhoven & Buhlmann, 2011) and recently ¨",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "proposed GANs-based method GAIN (Yoon et al., 2018). In the second set of experiments we use VAEAC",
+                                    "type": "text"
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+                            ],
+                            "index": 9
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+                                    "score": 1.0,
+                                    "content": "to solve image inpainting problem. We show inpainitngs generated by VAEAC and compare our model with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "models from papers Pathak et al. (2016), Yeh et al. (2017) and Li et al. (2017) in terms of peak signal-to-noise",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
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+                                    "score": 1.0,
+                                    "content": "ratio (PSNR) of obtained inpaintings on CelebA dataset (Liu et al., 2015) . And finally, we evaluate VAEAC",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
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+                                    "score": 1.0,
+                                    "content": "against the competing method called Universal Marginalizer (Douglas et al., 2017). Additional experiments",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "can be found in appendices C and D. The code is available at https://github.com/tigvarts/",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
+                                    "content": "vaeac.",
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+                                    "content": "5.1 MISSING FEATURES IMPUTATION",
+                                    "type": "text"
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+                                    "content": "x _ { i } ~ = ~ \\omega",
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+                                    "score": 1.0,
+                                    "content": ") and their target values",
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+                                    "score": 1.0,
+                                    "content": ". The majority of",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "discriminative methods do not support missing values in the objects. The procedure of filling in the missing",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
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+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "In this section we evaluate the quality of imputations produced by VAEAC. For evaluation we use datasets",
+                                    "type": "text"
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+                            ],
+                            "index": 21
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+                        {
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+                                    "score": 1.0,
+                                    "content": "from UCI repository (Lichman, 2013). Before training we drop randomly",
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+                                    "content": "50 \\%",
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+                                    "score": 1.0,
+                                    "content": "of values both in train and test",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "set. After that we impute missing features using MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
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+                                    "score": 1.0,
+                                    "content": "(Stekhoven & Buhlmann, 2011), GAIN (Yoon et al., 2018) and VAEAC trained on the observed data. The ¨",
+                                    "type": "text"
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+                            "index": 24
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+                                    "score": 1.0,
+                                    "content": "details of GAIN implementation are described in appendix A.4.",
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+                                    "content": "objects with sampled imputations, so the size of the dataset",
+                                    "type": "text"
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+                            "index": 27
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+                        {
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+                                    "score": 1.0,
+                                    "content": "increases by",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "times. This procedure is called missing features multiple imputation. MICE and GAIN are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "also capable of multiple imputation (we use",
+                                    "type": "text"
+                                },
+                                {
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+                                    "content": "n = 1 0",
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+                                {
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+                                    "score": 1.0,
+                                    "content": "for them in experiments as well), but MissForest is not.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        }
+                    ],
+                    "index": 27.5,
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "For more details about the experimental setup see appendices A.1, A.2, and A.4.",
+                                    "type": "text"
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+                            ],
+                            "index": 30
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "In table 1 we report NRMSE (i.e. RMSE normalized by the standard deviation of each feature and then",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "averaged over all features) of imputations for continuous datasets and proportion of falsely classified (PFC)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for categorical ones. For multiple imputation methods we average imputations of continuous variables and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "take most frequent imputation for categorical ones for each object.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        }
+                    ],
+                    "index": 32.5,
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+                                    "content": "We also learn linear or logistic regression and report the regression or classification performance after ap-",
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+                                            "html": "<table><tr><td>Method /Dataset</td><td>WhiteWine</td><td>Yeast</td><td>Mushroom</td><td>Z00</td><td>Phishing</td></tr><tr><td>MICE</td><td>0.13±0.02</td><td>0.41 ±0.02</td><td>0.92± 0.01</td><td>0.78± 0.05</td><td>0.75 ±0.02</td></tr><tr><td>MissForest</td><td>0.17 ± 0.01</td><td>0.42 ± 0.02</td><td>0.972 ± 0.003</td><td>0.71 ± 0.07</td><td>0.73 ± 0.02</td></tr><tr><td>GAIN</td><td>0.11 ± 0.01</td><td>0.39 ± 0.06</td><td>0.969 ± 0.005</td><td>0.67 ± 0.06</td><td>0.74 ± 0.03</td></tr><tr><td>VAEAC</td><td>0.17 ± 0.01</td><td>0.43 ± 0.01</td><td>0.983 ± 0.002</td><td>0.8 ± 0.1</td><td>0.74 ± 0.02</td></tr></table>",
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+                                    "content": "5.2 IMAGE INPAINTING",
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+                                    "score": 1.0,
+                                    "content": ". Nevertheless, such construction is out of",
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+                                    "content": "In figures 1, 2, 3 and 4 we visualize image inpaintings produced by VAEAC on binarized MNIST (LeCun",
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+                                    "score": 1.0,
+                                    "content": "and description of datasets are available in appendixes A.1 and A.3.",
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+                                    "score": 1.0,
+                                    "content": "with these models. Nevertheless, we compute peak signal-to-noise ratio (PSNR) for one random inpainting",
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+                                    "content": "from VAEAC and the best PSNR among 10 random inpaintings from VAEAC. One inpainting might not",
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+                                    "score": 1.0,
+                                    "content": "be similar to the original image, so we also measure how good the inpainting which is most similar to the",
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+                                    "score": 1.0,
+                                    "content": "original image reconstructs it. We compare these two metrics computed for certain masks with the PSNRs",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "for the same masks on CelebA from papers Yeh et al. (2017) and Li et al. (2017). The results are available",
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+                                    "score": 1.0,
+                                    "content": "for the same masks on CelebA from papers Yeh et al. (2017) and Li et al. (2017). The results are available",
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+                                    "score": 1.0,
+                                    "content": "but assumes conditional independence of the outputs. Nevertheless, there are a number of cases where",
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+                        },
+                        {
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "marginal distributions have many local optima, there is no straightforward parametrization which allows UM",
+                                    "type": "text"
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+                            ],
+                            "index": 9
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "to approximate them, and, therefore also the conditioned joint distribution. An example of such distribution",
+                                    "type": "text"
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+                            ],
+                            "index": 10
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and more illustrations for comparison of VAEAC and UM are available in appendix D.5.",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "6 CONCLUSION",
+                                    "type": "text"
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+                                    "content": "A EXPERIMENTAL DETAILS",
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+                                    "content": "et al., 2015), and convolutional neural networks based on ResNet blocks (He et al., 2016).",
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+                                    "content": ". As in previous section, the brightness of each pixel is treated as a Bernoulli probability of",
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+                                    "content": "In Li et al. (2017) six different masks O1–O6 are used on the testing stage. We reconstruct the positions",
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+                                    "content": "below. We use 32 latent variables.",
+                                    "type": "text"
+                                }
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+                    ],
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+                                    "content": "of masks from the illustrations in the paper and give their coordinates in table 6. The visualizations of the",
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+                                    "content": "masks are available in figure 10.",
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+                                    "type": "text"
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+                                    "cross_page": true
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+                            "index": 0
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+                                    "score": 1.0,
+                                    "content": "et al. (2016). As we deduced from the code 3, the generation process is follows: firstly we generate",
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+                                    "cross_page": true
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+                            "index": 1
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+                                    "cross_page": true
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+                                    "score": 1.0,
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+                                    "type": "text",
+                                    "cross_page": true
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+                                    "score": 1.0,
+                                    "content": "are considered as unobserved). To sample a mask we sample a random position in this",
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+                                    "cross_page": true
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+                                    "type": "text",
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+                                    "score": 0.87,
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+                                    "score": 0.87,
+                                    "content": "30 \\%",
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+                                    "cross_page": true
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+                                    "type": "text",
+                                    "cross_page": true
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "are considered as unobserved). To sample a mask we sample a random position in this",
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+                                    "content": "The theoretical guarantees that VAEAC can model arbitrary distribution are based on the same guarantees",
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+                                    "content": "for Condtitional Variational Autoencoder (CVAE). We prove below that if CVAE can model each of the",
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+                                    "content": ", then VAEAC can model all of them.",
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+                                    "score": 1.0,
+                                    "content": "where each feature may be missing (which we denote by",
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+                                    "score": 1.0,
+                                    "content": ") and their target values",
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+                                    "score": 1.0,
+                                    "content": ". In this section we show that the better results are achieved when our",
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+                                    "score": 1.0,
+                                    "content": "and targets",
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "with MICE (Buuren & Groothuis-Oudshoorn, 2010), MissForest (Stekhoven & Buhlmann, 2011), and GAIN ¨",
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diff --git a/parse/train/TScOKQW67Mj/images/07a3724823bcd540052ed15c8c366ca29e6a1843d56d7cce4940e24c70fd634b.jpg b/parse/train/TScOKQW67Mj/images/07a3724823bcd540052ed15c8c366ca29e6a1843d56d7cce4940e24c70fd634b.jpg
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+# Neural Contextual Bandits with Deep Representation and Shallow Exploration
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We study neural contextual bandits, a general class of contextual bandits, where   
+2 each context-action pair is associated with a raw feature vector, but the specific   
+3 reward generating function is unknown. We propose a novel learning algorithm   
+4 that transforms the raw feature vector using the last hidden layer of a deep ReLU   
+5 neural network (deep representation learning), and uses an upper confidence bound   
+6 (UCB) approach to explore in the last linear layer (shallow exploration). We prove   
+7 that under standard assumptions, our proposed algorithm achieves $\widetilde { O } ( \sqrt { T } )$ finite  
+8 time regret, where $T$ is the learning time horizon. Compared with existing neural   
+9 contextual bandit algorithms, our approach is computationally much more efficient   
+10 since it only needs to explore in the last layer of the deep neural network.
+
+# 11 1 Introduction
+
+12 Multi-armed bandits (MAB) [9, 8, 30] are a class of online decision-making problems where an   
+13 agent needs to learn to maximize its expected cumulative reward while repeatedly interacting with a   
+14 partially known environment. Based on a bandit algorithm (also called a strategy or policy), in each   
+15 round, the agent adaptively chooses an arm, and then observes and receives a reward associated with   
+16 that arm. Since only the reward of the chosen arm will be observed (bandit information feedback),   
+17 a good bandit algorithm has to deal with the exploration-exploitation dilemma: trade-off between   
+18 pulling the best arm based on existing knowledge/history data (exploitation) and trying the arms that   
+19 have not been fully explored (exploration).   
+20 In many real-world applications, the agent will also be able to access detailed contexts associated   
+21 with the arms. For example, when a company wants to choose an advertisement to present to a user,   
+22 the recommendation will be much more accurate if the company takes into consideration the contents,   
+23 specifications, and other features of the advertisements in the arm set as well as the profile of the user.   
+24 To encode the contextual information, contextual bandit models and algorithms have been developed,   
+25 and widely studied both in theory and in practice [19, 39, 34, 16, 1]. Most existing contextual bandit   
+26 algorithms assume that the expected reward of an arm at a context is a linear function in a known   
+27 context-action feature vector, which leads to many useful algorithms such as LinUCB [16], OFUL [1],   
+28 etc. The representation power of the linear model can be limited in applications such as marketing,   
+29 social networking, clinical studies, etc., where the rewards are usually counts or binary variables. The   
+30 linear contextual bandit problem has also been extended to richer classes of parametric bandits such   
+31 as the generalized linear bandits [24, 35] and kernelised bandits [44, 15].   
+32 With the prevalence of deep neural networks (DNNs) and their phenomenal performances in many   
+33 machine learning tasks [32, 25], there has emerged a line of work that employs DNNs to increase the   
+34 representation power of contextual bandit algorithms [5, 38, 17, 49, 52, 20, 51]. The problems they   
+35 solve are usually referred to as neural contextual bandits. For example, Zhou et al. [52] developed   
+36 the NeuralUCB algorithm, which can be viewed as a natural extension of LinUCB [16, 1], where they   
+37 use the output of a deep neural network with the feature vector as input to approximate the reward.   
+38 Zhang et al. [51] adapted neural networks in Thompson Sampling [43, 14, 40] for both exploration   
+39 and exploitation and proposed NeuralTS . For a fixed time horizon $T$ , it has been proved that both   
+40 NeuralUCB and NeuralTS achieve a $O ( \widetilde { d } \sqrt { T } )$ regret bound, where $\hat { d }$ is the effective dimension of a   
+41 neural tangent kernel matrix which can potentially scale with $O ( T K )$ for $K$ -armed bandits. This   
+42 high complexity is mainly due to that the exploration is performed over the entire huge neural network   
+43 parameter space, which is inefficient and even infeasible when the number of neurons is large. A more   
+44 realistic and efficient way of learning neural contextual bandits may be to just explore different arms   
+45 using the last layer as the exploration parameter. More specifically, Riquelme et al. [38] provided   
+46 an extensive empirical study of benchmark algorithms for contextual-bandits through the lens of   
+47 Thompson Sampling, which suggests decoupling representation learning and uncertainty estimation   
+48 improves performance.   
+49 In this paper, we show that the decoupling of representation learning and the exploration can be   
+50 theoretically validated. We study a new neural contextual bandit algorithm, which learns a mapping   
+51 to transform the raw features associated with each context-action pair using a deep neural network   
+52 (deep representation), and then performs an upper confidence bound (UCB)-type exploration over the   
+53 linear output layer of the network (shallow exploration). We prove a sublinear regret of the proposed   
+54 algorithm by exploiting the UCB exploration techniques in linear contextual bandits [1] and the   
+55 analysis of deep overparameterized neural networks using neural tangent kernels [27]. Our theory   
+56 confirms the empirically observed effectiveness of decoupling the deep representation learning and   
+57 the UCB exploration in contextual bandits [38, 49].
+
+58 Contributions we summarize the main contributions of this paper as follows.
+
+• We propose a contextual bandit algorithm, Neural-LinUCB, for solving a general class of contextual bandit problems without knowing the specific reward generating function. The proposed algorithm learns a deep representation to transform the raw feature vectors and performs UCB-type exploration in the last layer of the neural network, which we refer to as deep representation and shallow exploration. Compared with LinUCB [34, 16] and neural bandits such as NeuralUCB [52] and NeuralTS [51], our algorithm enjoys the best of two worlds: strong expressiveness due to the deep representation and computational efficiency due to the shallow exploration.
+
+66 • Despite the usage of a DNN as the feature mapping, we prove a $\widetilde { O } ( \sqrt { T } )$ regret for the proposed   
+67 Neural-LinUCB algorithm, which matches the regret bound of linear contextual bandits [16, 1].   
+68 To the best of our knowledge, this is the first work that theoretically shows the convergence of   
+69 bandits algorithms under the scheme of deep representation and shallow exploration. It is notable   
+70 that a similar scheme called Neural-Linear was proposed by Riquelme et al. [38] for Thompson   
+71 sampling algorithms, and they empirically showed that decoupling representation learning and   
+72 uncertainty estimation improves the performance. Our work confirms this observation from a   
+73 theoretical perspective.
+
+• We conduct experiments on contextual bandit problems based on real-world datasets, demonstrating a better performance and computational efficiency of Neural-LinUCB over LinUCB and NeuralUCB, which well aligns with our theory.
+
+# 77 1.1 Additional related work
+
+78 There is a line of related work to ours on the recent advance in the optimization and generalization   
+79 analysis of deep neural networks. In particular, Jacot et al. [27] first introduced the neural tangent   
+80 kernel (NTK) to characterize the training dynamics of network outputs in the infinite width limit.   
+81 From the notion of NTK, a fruitful line of research emerged and showed that loss functions of   
+82 deep neural networks trained by (stochastic) gradient descent can converge to the global minimum   
+83 [22, 4, 21, 54, 53]. The generalization bounds for overparameterized deep neural networks are also   
+84 established in Arora et al. [6, 7], Allen-Zhu et al. [3], Cao and Gu [12, 13]. Recently, the NTK based   
+85 analysis is also extended to the study of sequential decision problems including bandits [52, 51], and   
+86 reinforcement learning algorithms [11, 36, 45, 47].   
+87 Our algorithm is also different from Langford and Zhang [29], Agarwal et al. [2] which reduce the   
+88 bandit problem to supervised learning. Moreover, their algorithms need to access an oracle that   
+89 returns the optimal policy in a policy class given a sequence of context and reward vectors, whose   
+90 regret depends on the VC-dimension of the policy class.   
+91 Notation We use $[ k ]$ to denote a set $\{ 1 , \ldots , k \}$ , $k \in \mathbb { N } ^ { + }$ . $\| \mathbf { x } \| _ { 2 } = \sqrt { \mathbf { x } ^ { \top } \mathbf { x } }$ is the Euclidean norm of   
+92 a vector $\mathbf { x } \in \mathbb { R } ^ { d }$ . For a matrix $\mathbf { W } \in \mathbb { R } ^ { m \times n }$ , we denote by $\lVert \mathbf { W } \rVert _ { 2 }$ and $\| \mathbf { W } \| _ { F }$ its operator norm   
+93 and Frobenius norm respectively. For a semi-definite matrix $\mathbf { A } \in \mathbb { R } ^ { d \times d }$ and a vector $\mathbf { x } \in \mathbb { R } ^ { d }$ , we   
+94 denote the Mahalanobis norm as $\| \mathbf { x } \| _ { \mathbf { A } } = \sqrt { \mathbf { x } ^ { \top } \mathbf { A } \mathbf { x } }$ . Throughout this paper, we reserve the notations   
+95 $\{ C _ { i } \} _ { i = 0 , 1 , \ldots }$ to represent absolute positive constants that are independent of problem parameters such   
+96 as dimension, sample size, iteration number, step size, network length and so on. The specific values   
+97 of $\{ C _ { i } \} _ { i = 0 , 1 , \ldots }$ . can be different in different context. For a parameter of interest $T$ and a function   
+98 $f ( T )$ , we use notations such as $O ( f ( T ) )$ and $\Omega ( f ( T ) )$ to hide constant factors and ${ \widetilde { O } } ( f ( T ) )$ to hide   
+99 constant and logarithmic dependence of $T$ .
+
+# 2 Preliminaries
+
+In this section, we provide the background of contextual bandits and deep neural networks.
+
+# 2.1 Linear contextual bandits
+
+103 A contextual bandit is characterized by a tuple $( S , A , r )$ , where $s$ is the context (state) space, $\mathcal { A }$ is the   
+104 arm (action) space, and $r$ encodes the unknown reward generating function at all context-arm pairs.   
+105 A learning agent, who knows $s$ and $\mathcal { A }$ but does not know the true reward $r$ (values bounded in $( 0 , 1 )$   
+106 for simplicity), needs to interact with the contextual bandit for $T$ rounds. At each round $t = 1 , \dots , T$ ,   
+107 the agent first observes a context $s _ { t } \in S$ chosen by the environment; then it needs to adaptively select   
+108 an arm $a _ { t } \in \mathcal A$ based on its past observations; finally it receives a reward $\widehat { r } _ { t } ( \mathbf { x } _ { s , a _ { t } } ) = r ( \mathbf { x } _ { s , a _ { t } } ) + \xi _ { t }$ ,   
+109 where $\mathbf { x } _ { s , a } \in \mathbb { R } ^ { d }$ is a known feature vector for context-arm pair $( s , a ) \in S \times A$ , and $\xi _ { t }$ is a random   
+110 noise with zero mean. The agent’s objective is to maximize its expected total reward over these $T$   
+111 rounds, which is equivalent to minimizing the pseudo regret [8]:
+
+$$
+R _ { T } = \mathbb { E } \bigg [ \sum _ { t = 1 } ^ { T } \big ( \widehat { r } ( \mathbf { x } _ { s _ { t } , a _ { t } ^ { * } } ) - \widehat { r } ( \mathbf { x } _ { s _ { t } , a _ { t } } ) \big ) \bigg ] ,
+$$
+
+where 112 $a _ { t } ^ { * } \in \operatorname { a r g m a x } _ { a \in \mathcal { A } } \{ r ( \mathbf { x } _ { s _ { t } , a } ) = \mathbb { E } [ \widehat { r } ( \mathbf { x } _ { s _ { t } , a } ) ] \}$ . To simplify the exposition, we use $\mathbf { x } _ { t , a }$ to denote 113 $\mathbf { x } _ { s _ { t } , a }$ bsince it only depends on the round index $t$ in most bandit problems, and we assume $A = [ K ]$ .
+
+114 In some practical problems, the agent has a prior knowledge that the reward-generating function   
+115 $r$ has some specific parametric form. For instance, in linear contextual bandits, the agent knows   
+116 that $r ( \mathbf { x } _ { s , a } ) \stackrel { * } { = } \mathbf { x } _ { s , a } ^ { \top } \pmb { \theta } ^ { * }$ for some unknown weight vector $\pmb { \theta } ^ { * } \in \mathbb { R } ^ { d }$ . One provably sample efficient   
+117 algorithm for linear contextual bandits is Linear Upper Confidence Bound (LinUCB) [1]. Specifically,   
+118 at each round $t$ , LinUCB chooses action by the following strategy
+
+$$
+a _ { t } = \underset { a \in [ K ] } { \operatorname { a r g m a x } } \left. \mathbf { x } _ { t , a } ^ { \top } \pmb { \theta } _ { t } + \alpha _ { t } \Vert \mathbf { x } _ { t , a } \Vert _ { \mathbf { A } _ { t } ^ { - 1 } } \right. ,
+$$
+
+119 where θt is a point estimate of θ∗, At = λI + Pti=1 xi,ai x> i,ai with some λ > 0 is a matrix   
+120 defined based on the historical context-arm pairs, and $\alpha _ { t } > 0$ is a tuning parameter that controls the   
+121 exploration rate in LinUCB.
+
+# 2.2 Deep neural networks
+
+In this paper, we use 123 $f ( \mathbf { x } )$ to denote a neural network with input data $\mathbf { x } \in \mathbb { R } ^ { d }$ . Let $L$ be the number 124 of hidden layers and $\mathbf { W } _ { l } \in \mathbb { R } ^ { m _ { l } \times m _ { l - 1 } }$ be the weight matrices in the $l$ -th layer, where $l = 1 , \ldots , L$ 125 $m _ { 1 } = . . . = m _ { L - 1 } = m$ and $m _ { 0 } = m _ { L } = d$ . Then a $L$ -hidden layer neural network is defined as
+
+$$
+f ( \mathbf { x } ) = \sqrt { m } \pmb { \theta } ^ { \ast \top } \sigma _ { L } ( \mathbf { W } _ { L } \sigma _ { L - 1 } ( \mathbf { W } _ { L - 1 } \cdot \cdot \cdot \sigma _ { 1 } ( \mathbf { W } _ { 1 } \mathbf { x } ) \cdot \cdot \cdot ) ) ,
+$$
+
+126 where $\sigma _ { l }$ is an activation function and $\pmb { \theta } ^ { * } \in \mathbb { R } ^ { d }$ is the weight of the output layer. To simplify the   
+127 presentation, we will assume $\sigma _ { 1 } = \sigma _ { 2 } = . . . = \sigma _ { L } = \sigma$ is the ReLU activation function, i.e.,   
+128 ${ \bar { \sigma } } ( x ) = \operatorname* { m a x } \{ 0 , x \}$ for $x \in \mathbb { R }$ . We denote $\mathbf { w } = ( \mathrm { v e c } ( \mathbf { W } _ { 1 } ) ^ { \top } , \ldots , \mathrm { v e c } ( \mathbf { W } _ { L } ) ^ { \top } ) ^ { \top }$ , which is the   
+129 concatenation of the vectorized weight parameters of all hidden layers of the neural network. We also   
+130 write $f ( \mathbf { x } ; \pmb { \theta } ^ { * } , \mathbf { w } ) = f ( \mathbf { x } )$ in order to explicitly specify the weight parameters of neural network $f$ . It   
+131 is easy to show that the dimension $p$ of vector w satisfies $p = ( L - 2 ) m ^ { 2 } + 2 m d .$ . To simplify the   
+132 notation, we define $\phi ( \mathbf { x } ; \mathbf { w } )$ as the output of the $L$ -th hidden layer of neural network $f$ .
+
+$$
+\boldsymbol { \phi } ( \mathbf { x } ; \mathbf { w } ) = \sqrt { m } \sigma ( \mathbf { W } _ { L } \sigma ( \mathbf { W } _ { L - 1 } \cdot \cdot \cdot \sigma ( \mathbf { W } _ { 1 } \mathbf { x } ) \cdot \cdot \cdot \cdot ) ) .
+$$
+
+33 Note that $\phi ( \mathbf { x } ; \mathbf { w } )$ itself can also be viewed as a neural network with vector-valued outputs.
+
+# 134 3 Deep Representation and Shallow Exploration
+
+135 The linear parametric form in linear contextual bandits might produce biased estimates of the reward   
+136 due to the lack of representation power [42, 38]. In contrast, it is well known that deep neural networks   
+137 are powerful enough to approximate an arbitrary function [18]. Therefore, a natural extension of   
+138 linear contextual bandits is to use a deep neural network to approximate the reward generating   
+139 function $r ( \cdot )$ . Nonetheless, DNNs usually have a prohibitively large dimension for weight parameters,   
+140 which makes the exploration in neural networks based UCB algorithm inefficient [28, 52].   
+141 In this work, we study a neural contextual bandit algorithm, where the hidden layers of a deep neural   
+142 network are used to represent the features and the exploration is only performed in the last layer of the   
+143 neural network. In particular, we assume that the reward generating function $r ( \cdot )$ can be expressed as   
+144 the inner product between a deep represented feature vector and an exploration weight parameter,   
+145 namely, $\bar { r ( \cdot ) } = \langle \theta ^ { * } , \psi ( \cdot ) \rangle$ , where $\pmb { \theta } ^ { * } \in \mathbb { R } ^ { d }$ is some weight parameter and $\psi ( \cdot )$ is an unknown feature   
+146 mapping. This decoupling of the representation and the exploration will achieve the best of both   
+147 worlds: efficient exploration in shallow (linear) models and high expressive power of deep models.   
+148 To learn the unknown feature mapping, we propose to use a neural network to approximate it. In   
+149 what follows, we will describe a neural contextual bandit algorithm that uses the output of the last   
+150 hidden layer of a neural network to transform the raw feature vectors (deep representation) and   
+151 performs UCB-type exploration in the last layer of the neural network (shallow exploration). Since   
+152 the exploration is performed only in the last linear layer, we call this procedure Neural-LinUCB,   
+153 which is displayed in Algorithm 1.   
+154 Specifically, in round $t$ , the agent receives an action set with raw features $\mathcal { X } _ { t } = \{ \mathbf { x } _ { t , 1 } , . . . , \mathbf { x } _ { t , K } \}$ .   
+155 Then the agent chooses an arm $a _ { t }$ that maximizes the following upper confidence bound:
+
+$$
+a _ { t } = \underset { k \in [ K ] } { \operatorname { a r g m a x } } \Big \{ \langle \phi ( \mathbf { x } _ { t , k } ; \mathbf { w } _ { t - 1 } ) , \theta _ { t - 1 } \rangle + \alpha _ { t } \| \phi ( \mathbf { x } _ { t , k } ; \mathbf { w } _ { t - 1 } ) \| _ { \mathbf { A } _ { t - 1 } ^ { - 1 } } \Big \} ,
+$$
+
+156 where $\pmb { \theta } _ { t - 1 }$ is a point estimate of the unknown weight in the last layer, $\phi ( \mathbf { x } ; \mathbf { w } )$ is defined as in (2.3),   
+157 $\mathbf { w } _ { t - 1 }$ is an estimate of all the weight parameters in the hidden layers of the neural network, $\alpha _ { t } > 0$ is   
+158 the algorithmic parameter controlling the exploration, and ${ \bf A } _ { t }$ is a matrix defined based on historical   
+159 transformed features:
+
+$$
+\mathbf { A } _ { t } = \lambda \mathbf { I } + \sum _ { i = 1 } ^ { t } \phi ( \mathbf { x } _ { i , a _ { i } } ; \mathbf { w } _ { i - 1 } ) \phi ( \mathbf { x } _ { i , a _ { i } } ; \mathbf { w } _ { i - 1 } ) ^ { \top } ,
+$$
+
+and 160 $\lambda > 0$ . After pulling arm $a _ { t }$ , the agent will observe a noisy reward $\widehat { r } _ { t } : = \widehat { r } ( \mathbf { x } _ { t , a _ { t } } )$ defined as
+
+$$
+\widehat { r } ( \mathbf { x } _ { t , k } ) = r ( \mathbf { x } _ { t , k } ) + \xi _ { t } ,
+$$
+
+161 where $\xi _ { t }$ is an independent $\nu$ -subGaussian random noise for some $\nu > 0$ and $r ( \cdot )$ is an unknown   
+162 reward function. In this paper, we will interchangeably use notation $\widehat { r _ { t } }$ to denote the reward received   
+163 at the $t$ -th step and an equivalent notation $\widehat { r } ( \mathbf { x } )$ b to express its dependence on the feature vector $\mathbf { x }$ .
+
+164 Upon receiving the reward $\widehat { r _ { t } }$ , the agent updates its estimate $\theta _ { t }$ of the output layer weight by using the same 165 $\ell ^ { 2 }$ b-regularized least-squares estimate in linear contextual bandits [1]. In particular, we have
+
+$$
+\pmb { \theta } _ { t } = \mathbf { A } _ { t } ^ { - 1 } \mathbf { b } _ { t } ,
+$$
+
+where 166 $\begin{array} { r } { \mathbf { b } _ { t } = \sum _ { i = 1 } ^ { t } \widehat { r } _ { i } \phi ( \mathbf { x } _ { i , a _ { i } } ; \mathbf { w } _ { i - 1 } ) } \end{array}$ .
+
+167 To save the computation, the neural network $\phi ( \cdot ; { \mathbf w } _ { t } )$ will be updated once every $H$ steps. Therefore,   
+168 we have $\mathbf { w } _ { ( q - 1 ) H + 1 } = . . . = \mathbf { w } _ { q H }$ for $q = 1 , 2 , \ldots$ We call the time steps $\{ ( q - 1 ) H + 1 , \ldots , q H \}$   
+169 an epoch with length $H$ . At time step $t = H q$ , for any $q = 1 , 2 , \ldots$ , Algorithm 1 will retrain the   
+170 neural network based on all the historical data. In Algorithm 2, our goal is to minimize the following   
+171 empirical loss function:
+
+$$
+\mathcal { L } _ { q } ( \mathbf { w } ) = \sum _ { i = 1 } ^ { q H } \big ( \pmb { \theta } _ { i } ^ { \top } \phi ( \mathbf { x } _ { i , a _ { i } } ; \mathbf { w } ) - \widehat { r } _ { i } \big ) ^ { 2 } .
+$$
+
+In practi172 i=1e, one can further save computational cost by onl eding data $\{ \mathbf { x } _ { i , a _ { i } } , \widehat { r } _ { i } , \pmb { \theta } _ { i } \} _ { i = ( q - 1 ) H + 1 } ^ { q H }$ $q$ $\mathbf { w } _ { t }$ b, which does not hurt the performance 174 since the historical information has been encoded into the estimate of $\theta _ { i }$ . In this paper, we will 175 perform the following gradient descent step
+
+$$
+\mathbf { w } _ { q } ^ { ( s ) } = \mathbf { w } _ { q } ^ { ( s - 1 ) } - \eta _ { q } \nabla _ { \mathbf { w } } \mathcal { L } _ { q } \big ( \mathbf { w } ^ { ( s - 1 ) } \big ) .
+$$
+
+176 for $s = 1 , \ldots , n$ , where $\mathbf { w } _ { q } ^ { ( 0 ) } = \mathbf { w } ^ { ( 0 ) }$ is chosen as the same random initialization point. We will   
+177 discuss more about the initial point $\mathbf { w } ^ { ( 0 ) }$ in the next paragraph. Then Algorithm 2 outputs $\mathbf { w } _ { q } ^ { ( n ) }$ and   
+178 we set it as the updated weight parameter $\mathbf { w } _ { H q + 1 }$ in Algorithm 1. In the next round, the agent will   
+179 receive another action set $\mathcal { X } _ { t + 1 }$ with raw feature vectors and repeat the above steps to choose the   
+180 sub-optimal arm and update estimation for contextual parameters.   
+181 Initialization: Recall that w is the collection of all hidden layer weight parameters of the neural   
+182 network. We will follow the same initialization scheme as used in Zhou et al. [52], where each entry   
+183 of the weight matrices follows some Gaussian distribution. Specifically, for any $l \in \{ 1 , \ldots , L - 1 \}$ ,   
+184 we set $\mathbf { W } _ { l } = \left[ \begin{array} { c c } { \mathbf { W } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { W } } \end{array} \right]$ , where each entry of $\mathbf { W }$ follows distribution $N ( 0 , 4 / m )$ independently; for   
+185 $\mathbf { W } _ { L }$ , we set it as $\begin{array} { r l } { [ \mathbf { V } } & { { } - \mathbf { V } ] } \end{array}$ , where each entry of $\mathbf { V }$ follows distribution $N ( 0 , 2 / m )$ independently.
+
+Comparison with LinUCB and NeuralUCB: Compared with linear contextual bandits in Section 2.1, Algorithm 1 has a distinct feature that it learns a deep neural network to obtain a deep representation of the raw data vectors and then performs UCB exploration. This deep representation allows our algorithm to characterize more intrinsic and latent information about the raw data $\left\{ \mathbf { x } _ { t , k } \right\} _ { t \in [ T ] , k \in [ K ] } \subset \mathbb { R } ^ { d }$ . However, the increased complexity of the feature mapping $\phi ( \cdot ; { \mathbf { w } } )$ also introduces great hardness in training. For instance, a recent work by Zhou et al. [52] also studied the neural contextual bandit problem, but different from (3.1), their algorithm (NeuralUCB) performs the UCB exploration on the entire network parameter space, which is $\dot { \mathbb { R } } ^ { \widetilde { p } + d }$ , where $\ddot { \tilde { p } } = m + m d + ( L \dot { \bar { \mathbf { \alpha } } } 1 ) m ^ { 2 }$ . Note that in Zhou et al. [52], they need to compute the inverse eof a matrix $\mathbf { Z } _ { t } \in \mathbb { R } ^ { ( \widetilde { p } + d ) \times ( \widetilde { p } + d ) }$ , which is defined in a similar way to the matrix ${ \bf A } _ { t }$ in our paper except that $\mathbf { Z } _ { t }$ is defined based on the gradient of the network instead of the output of the last hidden layer as in (3.2). In sharp contrast, ${ \bf A } _ { t }$ in our paper is only of size $d \times d$ and thus is much more efficient and practical in implementation, which will be seen from our experiments in later sections.
+
+199 We note that there is also a similar algorithm to our Neural-LinUCB presented in Deshmukh et al.   
+200 [20], where they studied the self-supervised learning loss in contextual bandits with neural network   
+201 representation for computer vision problems. However, no regret analysis has been provided. When   
+202 the feature mapping $\phi ( \cdot ; { \mathbf { w } } )$ is an identity function, the problem reduces to linear contextual bandits   
+203 where we directly use $\mathbf { x } _ { t }$ as the feature vector. In this case, it is easy to see that Algorithm 1 reduces   
+204 to LinUCB [16] since we do not need to learn the representation parameter w anymore.   
+205 Comparison with Neural-Linear: The high-level idea of decoupling the representation and explo  
+206 ration in our algorithm is also similar to that of the Neural-Linear algorithm [38, 49], which trains a   
+207 deep neural network to learn a representation of the raw feature vectors, and then uses a Bayesian   
+208 linear regression to estimate the uncertainty in the bandit problem. However, these two algorithms   
+209 are significantly different since Neural-Linear [38] is a Thompson sampling based algorithm that   
+210 uses posterior sampling to estimate the weight parameter $\pmb { \theta } ^ { * }$ via Bayesian linear regression, whereas   
+211 Neural-LinUCB adopts upper confidence bound based techniques to estimate the weight $\pmb { \theta } ^ { * }$ . Never  
+212 theless, both algorithms share the same idea of deep representation and shallow exploration, and we   
+213 view our Neural-LinUCB algorithm as one instantiation of the Neural-Linear scheme.
+
+# 214 4 Main Results
+
+215 To analyze the regret bound of Algorithm 1, we first lay down some important assumptions on the   
+216 neural contextual bandit model.
+
+# Algorithm 1 Deep Representation and Shallow Exploration (Neural-LinUCB)
+
+1: Input: regularization parameter $\lambda > 0$ , number of total steps $T$ , episode length $H$ , exploration parameters $\{ \alpha _ { t } > 0 \} _ { t \in [ T ] }$   
+2: Initialization: $\mathbf { A } _ { 0 } = \lambda \mathbf { I }$ , $\mathbf { b } _ { 0 } = \mathbf { 0 }$ ; entries of $\pmb { \theta } _ { 0 }$ follow $N ( 0 , 1 / d )$ , and $\mathbf { w } ^ { ( 0 ) }$ is initialized as described in Section 3; $q = 1$ ; $\mathbf { w } _ { 0 } = \mathbf { w } ^ { ( 0 ) }$   
+3: for $t = 1 , \dots , T$ do   
+4: receive feature vectors $\left\{ \mathbf { x } _ { t , 1 } , \ldots , \mathbf { x } _ { t , K } \right\}$   
+5: choose arm $\begin{array} { r } { a _ { t } \ = \ \mathrm { a r g m a x } _ { k \in [ K ] } \pmb { \theta } _ { t - 1 } ^ { \top } \phi ( \mathbf { x } _ { t , k } ; \mathbf { w } _ { t - 1 } ) \ + \alpha _ { t } \| \phi ( \mathbf { x } _ { t , k } ; \mathbf { w } _ { t - 1 } ) \| _ { \mathbf { A } _ { t - 1 } ^ { - 1 } } } \end{array}$ , and obtain reward $\widehat { r } _ { t }$   
+6: update ${ \bf A } _ { t }$ and $\mathbf { b } _ { t }$ as follows: $\begin{array} { r l } & { \mathbf { \Phi } ^ { \mathbf { A } } t = \tilde { \mathbf { A } } _ { t - 1 } + \tilde { \phi } ( \mathbf { x } _ { t , a _ { t } } ; \mathbf { w } _ { t - 1 } ) \phi ( \mathbf { x } _ { t , a _ { t } } ; \mathbf { w } _ { t - 1 } ) ^ { \top } } \\ & { \mathbf { b } _ { t } = \mathbf { b } _ { t - 1 } + \hat { r } _ { t } \phi ( \mathbf { x } _ { t , a _ { t } } ; \mathbf { w } _ { t - 1 } ) , } \end{array}$ ,   
+7: update $\pmb { \theta } _ { t } = \mathbf { A } _ { t } ^ { - 1 } \mathbf { b } _ { t }$   
+8: if $\mathrm { n o d } ( t , H ) = 0$ then   
+9: $\mathbf { w } _ { t } \gets$ output of Algorithm 2   
+10: $q = q + 1$   
+11: else   
+12: $\mathbf { w } _ { t } = \mathbf { w } _ { t - 1 }$   
+13: end if   
+14: end for   
+15: Output $\mathbf { w } _ { T }$
+
+# Algorithm 2 Update Weight Parameters with Gradient Descent
+
+1: Input: initial point $\mathbf { w } _ { q } ^ { ( 0 ) } = \mathbf { w } ^ { ( 0 ) }$ , maximum iteration number $n$ , step size $\eta _ { q }$ , and loss function   
+defined in (3.5).   
+2: for $s = 1 , \ldots , n$ do   
+3: $\begin{array} { r } { \mathbf { w } _ { q } ^ { ( s ) } = \mathbf { w } _ { q } ^ { ( s - 1 ) } - \eta _ { q } \nabla _ { \mathbf { w } } \mathcal { L } _ { q } ( \mathbf { w } _ { q } ^ { ( s - 1 ) } ) . } \end{array}$   
+4: end for   
+5: Output w(n)q   
+7 Assumption 4.1. For all $i \geq 1$ and $k \in [ K ]$ , we assume that $\| \mathbf { x } _ { i , k } \| _ { 2 } = 1$ and its entries satisfy   
+8 $[ { \bf { x } } _ { i , k } ] _ { j } \stackrel { - } { = } [ { \bf { x } } _ { j , k } ] _ { j + d / 2 }$ .
+
+19 The assumption that $\| \mathbf { x } _ { i , k } \| _ { 2 } = 1$ is not essential and is only imposed for simplicity, which is also 20 used in Zou and $\mathrm { G u }$ [53], Zhou et al. [52]. Finally, the condition on the entries of $\mathbf { x } _ { i , k }$ is also mild since otherwise we could always construct 21 $\mathbf { x } _ { i , k } ^ { \prime } = [ \mathbf { x } _ { i , k } ^ { \top } , \mathbf { x } _ { i , k } ^ { \top } ] ^ { \top } / \sqrt { 2 }$ to replace it. An implication of
+
+Assumption 4.1 is that the initialization scheme in Algorithm 1 results in $\phi ( \mathbf { x } _ { i , k } ; \mathbf { w } ^ { ( 0 ) } ) = \mathbf { 0 }$ for all $i \in [ T ]$ and $k \in [ K ]$ .
+
+24 We assume the following stability condition on the spectral norm of the neural network gradient:
+
+225 Assumption 4.2. There is a constant $\ell _ { \mathrm { L i p } } > 0$ such that it holds
+
+$$
+\left\| \frac { \partial \phi } { \partial \mathbf { w } } ( \mathbf { x } ; \mathbf { w } _ { 0 } ) - \frac { \partial \phi } { \partial \mathbf { w } } ( \mathbf { x } ^ { \prime } ; \mathbf { w } _ { 0 } ) \right\| _ { 2 } \leq \ell _ { \mathrm { L i p } } \| \mathbf { x } - \mathbf { x } ^ { \prime } \| _ { 2 } ,
+$$
+
+for all 226 $\mathbf { x } , \mathbf { x } ^ { \prime } \in \{ \mathbf { x } _ { i , k } \} _ { i \in [ T ] , k \in [ K ] }$
+
+227 The inequality in Assumption 4.2 resembles the Lipschitz condition on the gradient of the neural   
+228 network. However, it is essentially different from the smoothness condition since here the gradient   
+229 is taken with respect to the neural network weights while the Lipschitz condition is imposed on the   
+230 feature parameter x. Similar conditions are widely made in nonconvex optimization [46, 10, 48], in   
+231 the name of first-order stability, which is essential to derive the convergence of alternating optimization   
+232 algorithms. Furthermore, Assumption 4.2 is only required on the $T K$ training data points and a   
+233 specific weight parameter $\mathbf { w } _ { 0 }$ . Therefore, the condition will hold if the raw feature data lie in a   
+234 certain subspace of $\mathbb { R } ^ { d }$ . We provided some further discussions in the supplementary material about   
+235 this assumption for interested readers.   
+236 In order to analyze the regret bound of Algorithm 1, we need to characterize the properties of the   
+237 deep neural network in (2.2) that is used to represent the feature vectors. Following a recent line of   
+238 research [27, 12, 7, 52], we define the covariance between two data point $\mathbf { x } , \mathbf { y } \in \mathbb { R } ^ { \bar { d } }$ as follows.
+
+$$
+\begin{array} { r l } & { \widetilde { \pmb { \Sigma } } ^ { ( 0 ) } ( \mathbf x , \mathbf y ) = \pmb { \Sigma } ^ { ( 0 ) } ( \mathbf x , \mathbf y ) = \mathbf x ^ { \top } \mathbf y , } \\ & { \pmb { \Lambda } ^ { ( l ) } ( \mathbf x , \mathbf y ) = \left[ \pmb { \Sigma } ^ { l - 1 } ( \mathbf x , \mathbf x ) \quad \pmb { \Sigma } ^ { l - 1 } ( \mathbf x , \mathbf y ) \right] , } \\ & { \pmb { \Sigma } ^ { ( l ) } ( \mathbf x , \mathbf y ) = 2 \mathbb { E } _ { ( u , v ) \sim N ( \mathbf 0 , \mathbf { A } ^ { ( l - 1 ) } ( \mathbf x , \mathbf y ) ) } [ \sigma ( u ) \sigma ( v ) ] , } \\ & { \widetilde { \pmb { \Sigma } } ^ { ( l ) } ( \mathbf x , \mathbf y ) = 2 \widetilde { \pmb { \Sigma } } ^ { ( l - 1 ) } ( \mathbf x , \mathbf y ) \mathbb { E } _ { u , v } [ \dot { \sigma } ( u ) \dot { \sigma } ( v ) ] + \pmb { \Sigma } ^ { ( l ) } ( \mathbf x , \mathbf y ) , } \end{array}
+$$
+
+where 239 the ne240 $( u , v ) \sim N ( \mathbf { 0 } , \mathbf { \Lambda } \Lambda ^ { ( l - 1 ) } ( \mathbf { x } , \mathbf { y } ) )$ , arix $\dot { \sigma } ( \cdot )$ rivative of activation functiobased on all feature vectors $\sigma ( \cdot )$ $\mathbf { H } \in \mathbb { R } ^ { T K \times T K }$ $\{ \mathbf { x } _ { t , k } \} _ { t \in [ T ] , k \in [ K ] }$ 241 Renumbering $\{ \mathbf { x } _ { t , k } \} _ { t \in [ T ] , k \in [ K ] }$ as $\{ \mathbf { x } _ { i } \} _ { i = 1 , \dots , T K }$ , then each entry $\mathbf { H } _ { i j }$ is defined as
+
+$$
+\mathbf { H } _ { i j } = \frac { 1 } { 2 } \big ( \widetilde { \boldsymbol { \Sigma } } ^ { ( L ) } ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) + \boldsymbol { \Sigma } ^ { ( L ) } ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) \big ) ,
+$$
+
+42 for all $i , j \in [ T K ]$ . Based on the above definition, we impose the following assumption on $\mathbf { H }$
+
+43 Assumption 4.3. The neural tangent kernel defined in (4.2) is positive definite, i.e., $\lambda _ { \operatorname* { m i n } } ( \mathbf { H } ) \geq \lambda _ { 0 }$   
+244 for some constant $\lambda _ { 0 } > 0$ .   
+245 Assumption 4.3 essentially requires the neural tangent kernel matrix $\mathbf { H }$ to be non-singular, which is   
+246 a mild condition and also imposed in other related work [21, 7, 12, 52]. Moreover, it is shown that   
+247 Assumption 4.3 can be easily derived from Assumption 4.1 for two-layer ReLU networks [37, 53].   
+248 Therefore, Assumption 4.3 is mild or even negligible given the non-degeneration assumption on the   
+249 feature vectors. Also note that matrix $\mathbf { H }$ is only defined based on layers $l = 1 , \ldots , L$ of the neural   
+250 network, and does not depend on the output layer $\pmb \theta$ . It is easy to extend the definition of $\mathbf { H }$ to the   
+251 NTK matrix defined on all layers including the output layer $\pmb \theta$ , which would also be positive definite   
+252 by Assumption 4.3 and the recursion in (4.2).   
+253 Before we present the regret analysis of the neural contextual bandit, we need to modify the regret   
+254 defined in (2.1) to account for the randomness of the neural network initialization. For a fixed time   
+255 horizon $T$ , we define the regret of Algorithm 1 as follows.
+
+$$
+R _ { T } = \mathbb { E } \bigg [ \sum _ { t = 1 } ^ { T } \big ( \widehat { r } ( \mathbf { x } _ { t , a _ { t } ^ { * } } ) - \widehat { r } ( \mathbf { x } _ { t , a _ { t } } ) \big ) \big | \mathbf { w } ^ { ( 0 ) } \bigg ] ,
+$$
+
+256 where the expectation is taken over the randomness of the reward noise. Note that $R _ { T }$ defined in (4.3)   
+257 is still a random variable since the initialization of Algorithm 2 is randomly generated.
+
+258 Now we are going to present the regret bound of the proposed algorithm.
+
+59 Theorem 4.4. Suppose Assumptions 4.1, 4.2 and 4.3 hold. Assume that $\lVert \pmb { \theta } ^ { * } \rVert _ { 2 } \leq M$ for some   
+0 positive constant $M > 0$ . For any $\delta \in ( 0 , 1 )$ , let us choose $\alpha _ { t }$ in Neural-LinUCB as
+
+$$
+\alpha _ { t } = \nu \sqrt { 2 \big ( d \log ( 1 + t \log ( H K ) / \lambda ) + \log ( 1 / \delta ) \big ) } + \lambda ^ { 1 / 2 } M .
+$$
+
+261 We choose the step size $\eta _ { q }$ of Algorithm 2 as
+
+$$
+\eta _ { q } \leq C _ { 0 } \big ( d ^ { 2 } m n T ^ { 5 . 5 } L ^ { 6 } \log ( T K / \delta ) \big ) ^ { - 1 } ,
+$$
+
+262 and the width of the neural network satisfies $m = \mathrm { p o l y } ( L , d , 1 / \delta , H , \log ( T K / \delta ) )$ . With probability   
+263 at least $1 - \delta$ over the randomness of the initialization of the neural network, it holds that
+
+$$
+R _ { T } \leq C _ { 1 } \alpha _ { T } \sqrt { T d \log \left( 1 + \frac { T G ^ { 2 } } { \lambda d } \right) } + \frac { C _ { 2 } \ell _ { \mathrm { L i p } } L ^ { 3 } d ^ { 5 / 2 } T \sqrt { \log m \log ( \frac { 1 } { \delta } ) \log ( \frac { T K } { \delta } ) } \| \mathbf { r } - \widetilde { \mathbf { r } } \| _ { \mathbf { H } ^ { - 1 } } } { m ^ { 1 / 6 } } ,
+$$
+
+264 where $\{ C _ { i } \} _ { i = 0 , 1 , 2 }$ are absolute constants independent of the problem parameters, $\begin{array} { r l } { \mathbf { r } } & { { } = } \end{array}$   
+265 $( r ( \mathbf { x } _ { 1 } ) , r ( \mathbf { x } _ { 2 } ) , \ldots , r ( \mathbf { x } _ { T K } ) ) ^ { \top } \ \in \ \mathbb { R } ^ { T K }$ and $\widetilde { \textbf { r } } = ( f ( \mathbf { x } _ { 1 } ; \pmb { \theta } _ { 0 } , \mathbf { w } _ { 0 } ) , \dots , f ( \mathbf { x } _ { T K } ; \pmb { \theta } _ { T - 1 } , \mathbf { w } _ { T - 1 } ) ) ^ { \top } \ \in$   
+266 $\mathbb { R } ^ { T K }$ , and $\| \mathbf { r } \| _ { \mathbf { A } } = \sqrt { \mathbf { r } ^ { \top } \mathbf { A } \mathbf { r } }$ .   
+267 Remark 4.5. Theorem 4.4 shows that the regret of Algorithm 1 can be bounded by two parts: the   
+268 first part is of order $\widetilde { O } ( \sqrt { T } )$ , which resembles the regret bound of linear contextual bandits [1]; the   
+269 second part is of order $\widetilde { O } ( m ^ { - 1 / 6 } T \sqrt { ( \mathbf { r } - \widetilde { \mathbf { r } } ) ^ { \top } \mathbf { H } ^ { - 1 } ( \mathbf { r } - \widetilde { \mathbf { r } } ) } )$ , which depends on the estimation error   
+270 of the neural network $f$ e efor the reward generating function $r$ and the neural tangent kernel $\mathbf { H }$ .
+
+It is worth noting that our theoretical analysis depends on the reward structure assumption that $r ( \cdot ) = \langle \theta \ast , \psi ( \cdot ) \rangle$ . However, the linear structure between $\pmb { \theta } \ast$ and $\psi ( \cdot )$ is not essential. As long as the deep representation of the feature vector and the uncertainty weight parameter can be decoupled, Algorithm 1 can be easily extended to settings with milder assumptions on the reward structure such as generalized linear models [41, 24, 35, 28]. For more general bandit models where no assumption is imposed to the reward generating function, it is still unclear whether the decoupled deep representation and shallow exploration would work especially in cases a thorough exploration may be needed.
+
+Based on the result in Theorem 4.4, we can easily verify the following conclusion:
+
+Corollary 4.6. Under the same conditions of Theorem 4.4, if we choose a sufficiently overparameterized neural network mapping $\phi ( \cdot )$ such that $m \geq T ^ { 3 }$ , then the regret of Algorithm 1 is $R _ { T } = { \widetilde { O } } ( { \sqrt { T } } { \sqrt { ( \mathbf { r } - { \widetilde { \mathbf { r } } } ) ^ { \top } \mathbf { H } ^ { - 1 } ( \mathbf { r } - { \widetilde { \mathbf { r } } } ) } } )$ .
+
+Remark 4.7. For the ease of presentation, let us denote $\mathcal { E } : = \| \mathbf { r } - \widetilde { \mathbf { r } } \| _ { \mathbf { H } ^ { - 1 } }$ . If we have $\mathcal { E } = O ( 1 )$ , the total regret in Theorem 4.4 becomes $\widetilde { O } ( \sqrt { T } )$ which matches the regret of linear contextual bandits [1]. We remark that there is a similar assumption in [52] where they assume that $\mathbf { r } ^ { \top } \mathbf { H } ^ { - 1 } \mathbf { r }$ can be upper bounded by a constant. They show that this term can be bounded by the RKHS norm of $\mathbf { r }$ if it belongs to the RKHS induced by the neural tangent kernel [6, 7, 33]. In addition, $\mathcal { E }$ here is the difference between the true reward function and the neural network function, which can also be small if the deep neural network function well approximates the reward generating function $r ( \cdot )$ .
+
+# 290 5 Experiments
+
+291   
+292   
+293   
+294   
+295   
+296   
+297   
+298   
+299   
+300   
+301   
+302   
+303   
+304   
+305
+
+In this section, we provide empirical evaluations of Neural-LinUCB on real-world datasets. As we have discussed in Section 3, Neural-LinUCB could be viewed as an instantiation of the NeuralLinear scheme studied in Riquelme et al. [38] except that we use the UCB exploration instead of the posterior sampling exploration therein. Note that there has been an extensive comparison [38] of the Neural-Linear methods with many other baselines such as greedy algorithms, Variational Inference, Expectation-Propagation, Bayesian Non-parametrics and so on. Therefore, we do not seek a thorough empirical comparison of Neural-LinUCB with all existing bandits algorithms. We refer readers who are interested in the performance of Neural-Linear methods with deep representation and shallow exploration compared with a vast of baselines in the literature to the benchmark study by Riquelme et al. [38]. In this experiment, we only aim to show the advantages of our algorithm over the following baselines: (1) Neural-Linear [38]; (2) LinUCB [16], which does not have a deep representation of the feature vectors; and (3) NeuralUCB [52], which performs UCB exploration on all the parameters of the neural network instead of the shallow exploration used in our paper. All numerical experiments were run on a workstation with Intel(R) Xeon(R) CPU E5-2637 v4 $@$ 3.50GHz.
+
+Datasets: we evaluate the performances of all algorithms on bandit problems created from real-world data. Specifically, following the experimental setting in Zhou et al. [52],we use datasets (Shuttle) Statlog, Magic and Covertype from UCI machine learning repository [23], and the MINST dataset from LeCun et al. [31]. The details of these datasets are presented in Table 1. In Table 1, each instance represents a feature vector $\mathbf { x } \in \mathbb { R } ^ { d }$ that is associated with one of the $K$ arms, and dimension $d$ is the number of attributes in each instance.
+
+Table 1: Specifications of datasets from the UCI machine learning repository used in this paper.   
+
+<table><tr><td></td><td>Statlog</td><td>Magic</td><td>Covertype</td><td>MNIST</td></tr><tr><td>Number of attributes</td><td>9</td><td>11</td><td>54</td><td>784</td></tr><tr><td>Number of arms</td><td>7</td><td>2</td><td>7</td><td>10</td></tr><tr><td>Number of instances</td><td>58,000</td><td>19,020</td><td>581,012</td><td>60,000</td></tr></table>
+
+![](images/943bd96058f4439657901f76b8c00684d91994e2afa724f14d33961644a23aaf.jpg)  
+Figure 1: The cumulative regrets of LinUCB, NeuralUCB, Neural-Linear and Neural-LinUCB over 15, 000 rounds. Experiments are averaged over 10 repetitions.
+
+311 Implementations: for LinUCB, we follow the setting in Li et al. [34] to use disjoint models   
+312 for different arms. For neural network based algorithms such as NeuralUCB, Neural-Linear and   
+313 Neural-LinUCB, we use a ReLU neural network defined as in (2.2) with $L = 2$ and 2000 for the   
+314 UCI datasets (Statlog, Magic, Covertype). Thus the neural network weights are $\mathbf { W } _ { 1 } \in \mathbb { R } ^ { m \times d }$   
+315 $\mathbf { W } _ { 2 } \in \mathbb { R } ^ { k \times m }$ , and $\pmb { \theta } \in \mathbb { R } ^ { \widetilde { k } }$ respectively, where $k = 1 0 0$ , $m = 2 0 0 0$ , and $d$ is the dimension of   
+316 features in the corresponding task. Since the problem size of the MNIST dataset is larger, inspired   
+317 by Hinton and Salakhutdinov [26], we use a deeper NN and set $L = 3$ , $k = 1 0 0$ and $m = 1 0 0$ ,   
+318 with weights $\mathbf { W } _ { 1 } \in \mathbb { R } ^ { m \times d }$ , $\mathbf { W } _ { 2 } \in \mathbb { R } ^ { m \times m }$ , $\mathbf { W _ { 3 } } \in \mathbb { R } ^ { k \times m }$ , and $\pmb \theta \in \mathbb { R } ^ { k }$ . We set the time horizon   
+319 $T = 1 5 , 0 0 0$ , which is the total number of rounds for each algorithm on each dataset. We use   
+320 gradient decent to optimize the network weights, with a step size $\eta _ { q } = 1 \mathrm { e } { - 5 }$ and maximum iteration   
+321 number $n = 1 , 0 0 0$ . To speed up the training process, the network parameter w is updated every   
+322 $H = 1 0 0$ rounds starting from round 2000. We also apply early stopping when the loss difference   
+323 of two consecutive iterations is smaller than a threshold of 1e-6. We set $\lambda = 1$ and $\alpha _ { t } = 0 . 0 2$   
+324 for all algorithms, $t \in [ T ]$ . Following the setting in Riquelme et al. [38], we use round-robin to   
+325 independently select each arm for 3 times at the beginning of each algorithm. For NeuralUCB, since   
+326 it is computationally unaffordable to perform the original UCB exploration as displayed in Zhou et al.   
+327 [52], we follow their experimental setting to replace the matrix $\mathbf { Z } _ { t } \in \mathbb { R } ^ { ( d + \widetilde { p } ) \times \widetilde { ( } d + \widetilde { p } ) }$ in Zhou et al.   
+328 [52] with its diagonal matrix.
+
+Results: we plot the cumulative regret of all algorithms versus round in Figures 1(a), 1(b) and 1(c) for UCI datasets and in Figure 1(d) for MNIST. The results are reported based on the average of 10 repetitions over different random shuffles of the datasets. It can be seen that algorithms based on neural network representations (NeuralUCB, Neural-Linear and Neural-LinUCB) consistently outperform the linear contextual bandit method LinUCB, which shows that linear models may lack representation power and find biased estimates for the underlying reward generating function. Furthermore, our proposed Neural-LinUCB achieves a comparable regret with NeuralUCB in all experiments despite the fact that our algorithm only explores in the output layer of the neural network, which is more computationally efficient as we will show in the sequel.The results in our experiment are well aligned with our theory that deep representation and shallow exploration are sufficient to guarantee a good performance of neural contextual bandit algorithms, which is also consistent with the findings in existing literature [38] that decoupling the representation learning and uncertainty estimation improves the performance.
+
+We also conducted experiments to study the effects of different widths of deep neural networks on the regret performance and to show the computational efficiency of Neural-LinUCB compared with existing neural bandit algorithms. Due to the space limit, we defer the results to Appendix A.
+
+# 6 Conclusions
+
+In this paper, we propose a new neural contextual bandit algorithm called Neural-LinUCB, which uses the hidden layers of a ReLU neural network as a deep representation of the raw feature vectors and performs UCB type exploration on the last layer of the neural network. By incorporating techniques in liner contextual bandits and neural tangent kernels, we prove that the proposed algorithm achieves a sublinear regret when the width of the network is sufficiently large. This is the first regret analysis of neural contextual bandit algorithms with deep representation and shallow exploration, which have been observed in practice to work well on many benchmark bandit problems [38]. We also conducted experiments on real-world datasets to demonstrate the advantage of the proposed algorithm over LinUCB and existing neural contextual bandit algorithms.
+
+355 References [1] Yasin Abbasi-Yadkori, Dávid Pál, and Csaba Szepesvári. Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pages 2312–2320, 2011. [2] Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning, pages 1638–1646, 2014. [3] Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. In Advances in neural information processing systems, pages 6155–6166, 2019. [4] Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over-parameterization. In International Conference on Machine Learning, pages 242–252, 2019. [5] Robin Allesiardo, Raphaël Féraud, and Djallel Bouneffouf. A neural networks committee for the contextual bandit problem. In International Conference on Neural Information Processing, pages 374–381. Springer, 2014. [6] Sanjeev Arora, Simon Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks. In International Conference on Machine Learning, pages 322–332, 2019. [7] Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, Russ R Salakhutdinov, and Ruosong Wang. On exact computation with an infinitely wide neural net. In Advances in Neural Information Processing Systems, pages 8139–8148, 2019. [8] Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Exploration–exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science, 410(19): 1876–1902, 2009. [9] Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine learning, 47(2-3):235–256, 2002.   
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] We discussed the limitation of the assumptions made in this paper. We also admit in the experiment that the theory maybe conservative since our experiment does not require a very wide neural network to achieve good performance.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work focuses on a general methodology in bandit problems and its theoretical analysis. It does not cause any negative social impact.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See the assumptions listed in Section 4   
+(b) Did you include complete proofs of all theoretical results? [Yes] Proofs are provided in the appendix.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We provide them in the supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We specify all the details in the Implementations paragraph of Section 5.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All the figures are plotted with the standard error with respect to random repetitions.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We stated the type of workstation at the end of the first paragraph of Section 5.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] As we mentioned in Section 5, we used codes from baseline algorithms and public available datasets. All the assets were properly cited.   
+(b) Did you mention the license of the assets? [N/A] All the codes and datasets are open-source.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code in the supplementary for reproduction.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] The data does not contain any personally identifiable information or offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
\ No newline at end of file
diff --git a/parse/train/jCxDyge46t2/jCxDyge46t2_content_list.json b/parse/train/jCxDyge46t2/jCxDyge46t2_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..f48b4cf3239a10dac4f495dc997c38d26202aab5
--- /dev/null
+++ b/parse/train/jCxDyge46t2/jCxDyge46t2_content_list.json
@@ -0,0 +1,1356 @@
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+        "text": "Neural Contextual Bandits with Deep Representation and Shallow Exploration ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 We study neural contextual bandits, a general class of contextual bandits, where   \n2 each context-action pair is associated with a raw feature vector, but the specific   \n3 reward generating function is unknown. We propose a novel learning algorithm   \n4 that transforms the raw feature vector using the last hidden layer of a deep ReLU   \n5 neural network (deep representation learning), and uses an upper confidence bound   \n6 (UCB) approach to explore in the last linear layer (shallow exploration). We prove   \n7 that under standard assumptions, our proposed algorithm achieves $\\widetilde { O } ( \\sqrt { T } )$ finite  \n8 time regret, where $T$ is the learning time horizon. Compared with existing neural   \n9 contextual bandit algorithms, our approach is computationally much more efficient   \n10 since it only needs to explore in the last layer of the deep neural network. ",
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+        "text": "11 1 Introduction ",
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+        "text": "12 Multi-armed bandits (MAB) [9, 8, 30] are a class of online decision-making problems where an   \n13 agent needs to learn to maximize its expected cumulative reward while repeatedly interacting with a   \n14 partially known environment. Based on a bandit algorithm (also called a strategy or policy), in each   \n15 round, the agent adaptively chooses an arm, and then observes and receives a reward associated with   \n16 that arm. Since only the reward of the chosen arm will be observed (bandit information feedback),   \n17 a good bandit algorithm has to deal with the exploration-exploitation dilemma: trade-off between   \n18 pulling the best arm based on existing knowledge/history data (exploitation) and trying the arms that   \n19 have not been fully explored (exploration).   \n20 In many real-world applications, the agent will also be able to access detailed contexts associated   \n21 with the arms. For example, when a company wants to choose an advertisement to present to a user,   \n22 the recommendation will be much more accurate if the company takes into consideration the contents,   \n23 specifications, and other features of the advertisements in the arm set as well as the profile of the user.   \n24 To encode the contextual information, contextual bandit models and algorithms have been developed,   \n25 and widely studied both in theory and in practice [19, 39, 34, 16, 1]. Most existing contextual bandit   \n26 algorithms assume that the expected reward of an arm at a context is a linear function in a known   \n27 context-action feature vector, which leads to many useful algorithms such as LinUCB [16], OFUL [1],   \n28 etc. The representation power of the linear model can be limited in applications such as marketing,   \n29 social networking, clinical studies, etc., where the rewards are usually counts or binary variables. The   \n30 linear contextual bandit problem has also been extended to richer classes of parametric bandits such   \n31 as the generalized linear bandits [24, 35] and kernelised bandits [44, 15].   \n32 With the prevalence of deep neural networks (DNNs) and their phenomenal performances in many   \n33 machine learning tasks [32, 25], there has emerged a line of work that employs DNNs to increase the   \n34 representation power of contextual bandit algorithms [5, 38, 17, 49, 52, 20, 51]. The problems they   \n35 solve are usually referred to as neural contextual bandits. For example, Zhou et al. [52] developed   \n36 the NeuralUCB algorithm, which can be viewed as a natural extension of LinUCB [16, 1], where they   \n37 use the output of a deep neural network with the feature vector as input to approximate the reward.   \n38 Zhang et al. [51] adapted neural networks in Thompson Sampling [43, 14, 40] for both exploration   \n39 and exploitation and proposed NeuralTS . For a fixed time horizon $T$ , it has been proved that both   \n40 NeuralUCB and NeuralTS achieve a $O ( \\widetilde { d } \\sqrt { T } )$ regret bound, where $\\hat { d }$ is the effective dimension of a   \n41 neural tangent kernel matrix which can potentially scale with $O ( T K )$ for $K$ -armed bandits. This   \n42 high complexity is mainly due to that the exploration is performed over the entire huge neural network   \n43 parameter space, which is inefficient and even infeasible when the number of neurons is large. A more   \n44 realistic and efficient way of learning neural contextual bandits may be to just explore different arms   \n45 using the last layer as the exploration parameter. More specifically, Riquelme et al. [38] provided   \n46 an extensive empirical study of benchmark algorithms for contextual-bandits through the lens of   \n47 Thompson Sampling, which suggests decoupling representation learning and uncertainty estimation   \n48 improves performance.   \n49 In this paper, we show that the decoupling of representation learning and the exploration can be   \n50 theoretically validated. We study a new neural contextual bandit algorithm, which learns a mapping   \n51 to transform the raw features associated with each context-action pair using a deep neural network   \n52 (deep representation), and then performs an upper confidence bound (UCB)-type exploration over the   \n53 linear output layer of the network (shallow exploration). We prove a sublinear regret of the proposed   \n54 algorithm by exploiting the UCB exploration techniques in linear contextual bandits [1] and the   \n55 analysis of deep overparameterized neural networks using neural tangent kernels [27]. Our theory   \n56 confirms the empirically observed effectiveness of decoupling the deep representation learning and   \n57 the UCB exploration in contextual bandits [38, 49]. ",
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+        "text": "58 Contributions we summarize the main contributions of this paper as follows. ",
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+        "text": "• We propose a contextual bandit algorithm, Neural-LinUCB, for solving a general class of contextual bandit problems without knowing the specific reward generating function. The proposed algorithm learns a deep representation to transform the raw feature vectors and performs UCB-type exploration in the last layer of the neural network, which we refer to as deep representation and shallow exploration. Compared with LinUCB [34, 16] and neural bandits such as NeuralUCB [52] and NeuralTS [51], our algorithm enjoys the best of two worlds: strong expressiveness due to the deep representation and computational efficiency due to the shallow exploration. ",
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+        "text": "66 • Despite the usage of a DNN as the feature mapping, we prove a $\\widetilde { O } ( \\sqrt { T } )$ regret for the proposed   \n67 Neural-LinUCB algorithm, which matches the regret bound of linear contextual bandits [16, 1].   \n68 To the best of our knowledge, this is the first work that theoretically shows the convergence of   \n69 bandits algorithms under the scheme of deep representation and shallow exploration. It is notable   \n70 that a similar scheme called Neural-Linear was proposed by Riquelme et al. [38] for Thompson   \n71 sampling algorithms, and they empirically showed that decoupling representation learning and   \n72 uncertainty estimation improves the performance. Our work confirms this observation from a   \n73 theoretical perspective. ",
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+        "text": "• We conduct experiments on contextual bandit problems based on real-world datasets, demonstrating a better performance and computational efficiency of Neural-LinUCB over LinUCB and NeuralUCB, which well aligns with our theory. ",
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+        "type": "text",
+        "text": "77 1.1 Additional related work ",
+        "text_level": 1,
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+        "text": "78 There is a line of related work to ours on the recent advance in the optimization and generalization   \n79 analysis of deep neural networks. In particular, Jacot et al. [27] first introduced the neural tangent   \n80 kernel (NTK) to characterize the training dynamics of network outputs in the infinite width limit.   \n81 From the notion of NTK, a fruitful line of research emerged and showed that loss functions of   \n82 deep neural networks trained by (stochastic) gradient descent can converge to the global minimum   \n83 [22, 4, 21, 54, 53]. The generalization bounds for overparameterized deep neural networks are also   \n84 established in Arora et al. [6, 7], Allen-Zhu et al. [3], Cao and Gu [12, 13]. Recently, the NTK based   \n85 analysis is also extended to the study of sequential decision problems including bandits [52, 51], and   \n86 reinforcement learning algorithms [11, 36, 45, 47].   \n87 Our algorithm is also different from Langford and Zhang [29], Agarwal et al. [2] which reduce the   \n88 bandit problem to supervised learning. Moreover, their algorithms need to access an oracle that   \n89 returns the optimal policy in a policy class given a sequence of context and reward vectors, whose   \n90 regret depends on the VC-dimension of the policy class.   \n91 Notation We use $[ k ]$ to denote a set $\\{ 1 , \\ldots , k \\}$ , $k \\in \\mathbb { N } ^ { + }$ . $\\| \\mathbf { x } \\| _ { 2 } = \\sqrt { \\mathbf { x } ^ { \\top } \\mathbf { x } }$ is the Euclidean norm of   \n92 a vector $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ . For a matrix $\\mathbf { W } \\in \\mathbb { R } ^ { m \\times n }$ , we denote by $\\lVert \\mathbf { W } \\rVert _ { 2 }$ and $\\| \\mathbf { W } \\| _ { F }$ its operator norm   \n93 and Frobenius norm respectively. For a semi-definite matrix $\\mathbf { A } \\in \\mathbb { R } ^ { d \\times d }$ and a vector $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ , we   \n94 denote the Mahalanobis norm as $\\| \\mathbf { x } \\| _ { \\mathbf { A } } = \\sqrt { \\mathbf { x } ^ { \\top } \\mathbf { A } \\mathbf { x } }$ . Throughout this paper, we reserve the notations   \n95 $\\{ C _ { i } \\} _ { i = 0 , 1 , \\ldots }$ to represent absolute positive constants that are independent of problem parameters such   \n96 as dimension, sample size, iteration number, step size, network length and so on. The specific values   \n97 of $\\{ C _ { i } \\} _ { i = 0 , 1 , \\ldots }$ . can be different in different context. For a parameter of interest $T$ and a function   \n98 $f ( T )$ , we use notations such as $O ( f ( T ) )$ and $\\Omega ( f ( T ) )$ to hide constant factors and ${ \\widetilde { O } } ( f ( T ) )$ to hide   \n99 constant and logarithmic dependence of $T$ . ",
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+        "text": "2 Preliminaries ",
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+        "text": "In this section, we provide the background of contextual bandits and deep neural networks. ",
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+        "text": "2.1 Linear contextual bandits ",
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+        "text": "103 A contextual bandit is characterized by a tuple $( S , A , r )$ , where $s$ is the context (state) space, $\\mathcal { A }$ is the   \n104 arm (action) space, and $r$ encodes the unknown reward generating function at all context-arm pairs.   \n105 A learning agent, who knows $s$ and $\\mathcal { A }$ but does not know the true reward $r$ (values bounded in $( 0 , 1 )$   \n106 for simplicity), needs to interact with the contextual bandit for $T$ rounds. At each round $t = 1 , \\dots , T$ ,   \n107 the agent first observes a context $s _ { t } \\in S$ chosen by the environment; then it needs to adaptively select   \n108 an arm $a _ { t } \\in \\mathcal A$ based on its past observations; finally it receives a reward $\\widehat { r } _ { t } ( \\mathbf { x } _ { s , a _ { t } } ) = r ( \\mathbf { x } _ { s , a _ { t } } ) + \\xi _ { t }$ ,   \n109 where $\\mathbf { x } _ { s , a } \\in \\mathbb { R } ^ { d }$ is a known feature vector for context-arm pair $( s , a ) \\in S \\times A$ , and $\\xi _ { t }$ is a random   \n110 noise with zero mean. The agent’s objective is to maximize its expected total reward over these $T$   \n111 rounds, which is equivalent to minimizing the pseudo regret [8]: ",
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+        "img_path": "images/4ac3360b2010939a09fbbebb9b156b63a1082607a5e40bddbe053ae9cd752970.jpg",
+        "text": "$$\nR _ { T } = \\mathbb { E } \\bigg [ \\sum _ { t = 1 } ^ { T } \\big ( \\widehat { r } ( \\mathbf { x } _ { s _ { t } , a _ { t } ^ { * } } ) - \\widehat { r } ( \\mathbf { x } _ { s _ { t } , a _ { t } } ) \\big ) \\bigg ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where 112 $a _ { t } ^ { * } \\in \\operatorname { a r g m a x } _ { a \\in \\mathcal { A } } \\{ r ( \\mathbf { x } _ { s _ { t } , a } ) = \\mathbb { E } [ \\widehat { r } ( \\mathbf { x } _ { s _ { t } , a } ) ] \\}$ . To simplify the exposition, we use $\\mathbf { x } _ { t , a }$ to denote 113 $\\mathbf { x } _ { s _ { t } , a }$ bsince it only depends on the round index $t$ in most bandit problems, and we assume $A = [ K ]$ . ",
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+        "text": "114 In some practical problems, the agent has a prior knowledge that the reward-generating function   \n115 $r$ has some specific parametric form. For instance, in linear contextual bandits, the agent knows   \n116 that $r ( \\mathbf { x } _ { s , a } ) \\stackrel { * } { = } \\mathbf { x } _ { s , a } ^ { \\top } \\pmb { \\theta } ^ { * }$ for some unknown weight vector $\\pmb { \\theta } ^ { * } \\in \\mathbb { R } ^ { d }$ . One provably sample efficient   \n117 algorithm for linear contextual bandits is Linear Upper Confidence Bound (LinUCB) [1]. Specifically,   \n118 at each round $t$ , LinUCB chooses action by the following strategy ",
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+        "img_path": "images/7a82c47bf9fdeb914a8ed73004df11005f1391e8ddacd3616c91fada54d74cfa.jpg",
+        "text": "$$\na _ { t } = \\underset { a \\in [ K ] } { \\operatorname { a r g m a x } } \\left. \\mathbf { x } _ { t , a } ^ { \\top } \\pmb { \\theta } _ { t } + \\alpha _ { t } \\Vert \\mathbf { x } _ { t , a } \\Vert _ { \\mathbf { A } _ { t } ^ { - 1 } } \\right. ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "119 where θt is a point estimate of θ∗, At = λI + Pti=1 xi,ai x> i,ai with some λ > 0 is a matrix   \n120 defined based on the historical context-arm pairs, and $\\alpha _ { t } > 0$ is a tuning parameter that controls the   \n121 exploration rate in LinUCB. ",
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+        "type": "text",
+        "text": "2.2 Deep neural networks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we use 123 $f ( \\mathbf { x } )$ to denote a neural network with input data $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ . Let $L$ be the number 124 of hidden layers and $\\mathbf { W } _ { l } \\in \\mathbb { R } ^ { m _ { l } \\times m _ { l - 1 } }$ be the weight matrices in the $l$ -th layer, where $l = 1 , \\ldots , L$ 125 $m _ { 1 } = . . . = m _ { L - 1 } = m$ and $m _ { 0 } = m _ { L } = d$ . Then a $L$ -hidden layer neural network is defined as ",
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+        "img_path": "images/8e1edc132f0151005d0207b777a8d3f22e9f803748e11419052ffd4036ea6c87.jpg",
+        "text": "$$\nf ( \\mathbf { x } ) = \\sqrt { m } \\pmb { \\theta } ^ { \\ast \\top } \\sigma _ { L } ( \\mathbf { W } _ { L } \\sigma _ { L - 1 } ( \\mathbf { W } _ { L - 1 } \\cdot \\cdot \\cdot \\sigma _ { 1 } ( \\mathbf { W } _ { 1 } \\mathbf { x } ) \\cdot \\cdot \\cdot ) ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "126 where $\\sigma _ { l }$ is an activation function and $\\pmb { \\theta } ^ { * } \\in \\mathbb { R } ^ { d }$ is the weight of the output layer. To simplify the   \n127 presentation, we will assume $\\sigma _ { 1 } = \\sigma _ { 2 } = . . . = \\sigma _ { L } = \\sigma$ is the ReLU activation function, i.e.,   \n128 ${ \\bar { \\sigma } } ( x ) = \\operatorname* { m a x } \\{ 0 , x \\}$ for $x \\in \\mathbb { R }$ . We denote $\\mathbf { w } = ( \\mathrm { v e c } ( \\mathbf { W } _ { 1 } ) ^ { \\top } , \\ldots , \\mathrm { v e c } ( \\mathbf { W } _ { L } ) ^ { \\top } ) ^ { \\top }$ , which is the   \n129 concatenation of the vectorized weight parameters of all hidden layers of the neural network. We also   \n130 write $f ( \\mathbf { x } ; \\pmb { \\theta } ^ { * } , \\mathbf { w } ) = f ( \\mathbf { x } )$ in order to explicitly specify the weight parameters of neural network $f$ . It   \n131 is easy to show that the dimension $p$ of vector w satisfies $p = ( L - 2 ) m ^ { 2 } + 2 m d .$ . To simplify the   \n132 notation, we define $\\phi ( \\mathbf { x } ; \\mathbf { w } )$ as the output of the $L$ -th hidden layer of neural network $f$ . ",
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+        "type": "text",
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+    {
+        "type": "equation",
+        "img_path": "images/adb847137db9f7bc640f969b5c2162050750e1e37bd0bca12621f7b419713ad6.jpg",
+        "text": "$$\n\\boldsymbol { \\phi } ( \\mathbf { x } ; \\mathbf { w } ) = \\sqrt { m } \\sigma ( \\mathbf { W } _ { L } \\sigma ( \\mathbf { W } _ { L - 1 } \\cdot \\cdot \\cdot \\sigma ( \\mathbf { W } _ { 1 } \\mathbf { x } ) \\cdot \\cdot \\cdot \\cdot ) ) .\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "33 Note that $\\phi ( \\mathbf { x } ; \\mathbf { w } )$ itself can also be viewed as a neural network with vector-valued outputs. ",
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+    {
+        "type": "text",
+        "text": "134 3 Deep Representation and Shallow Exploration ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "135 The linear parametric form in linear contextual bandits might produce biased estimates of the reward   \n136 due to the lack of representation power [42, 38]. In contrast, it is well known that deep neural networks   \n137 are powerful enough to approximate an arbitrary function [18]. Therefore, a natural extension of   \n138 linear contextual bandits is to use a deep neural network to approximate the reward generating   \n139 function $r ( \\cdot )$ . Nonetheless, DNNs usually have a prohibitively large dimension for weight parameters,   \n140 which makes the exploration in neural networks based UCB algorithm inefficient [28, 52].   \n141 In this work, we study a neural contextual bandit algorithm, where the hidden layers of a deep neural   \n142 network are used to represent the features and the exploration is only performed in the last layer of the   \n143 neural network. In particular, we assume that the reward generating function $r ( \\cdot )$ can be expressed as   \n144 the inner product between a deep represented feature vector and an exploration weight parameter,   \n145 namely, $\\bar { r ( \\cdot ) } = \\langle \\theta ^ { * } , \\psi ( \\cdot ) \\rangle$ , where $\\pmb { \\theta } ^ { * } \\in \\mathbb { R } ^ { d }$ is some weight parameter and $\\psi ( \\cdot )$ is an unknown feature   \n146 mapping. This decoupling of the representation and the exploration will achieve the best of both   \n147 worlds: efficient exploration in shallow (linear) models and high expressive power of deep models.   \n148 To learn the unknown feature mapping, we propose to use a neural network to approximate it. In   \n149 what follows, we will describe a neural contextual bandit algorithm that uses the output of the last   \n150 hidden layer of a neural network to transform the raw feature vectors (deep representation) and   \n151 performs UCB-type exploration in the last layer of the neural network (shallow exploration). Since   \n152 the exploration is performed only in the last linear layer, we call this procedure Neural-LinUCB,   \n153 which is displayed in Algorithm 1.   \n154 Specifically, in round $t$ , the agent receives an action set with raw features $\\mathcal { X } _ { t } = \\{ \\mathbf { x } _ { t , 1 } , . . . , \\mathbf { x } _ { t , K } \\}$ .   \n155 Then the agent chooses an arm $a _ { t }$ that maximizes the following upper confidence bound: ",
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+        "type": "equation",
+        "img_path": "images/4e4e26795f0bc8675499c5a7e793d1a20254225213eaf68061c88a5bda924326.jpg",
+        "text": "$$\na _ { t } = \\underset { k \\in [ K ] } { \\operatorname { a r g m a x } } \\Big \\{ \\langle \\phi ( \\mathbf { x } _ { t , k } ; \\mathbf { w } _ { t - 1 } ) , \\theta _ { t - 1 } \\rangle + \\alpha _ { t } \\| \\phi ( \\mathbf { x } _ { t , k } ; \\mathbf { w } _ { t - 1 } ) \\| _ { \\mathbf { A } _ { t - 1 } ^ { - 1 } } \\Big \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "156 where $\\pmb { \\theta } _ { t - 1 }$ is a point estimate of the unknown weight in the last layer, $\\phi ( \\mathbf { x } ; \\mathbf { w } )$ is defined as in (2.3),   \n157 $\\mathbf { w } _ { t - 1 }$ is an estimate of all the weight parameters in the hidden layers of the neural network, $\\alpha _ { t } > 0$ is   \n158 the algorithmic parameter controlling the exploration, and ${ \\bf A } _ { t }$ is a matrix defined based on historical   \n159 transformed features: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb0921b282182ea1867f207b10a77c9e44b5aa08b66915662ab83d66b11081cc.jpg",
+        "text": "$$\n\\mathbf { A } _ { t } = \\lambda \\mathbf { I } + \\sum _ { i = 1 } ^ { t } \\phi ( \\mathbf { x } _ { i , a _ { i } } ; \\mathbf { w } _ { i - 1 } ) \\phi ( \\mathbf { x } _ { i , a _ { i } } ; \\mathbf { w } _ { i - 1 } ) ^ { \\top } ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "and 160 $\\lambda > 0$ . After pulling arm $a _ { t }$ , the agent will observe a noisy reward $\\widehat { r } _ { t } : = \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } } )$ defined as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b6a8c034c1ffd16f9b08b84c4acebf1cf345d72ed096ceeb6def98731c11d51.jpg",
+        "text": "$$\n\\widehat { r } ( \\mathbf { x } _ { t , k } ) = r ( \\mathbf { x } _ { t , k } ) + \\xi _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "161 where $\\xi _ { t }$ is an independent $\\nu$ -subGaussian random noise for some $\\nu > 0$ and $r ( \\cdot )$ is an unknown   \n162 reward function. In this paper, we will interchangeably use notation $\\widehat { r _ { t } }$ to denote the reward received   \n163 at the $t$ -th step and an equivalent notation $\\widehat { r } ( \\mathbf { x } )$ b to express its dependence on the feature vector $\\mathbf { x }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "164 Upon receiving the reward $\\widehat { r _ { t } }$ , the agent updates its estimate $\\theta _ { t }$ of the output layer weight by using the same 165 $\\ell ^ { 2 }$ b-regularized least-squares estimate in linear contextual bandits [1]. In particular, we have ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4298c842934c21749769fa88aac2870581f38ab0de80fecd1721b4d74a21a85c.jpg",
+        "text": "$$\n\\pmb { \\theta } _ { t } = \\mathbf { A } _ { t } ^ { - 1 } \\mathbf { b } _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where 166 $\\begin{array} { r } { \\mathbf { b } _ { t } = \\sum _ { i = 1 } ^ { t } \\widehat { r } _ { i } \\phi ( \\mathbf { x } _ { i , a _ { i } } ; \\mathbf { w } _ { i - 1 } ) } \\end{array}$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "167 To save the computation, the neural network $\\phi ( \\cdot ; { \\mathbf w } _ { t } )$ will be updated once every $H$ steps. Therefore,   \n168 we have $\\mathbf { w } _ { ( q - 1 ) H + 1 } = . . . = \\mathbf { w } _ { q H }$ for $q = 1 , 2 , \\ldots$ We call the time steps $\\{ ( q - 1 ) H + 1 , \\ldots , q H \\}$   \n169 an epoch with length $H$ . At time step $t = H q$ , for any $q = 1 , 2 , \\ldots$ , Algorithm 1 will retrain the   \n170 neural network based on all the historical data. In Algorithm 2, our goal is to minimize the following   \n171 empirical loss function: ",
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+    {
+        "type": "equation",
+        "img_path": "images/44b174f71bcc58f2a5d6084eeb0f52dce165c4ca6d6d1b01659c83c8da9c4d91.jpg",
+        "text": "$$\n\\mathcal { L } _ { q } ( \\mathbf { w } ) = \\sum _ { i = 1 } ^ { q H } \\big ( \\pmb { \\theta } _ { i } ^ { \\top } \\phi ( \\mathbf { x } _ { i , a _ { i } } ; \\mathbf { w } ) - \\widehat { r } _ { i } \\big ) ^ { 2 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In practi172 i=1e, one can further save computational cost by onl eding data $\\{ \\mathbf { x } _ { i , a _ { i } } , \\widehat { r } _ { i } , \\pmb { \\theta } _ { i } \\} _ { i = ( q - 1 ) H + 1 } ^ { q H }$ $q$ $\\mathbf { w } _ { t }$ b, which does not hurt the performance 174 since the historical information has been encoded into the estimate of $\\theta _ { i }$ . In this paper, we will 175 perform the following gradient descent step ",
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+        "img_path": "images/9b9d794581546b55e07caaecc79341427bc37d80fd6b763c6db383977553ad99.jpg",
+        "text": "$$\n\\mathbf { w } _ { q } ^ { ( s ) } = \\mathbf { w } _ { q } ^ { ( s - 1 ) } - \\eta _ { q } \\nabla _ { \\mathbf { w } } \\mathcal { L } _ { q } \\big ( \\mathbf { w } ^ { ( s - 1 ) } \\big ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "176 for $s = 1 , \\ldots , n$ , where $\\mathbf { w } _ { q } ^ { ( 0 ) } = \\mathbf { w } ^ { ( 0 ) }$ is chosen as the same random initialization point. We will   \n177 discuss more about the initial point $\\mathbf { w } ^ { ( 0 ) }$ in the next paragraph. Then Algorithm 2 outputs $\\mathbf { w } _ { q } ^ { ( n ) }$ and   \n178 we set it as the updated weight parameter $\\mathbf { w } _ { H q + 1 }$ in Algorithm 1. In the next round, the agent will   \n179 receive another action set $\\mathcal { X } _ { t + 1 }$ with raw feature vectors and repeat the above steps to choose the   \n180 sub-optimal arm and update estimation for contextual parameters.   \n181 Initialization: Recall that w is the collection of all hidden layer weight parameters of the neural   \n182 network. We will follow the same initialization scheme as used in Zhou et al. [52], where each entry   \n183 of the weight matrices follows some Gaussian distribution. Specifically, for any $l \\in \\{ 1 , \\ldots , L - 1 \\}$ ,   \n184 we set $\\mathbf { W } _ { l } = \\left[ \\begin{array} { c c } { \\mathbf { W } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { W } } \\end{array} \\right]$ , where each entry of $\\mathbf { W }$ follows distribution $N ( 0 , 4 / m )$ independently; for   \n185 $\\mathbf { W } _ { L }$ , we set it as $\\begin{array} { r l } { [ \\mathbf { V } } & { { } - \\mathbf { V } ] } \\end{array}$ , where each entry of $\\mathbf { V }$ follows distribution $N ( 0 , 2 / m )$ independently. ",
+        "bbox": [
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+        "type": "text",
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+        "text": "Comparison with LinUCB and NeuralUCB: Compared with linear contextual bandits in Section 2.1, Algorithm 1 has a distinct feature that it learns a deep neural network to obtain a deep representation of the raw data vectors and then performs UCB exploration. This deep representation allows our algorithm to characterize more intrinsic and latent information about the raw data $\\left\\{ \\mathbf { x } _ { t , k } \\right\\} _ { t \\in [ T ] , k \\in [ K ] } \\subset \\mathbb { R } ^ { d }$ . However, the increased complexity of the feature mapping $\\phi ( \\cdot ; { \\mathbf { w } } )$ also introduces great hardness in training. For instance, a recent work by Zhou et al. [52] also studied the neural contextual bandit problem, but different from (3.1), their algorithm (NeuralUCB) performs the UCB exploration on the entire network parameter space, which is $\\dot { \\mathbb { R } } ^ { \\widetilde { p } + d }$ , where $\\ddot { \\tilde { p } } = m + m d + ( L \\dot { \\bar { \\mathbf { \\alpha } } } 1 ) m ^ { 2 }$ . Note that in Zhou et al. [52], they need to compute the inverse eof a matrix $\\mathbf { Z } _ { t } \\in \\mathbb { R } ^ { ( \\widetilde { p } + d ) \\times ( \\widetilde { p } + d ) }$ , which is defined in a similar way to the matrix ${ \\bf A } _ { t }$ in our paper except that $\\mathbf { Z } _ { t }$ is defined based on the gradient of the network instead of the output of the last hidden layer as in (3.2). In sharp contrast, ${ \\bf A } _ { t }$ in our paper is only of size $d \\times d$ and thus is much more efficient and practical in implementation, which will be seen from our experiments in later sections. ",
+        "bbox": [
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+        "type": "text",
+        "text": "199 We note that there is also a similar algorithm to our Neural-LinUCB presented in Deshmukh et al.   \n200 [20], where they studied the self-supervised learning loss in contextual bandits with neural network   \n201 representation for computer vision problems. However, no regret analysis has been provided. When   \n202 the feature mapping $\\phi ( \\cdot ; { \\mathbf { w } } )$ is an identity function, the problem reduces to linear contextual bandits   \n203 where we directly use $\\mathbf { x } _ { t }$ as the feature vector. In this case, it is easy to see that Algorithm 1 reduces   \n204 to LinUCB [16] since we do not need to learn the representation parameter w anymore.   \n205 Comparison with Neural-Linear: The high-level idea of decoupling the representation and explo  \n206 ration in our algorithm is also similar to that of the Neural-Linear algorithm [38, 49], which trains a   \n207 deep neural network to learn a representation of the raw feature vectors, and then uses a Bayesian   \n208 linear regression to estimate the uncertainty in the bandit problem. However, these two algorithms   \n209 are significantly different since Neural-Linear [38] is a Thompson sampling based algorithm that   \n210 uses posterior sampling to estimate the weight parameter $\\pmb { \\theta } ^ { * }$ via Bayesian linear regression, whereas   \n211 Neural-LinUCB adopts upper confidence bound based techniques to estimate the weight $\\pmb { \\theta } ^ { * }$ . Never  \n212 theless, both algorithms share the same idea of deep representation and shallow exploration, and we   \n213 view our Neural-LinUCB algorithm as one instantiation of the Neural-Linear scheme. ",
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+        "type": "text",
+        "text": "214 4 Main Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "215 To analyze the regret bound of Algorithm 1, we first lay down some important assumptions on the   \n216 neural contextual bandit model. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Deep Representation and Shallow Exploration (Neural-LinUCB) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1: Input: regularization parameter $\\lambda > 0$ , number of total steps $T$ , episode length $H$ , exploration parameters $\\{ \\alpha _ { t } > 0 \\} _ { t \\in [ T ] }$   \n2: Initialization: $\\mathbf { A } _ { 0 } = \\lambda \\mathbf { I }$ , $\\mathbf { b } _ { 0 } = \\mathbf { 0 }$ ; entries of $\\pmb { \\theta } _ { 0 }$ follow $N ( 0 , 1 / d )$ , and $\\mathbf { w } ^ { ( 0 ) }$ is initialized as described in Section 3; $q = 1$ ; $\\mathbf { w } _ { 0 } = \\mathbf { w } ^ { ( 0 ) }$   \n3: for $t = 1 , \\dots , T$ do   \n4: receive feature vectors $\\left\\{ \\mathbf { x } _ { t , 1 } , \\ldots , \\mathbf { x } _ { t , K } \\right\\}$   \n5: choose arm $\\begin{array} { r } { a _ { t } \\ = \\ \\mathrm { a r g m a x } _ { k \\in [ K ] } \\pmb { \\theta } _ { t - 1 } ^ { \\top } \\phi ( \\mathbf { x } _ { t , k } ; \\mathbf { w } _ { t - 1 } ) \\ + \\alpha _ { t } \\| \\phi ( \\mathbf { x } _ { t , k } ; \\mathbf { w } _ { t - 1 } ) \\| _ { \\mathbf { A } _ { t - 1 } ^ { - 1 } } } \\end{array}$ , and obtain reward $\\widehat { r } _ { t }$   \n6: update ${ \\bf A } _ { t }$ and $\\mathbf { b } _ { t }$ as follows: $\\begin{array} { r l } & { \\mathbf { \\Phi } ^ { \\mathbf { A } } t = \\tilde { \\mathbf { A } } _ { t - 1 } + \\tilde { \\phi } ( \\mathbf { x } _ { t , a _ { t } } ; \\mathbf { w } _ { t - 1 } ) \\phi ( \\mathbf { x } _ { t , a _ { t } } ; \\mathbf { w } _ { t - 1 } ) ^ { \\top } } \\\\ & { \\mathbf { b } _ { t } = \\mathbf { b } _ { t - 1 } + \\hat { r } _ { t } \\phi ( \\mathbf { x } _ { t , a _ { t } } ; \\mathbf { w } _ { t - 1 } ) , } \\end{array}$ ,   \n7: update $\\pmb { \\theta } _ { t } = \\mathbf { A } _ { t } ^ { - 1 } \\mathbf { b } _ { t }$   \n8: if $\\mathrm { n o d } ( t , H ) = 0$ then   \n9: $\\mathbf { w } _ { t } \\gets$ output of Algorithm 2   \n10: $q = q + 1$   \n11: else   \n12: $\\mathbf { w } _ { t } = \\mathbf { w } _ { t - 1 }$   \n13: end if   \n14: end for   \n15: Output $\\mathbf { w } _ { T }$ ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Update Weight Parameters with Gradient Descent ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1: Input: initial point $\\mathbf { w } _ { q } ^ { ( 0 ) } = \\mathbf { w } ^ { ( 0 ) }$ , maximum iteration number $n$ , step size $\\eta _ { q }$ , and loss function   \ndefined in (3.5).   \n2: for $s = 1 , \\ldots , n$ do   \n3: $\\begin{array} { r } { \\mathbf { w } _ { q } ^ { ( s ) } = \\mathbf { w } _ { q } ^ { ( s - 1 ) } - \\eta _ { q } \\nabla _ { \\mathbf { w } } \\mathcal { L } _ { q } ( \\mathbf { w } _ { q } ^ { ( s - 1 ) } ) . } \\end{array}$   \n4: end for   \n5: Output w(n)q   \n7 Assumption 4.1. For all $i \\geq 1$ and $k \\in [ K ]$ , we assume that $\\| \\mathbf { x } _ { i , k } \\| _ { 2 } = 1$ and its entries satisfy   \n8 $[ { \\bf { x } } _ { i , k } ] _ { j } \\stackrel { - } { = } [ { \\bf { x } } _ { j , k } ] _ { j + d / 2 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "text",
+        "text": "19 The assumption that $\\| \\mathbf { x } _ { i , k } \\| _ { 2 } = 1$ is not essential and is only imposed for simplicity, which is also 20 used in Zou and $\\mathrm { G u }$ [53], Zhou et al. [52]. Finally, the condition on the entries of $\\mathbf { x } _ { i , k }$ is also mild since otherwise we could always construct 21 $\\mathbf { x } _ { i , k } ^ { \\prime } = [ \\mathbf { x } _ { i , k } ^ { \\top } , \\mathbf { x } _ { i , k } ^ { \\top } ] ^ { \\top } / \\sqrt { 2 }$ to replace it. An implication of ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.1 is that the initialization scheme in Algorithm 1 results in $\\phi ( \\mathbf { x } _ { i , k } ; \\mathbf { w } ^ { ( 0 ) } ) = \\mathbf { 0 }$ for all $i \\in [ T ]$ and $k \\in [ K ]$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "24 We assume the following stability condition on the spectral norm of the neural network gradient: ",
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+        "type": "text",
+        "text": "225 Assumption 4.2. There is a constant $\\ell _ { \\mathrm { L i p } } > 0$ such that it holds ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35b5d1a533a2cbb936460df839bd22f4a249c0863f54259893aff798251c8f90.jpg",
+        "text": "$$\n\\left\\| \\frac { \\partial \\phi } { \\partial \\mathbf { w } } ( \\mathbf { x } ; \\mathbf { w } _ { 0 } ) - \\frac { \\partial \\phi } { \\partial \\mathbf { w } } ( \\mathbf { x } ^ { \\prime } ; \\mathbf { w } _ { 0 } ) \\right\\| _ { 2 } \\leq \\ell _ { \\mathrm { L i p } } \\| \\mathbf { x } - \\mathbf { x } ^ { \\prime } \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "for all 226 $\\mathbf { x } , \\mathbf { x } ^ { \\prime } \\in \\{ \\mathbf { x } _ { i , k } \\} _ { i \\in [ T ] , k \\in [ K ] }$ ",
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+        "type": "text",
+        "text": "227 The inequality in Assumption 4.2 resembles the Lipschitz condition on the gradient of the neural   \n228 network. However, it is essentially different from the smoothness condition since here the gradient   \n229 is taken with respect to the neural network weights while the Lipschitz condition is imposed on the   \n230 feature parameter x. Similar conditions are widely made in nonconvex optimization [46, 10, 48], in   \n231 the name of first-order stability, which is essential to derive the convergence of alternating optimization   \n232 algorithms. Furthermore, Assumption 4.2 is only required on the $T K$ training data points and a   \n233 specific weight parameter $\\mathbf { w } _ { 0 }$ . Therefore, the condition will hold if the raw feature data lie in a   \n234 certain subspace of $\\mathbb { R } ^ { d }$ . We provided some further discussions in the supplementary material about   \n235 this assumption for interested readers.   \n236 In order to analyze the regret bound of Algorithm 1, we need to characterize the properties of the   \n237 deep neural network in (2.2) that is used to represent the feature vectors. Following a recent line of   \n238 research [27, 12, 7, 52], we define the covariance between two data point $\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { \\bar { d } }$ as follows. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1ca433195a3a22df3540af3ebe2aaaac2e2cd8d29ba8df51dd10b38683d4f874.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\widetilde { \\pmb { \\Sigma } } ^ { ( 0 ) } ( \\mathbf x , \\mathbf y ) = \\pmb { \\Sigma } ^ { ( 0 ) } ( \\mathbf x , \\mathbf y ) = \\mathbf x ^ { \\top } \\mathbf y , } \\\\ & { \\pmb { \\Lambda } ^ { ( l ) } ( \\mathbf x , \\mathbf y ) = \\left[ \\pmb { \\Sigma } ^ { l - 1 } ( \\mathbf x , \\mathbf x ) \\quad \\pmb { \\Sigma } ^ { l - 1 } ( \\mathbf x , \\mathbf y ) \\right] , } \\\\ & { \\pmb { \\Sigma } ^ { ( l ) } ( \\mathbf x , \\mathbf y ) = 2 \\mathbb { E } _ { ( u , v ) \\sim N ( \\mathbf 0 , \\mathbf { A } ^ { ( l - 1 ) } ( \\mathbf x , \\mathbf y ) ) } [ \\sigma ( u ) \\sigma ( v ) ] , } \\\\ & { \\widetilde { \\pmb { \\Sigma } } ^ { ( l ) } ( \\mathbf x , \\mathbf y ) = 2 \\widetilde { \\pmb { \\Sigma } } ^ { ( l - 1 ) } ( \\mathbf x , \\mathbf y ) \\mathbb { E } _ { u , v } [ \\dot { \\sigma } ( u ) \\dot { \\sigma } ( v ) ] + \\pmb { \\Sigma } ^ { ( l ) } ( \\mathbf x , \\mathbf y ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where 239 the ne240 $( u , v ) \\sim N ( \\mathbf { 0 } , \\mathbf { \\Lambda } \\Lambda ^ { ( l - 1 ) } ( \\mathbf { x } , \\mathbf { y } ) )$ , arix $\\dot { \\sigma } ( \\cdot )$ rivative of activation functiobased on all feature vectors $\\sigma ( \\cdot )$ $\\mathbf { H } \\in \\mathbb { R } ^ { T K \\times T K }$ $\\{ \\mathbf { x } _ { t , k } \\} _ { t \\in [ T ] , k \\in [ K ] }$ 241 Renumbering $\\{ \\mathbf { x } _ { t , k } \\} _ { t \\in [ T ] , k \\in [ K ] }$ as $\\{ \\mathbf { x } _ { i } \\} _ { i = 1 , \\dots , T K }$ , then each entry $\\mathbf { H } _ { i j }$ is defined as ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fb231f3d5356d34d8d8c49952e6f3a715cd685b08edbbe90de4ddb6668893cc.jpg",
+        "text": "$$\n\\mathbf { H } _ { i j } = \\frac { 1 } { 2 } \\big ( \\widetilde { \\boldsymbol { \\Sigma } } ^ { ( L ) } ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) + \\boldsymbol { \\Sigma } ^ { ( L ) } ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "42 for all $i , j \\in [ T K ]$ . Based on the above definition, we impose the following assumption on $\\mathbf { H }$ ",
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+        "type": "text",
+        "text": "43 Assumption 4.3. The neural tangent kernel defined in (4.2) is positive definite, i.e., $\\lambda _ { \\operatorname* { m i n } } ( \\mathbf { H } ) \\geq \\lambda _ { 0 }$   \n244 for some constant $\\lambda _ { 0 } > 0$ .   \n245 Assumption 4.3 essentially requires the neural tangent kernel matrix $\\mathbf { H }$ to be non-singular, which is   \n246 a mild condition and also imposed in other related work [21, 7, 12, 52]. Moreover, it is shown that   \n247 Assumption 4.3 can be easily derived from Assumption 4.1 for two-layer ReLU networks [37, 53].   \n248 Therefore, Assumption 4.3 is mild or even negligible given the non-degeneration assumption on the   \n249 feature vectors. Also note that matrix $\\mathbf { H }$ is only defined based on layers $l = 1 , \\ldots , L$ of the neural   \n250 network, and does not depend on the output layer $\\pmb \\theta$ . It is easy to extend the definition of $\\mathbf { H }$ to the   \n251 NTK matrix defined on all layers including the output layer $\\pmb \\theta$ , which would also be positive definite   \n252 by Assumption 4.3 and the recursion in (4.2).   \n253 Before we present the regret analysis of the neural contextual bandit, we need to modify the regret   \n254 defined in (2.1) to account for the randomness of the neural network initialization. For a fixed time   \n255 horizon $T$ , we define the regret of Algorithm 1 as follows. ",
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+        "type": "text",
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+        "bbox": [
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+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e138cc6ba15066bbaecd1cf6411285fc128bb09ba373e687791499a7b87b9e82.jpg",
+        "text": "$$\nR _ { T } = \\mathbb { E } \\bigg [ \\sum _ { t = 1 } ^ { T } \\big ( \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } ^ { * } } ) - \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } } ) \\big ) \\big | \\mathbf { w } ^ { ( 0 ) } \\bigg ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "256 where the expectation is taken over the randomness of the reward noise. Note that $R _ { T }$ defined in (4.3)   \n257 is still a random variable since the initialization of Algorithm 2 is randomly generated. ",
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+    {
+        "type": "text",
+        "text": "258 Now we are going to present the regret bound of the proposed algorithm. ",
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+        "type": "text",
+        "text": "59 Theorem 4.4. Suppose Assumptions 4.1, 4.2 and 4.3 hold. Assume that $\\lVert \\pmb { \\theta } ^ { * } \\rVert _ { 2 } \\leq M$ for some   \n0 positive constant $M > 0$ . For any $\\delta \\in ( 0 , 1 )$ , let us choose $\\alpha _ { t }$ in Neural-LinUCB as ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ccf3a022ad345f8b9e1e117135312146f7c078729730e4c25e7b626c7abe5a8.jpg",
+        "text": "$$\n\\alpha _ { t } = \\nu \\sqrt { 2 \\big ( d \\log ( 1 + t \\log ( H K ) / \\lambda ) + \\log ( 1 / \\delta ) \\big ) } + \\lambda ^ { 1 / 2 } M .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "261 We choose the step size $\\eta _ { q }$ of Algorithm 2 as ",
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a0c80ba7aa5ba80ea8ffcf392b283294329a6a569c484aefbd52aab86c2c2b9.jpg",
+        "text": "$$\n\\eta _ { q } \\leq C _ { 0 } \\big ( d ^ { 2 } m n T ^ { 5 . 5 } L ^ { 6 } \\log ( T K / \\delta ) \\big ) ^ { - 1 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "262 and the width of the neural network satisfies $m = \\mathrm { p o l y } ( L , d , 1 / \\delta , H , \\log ( T K / \\delta ) )$ . With probability   \n263 at least $1 - \\delta$ over the randomness of the initialization of the neural network, it holds that ",
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+    {
+        "type": "equation",
+        "img_path": "images/12ee80d826958d4079dc3a4c3a914f1252140256056fcf795f509a38f52cb3e7.jpg",
+        "text": "$$\nR _ { T } \\leq C _ { 1 } \\alpha _ { T } \\sqrt { T d \\log \\left( 1 + \\frac { T G ^ { 2 } } { \\lambda d } \\right) } + \\frac { C _ { 2 } \\ell _ { \\mathrm { L i p } } L ^ { 3 } d ^ { 5 / 2 } T \\sqrt { \\log m \\log ( \\frac { 1 } { \\delta } ) \\log ( \\frac { T K } { \\delta } ) } \\| \\mathbf { r } - \\widetilde { \\mathbf { r } } \\| _ { \\mathbf { H } ^ { - 1 } } } { m ^ { 1 / 6 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "264 where $\\{ C _ { i } \\} _ { i = 0 , 1 , 2 }$ are absolute constants independent of the problem parameters, $\\begin{array} { r l } { \\mathbf { r } } & { { } = } \\end{array}$   \n265 $( r ( \\mathbf { x } _ { 1 } ) , r ( \\mathbf { x } _ { 2 } ) , \\ldots , r ( \\mathbf { x } _ { T K } ) ) ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { T K }$ and $\\widetilde { \\textbf { r } } = ( f ( \\mathbf { x } _ { 1 } ; \\pmb { \\theta } _ { 0 } , \\mathbf { w } _ { 0 } ) , \\dots , f ( \\mathbf { x } _ { T K } ; \\pmb { \\theta } _ { T - 1 } , \\mathbf { w } _ { T - 1 } ) ) ^ { \\top } \\ \\in$   \n266 $\\mathbb { R } ^ { T K }$ , and $\\| \\mathbf { r } \\| _ { \\mathbf { A } } = \\sqrt { \\mathbf { r } ^ { \\top } \\mathbf { A } \\mathbf { r } }$ .   \n267 Remark 4.5. Theorem 4.4 shows that the regret of Algorithm 1 can be bounded by two parts: the   \n268 first part is of order $\\widetilde { O } ( \\sqrt { T } )$ , which resembles the regret bound of linear contextual bandits [1]; the   \n269 second part is of order $\\widetilde { O } ( m ^ { - 1 / 6 } T \\sqrt { ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) } )$ , which depends on the estimation error   \n270 of the neural network $f$ e efor the reward generating function $r$ and the neural tangent kernel $\\mathbf { H }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "It is worth noting that our theoretical analysis depends on the reward structure assumption that $r ( \\cdot ) = \\langle \\theta \\ast , \\psi ( \\cdot ) \\rangle$ . However, the linear structure between $\\pmb { \\theta } \\ast$ and $\\psi ( \\cdot )$ is not essential. As long as the deep representation of the feature vector and the uncertainty weight parameter can be decoupled, Algorithm 1 can be easily extended to settings with milder assumptions on the reward structure such as generalized linear models [41, 24, 35, 28]. For more general bandit models where no assumption is imposed to the reward generating function, it is still unclear whether the decoupled deep representation and shallow exploration would work especially in cases a thorough exploration may be needed. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Based on the result in Theorem 4.4, we can easily verify the following conclusion: ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Corollary 4.6. Under the same conditions of Theorem 4.4, if we choose a sufficiently overparameterized neural network mapping $\\phi ( \\cdot )$ such that $m \\geq T ^ { 3 }$ , then the regret of Algorithm 1 is $R _ { T } = { \\widetilde { O } } ( { \\sqrt { T } } { \\sqrt { ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) } } )$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark 4.7. For the ease of presentation, let us denote $\\mathcal { E } : = \\| \\mathbf { r } - \\widetilde { \\mathbf { r } } \\| _ { \\mathbf { H } ^ { - 1 } }$ . If we have $\\mathcal { E } = O ( 1 )$ , the total regret in Theorem 4.4 becomes $\\widetilde { O } ( \\sqrt { T } )$ which matches the regret of linear contextual bandits [1]. We remark that there is a similar assumption in [52] where they assume that $\\mathbf { r } ^ { \\top } \\mathbf { H } ^ { - 1 } \\mathbf { r }$ can be upper bounded by a constant. They show that this term can be bounded by the RKHS norm of $\\mathbf { r }$ if it belongs to the RKHS induced by the neural tangent kernel [6, 7, 33]. In addition, $\\mathcal { E }$ here is the difference between the true reward function and the neural network function, which can also be small if the deep neural network function well approximates the reward generating function $r ( \\cdot )$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "290 5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
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+        "text": "291   \n292   \n293   \n294   \n295   \n296   \n297   \n298   \n299   \n300   \n301   \n302   \n303   \n304   \n305 ",
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+        "text": "In this section, we provide empirical evaluations of Neural-LinUCB on real-world datasets. As we have discussed in Section 3, Neural-LinUCB could be viewed as an instantiation of the NeuralLinear scheme studied in Riquelme et al. [38] except that we use the UCB exploration instead of the posterior sampling exploration therein. Note that there has been an extensive comparison [38] of the Neural-Linear methods with many other baselines such as greedy algorithms, Variational Inference, Expectation-Propagation, Bayesian Non-parametrics and so on. Therefore, we do not seek a thorough empirical comparison of Neural-LinUCB with all existing bandits algorithms. We refer readers who are interested in the performance of Neural-Linear methods with deep representation and shallow exploration compared with a vast of baselines in the literature to the benchmark study by Riquelme et al. [38]. In this experiment, we only aim to show the advantages of our algorithm over the following baselines: (1) Neural-Linear [38]; (2) LinUCB [16], which does not have a deep representation of the feature vectors; and (3) NeuralUCB [52], which performs UCB exploration on all the parameters of the neural network instead of the shallow exploration used in our paper. All numerical experiments were run on a workstation with Intel(R) Xeon(R) CPU E5-2637 v4 $@$ 3.50GHz. ",
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+        "text": "Datasets: we evaluate the performances of all algorithms on bandit problems created from real-world data. Specifically, following the experimental setting in Zhou et al. [52],we use datasets (Shuttle) Statlog, Magic and Covertype from UCI machine learning repository [23], and the MINST dataset from LeCun et al. [31]. The details of these datasets are presented in Table 1. In Table 1, each instance represents a feature vector $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ that is associated with one of the $K$ arms, and dimension $d$ is the number of attributes in each instance. ",
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+        "table_caption": [
+            "Table 1: Specifications of datasets from the UCI machine learning repository used in this paper. "
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+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Statlog</td><td>Magic</td><td>Covertype</td><td>MNIST</td></tr><tr><td>Number of attributes</td><td>9</td><td>11</td><td>54</td><td>784</td></tr><tr><td>Number of arms</td><td>7</td><td>2</td><td>7</td><td>10</td></tr><tr><td>Number of instances</td><td>58,000</td><td>19,020</td><td>581,012</td><td>60,000</td></tr></table>",
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+            "Figure 1: The cumulative regrets of LinUCB, NeuralUCB, Neural-Linear and Neural-LinUCB over 15, 000 rounds. Experiments are averaged over 10 repetitions. "
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+        "text": "311 Implementations: for LinUCB, we follow the setting in Li et al. [34] to use disjoint models   \n312 for different arms. For neural network based algorithms such as NeuralUCB, Neural-Linear and   \n313 Neural-LinUCB, we use a ReLU neural network defined as in (2.2) with $L = 2$ and 2000 for the   \n314 UCI datasets (Statlog, Magic, Covertype). Thus the neural network weights are $\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }$   \n315 $\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { k \\times m }$ , and $\\pmb { \\theta } \\in \\mathbb { R } ^ { \\widetilde { k } }$ respectively, where $k = 1 0 0$ , $m = 2 0 0 0$ , and $d$ is the dimension of   \n316 features in the corresponding task. Since the problem size of the MNIST dataset is larger, inspired   \n317 by Hinton and Salakhutdinov [26], we use a deeper NN and set $L = 3$ , $k = 1 0 0$ and $m = 1 0 0$ ,   \n318 with weights $\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }$ , $\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { m \\times m }$ , $\\mathbf { W _ { 3 } } \\in \\mathbb { R } ^ { k \\times m }$ , and $\\pmb \\theta \\in \\mathbb { R } ^ { k }$ . We set the time horizon   \n319 $T = 1 5 , 0 0 0$ , which is the total number of rounds for each algorithm on each dataset. We use   \n320 gradient decent to optimize the network weights, with a step size $\\eta _ { q } = 1 \\mathrm { e } { - 5 }$ and maximum iteration   \n321 number $n = 1 , 0 0 0$ . To speed up the training process, the network parameter w is updated every   \n322 $H = 1 0 0$ rounds starting from round 2000. We also apply early stopping when the loss difference   \n323 of two consecutive iterations is smaller than a threshold of 1e-6. We set $\\lambda = 1$ and $\\alpha _ { t } = 0 . 0 2$   \n324 for all algorithms, $t \\in [ T ]$ . Following the setting in Riquelme et al. [38], we use round-robin to   \n325 independently select each arm for 3 times at the beginning of each algorithm. For NeuralUCB, since   \n326 it is computationally unaffordable to perform the original UCB exploration as displayed in Zhou et al.   \n327 [52], we follow their experimental setting to replace the matrix $\\mathbf { Z } _ { t } \\in \\mathbb { R } ^ { ( d + \\widetilde { p } ) \\times \\widetilde { ( } d + \\widetilde { p } ) }$ in Zhou et al.   \n328 [52] with its diagonal matrix. ",
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+        "text": "Results: we plot the cumulative regret of all algorithms versus round in Figures 1(a), 1(b) and 1(c) for UCI datasets and in Figure 1(d) for MNIST. The results are reported based on the average of 10 repetitions over different random shuffles of the datasets. It can be seen that algorithms based on neural network representations (NeuralUCB, Neural-Linear and Neural-LinUCB) consistently outperform the linear contextual bandit method LinUCB, which shows that linear models may lack representation power and find biased estimates for the underlying reward generating function. Furthermore, our proposed Neural-LinUCB achieves a comparable regret with NeuralUCB in all experiments despite the fact that our algorithm only explores in the output layer of the neural network, which is more computationally efficient as we will show in the sequel.The results in our experiment are well aligned with our theory that deep representation and shallow exploration are sufficient to guarantee a good performance of neural contextual bandit algorithms, which is also consistent with the findings in existing literature [38] that decoupling the representation learning and uncertainty estimation improves the performance. ",
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+        "text": "We also conducted experiments to study the effects of different widths of deep neural networks on the regret performance and to show the computational efficiency of Neural-LinUCB compared with existing neural bandit algorithms. Due to the space limit, we defer the results to Appendix A. ",
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+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
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+        "text": "In this paper, we propose a new neural contextual bandit algorithm called Neural-LinUCB, which uses the hidden layers of a ReLU neural network as a deep representation of the raw feature vectors and performs UCB type exploration on the last layer of the neural network. By incorporating techniques in liner contextual bandits and neural tangent kernels, we prove that the proposed algorithm achieves a sublinear regret when the width of the network is sufficiently large. This is the first regret analysis of neural contextual bandit algorithms with deep representation and shallow exploration, which have been observed in practice to work well on many benchmark bandit problems [38]. We also conducted experiments on real-world datasets to demonstrate the advantage of the proposed algorithm over LinUCB and existing neural contextual bandit algorithms. ",
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+        "text": "355 References [1] Yasin Abbasi-Yadkori, Dávid Pál, and Csaba Szepesvári. Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pages 2312–2320, 2011. [2] Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning, pages 1638–1646, 2014. [3] Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. In Advances in neural information processing systems, pages 6155–6166, 2019. [4] Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over-parameterization. In International Conference on Machine Learning, pages 242–252, 2019. [5] Robin Allesiardo, Raphaël Féraud, and Djallel Bouneffouf. A neural networks committee for the contextual bandit problem. 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URL https://openreview.net/forum?id=BJgQfkSYDS.   \n[46] Zhaoran Wang, Han Liu, and Tong Zhang. Optimal computational and statistical rates of convergence for sparse nonconvex learning problems. Annals of statistics, 42(6):2164, 2014.   \n[47] Pan Xu and Quanquan Gu. A finite-time analysis of q-learning with neural network function approximation. In International Conference on Machine Learning, 2020.   \n[48] Pan Xu, Jian Ma, and Quanquan Gu. Speeding up latent variable gaussian graphical model estimation via nonconvex optimization. In Advances in Neural Information Processing Systems, pages 1933–1944, 2017.   \n[49] Tom Zahavy and Shie Mannor. Deep neural linear bandits: Overcoming catastrophic forgetting through likelihood matching. arXiv preprint arXiv:1901.08612, 2019.   \n[50] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?id=Sy8gdB9xx.   \n[51] Weitong Zhang, Dongruo Zhou, Lihong Li, and Quanquan Gu. Neural thompson sampling. arXiv preprint arXiv:2010.00827, 2020.   \n[52] Dongruo Zhou, Lihong Li, and Quanquan Gu. Neural contextual bandits with ucb-based exploration. In International Conference on Machine Learning, 2020.   \n[53] Difan Zou and Quanquan Gu. An improved analysis of training over-parameterized deep neural networks. In Advances in Neural Information Processing Systems, pages 2053–2062, 2019.   \n[54] Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep relu networks. arXiv preprint arXiv:1811.08888, 2018. ",
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