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+# DRNAS: DIRICHLET NEURAL ARCHITECTURE SEARCH
+
+Xiangning Chen1∗ Ruochen Wang1∗ Minhao Cheng1∗ Xiaocheng Tang2 Cho-Jui Hsieh1
+
+1Department of Computer Science, UCLA, 2DiDi AI Labs   
+{xiangning, chohsieh}@cs.ucla.edu {ruocwang, mhcheng}@ucla.edu   
+xiaochengtang@didiglobal.com
+
+# ABSTRACT
+
+This paper proposes a novel differentiable architecture search method by formulating it into a distribution learning problem. We treat the continuously relaxed architecture mixing weight as random variables, modeled by Dirichlet distribution. With recently developed pathwise derivatives, the Dirichlet parameters can be easily optimized with gradient-based optimizer in an end-to-end manner. This formulation improves the generalization ability and induces stochasticity that naturally encourages exploration in the search space. Furthermore, to alleviate the large memory consumption of differentiable NAS, we propose a simple yet effective progressive learning scheme that enables searching directly on large-scale tasks, eliminating the gap between search and evaluation phases. Extensive experiments demonstrate the effectiveness of our method. Specifically, we obtain a test error of $2 . 4 6 \%$ for CIFAR-10, $2 3 . 7 \%$ for ImageNet under the mobile setting. On NASBench-201, we also achieve state-of-the-art results on all three datasets and provide insights for the effective design of neural architecture search algorithms.
+
+# 1 INTRODUCTION
+
+Recently, Neural Architecture Search (NAS) has attracted lots of attentions for its potential to democratize deep learning. For a practical end-to-end deep learning platform, NAS plays a crucial role in discovering task-specific architecture depending on users’ configurations (e.g., dataset, evaluation metric, etc.). Pioneers in this field develop prototypes based on reinforcement learning (Zoph & Le, 2017), evolutionary algorithms (Real et al., 2019) and Bayesian optimization (Liu et al., 2018). These works usually incur large computation overheads, which make them impractical to use. More recent algorithms significantly reduce the search cost including one-shot methods (Pham et al., 2018; Bender et al., 2018), a continuous relaxation of the space (Liu et al., 2019) and network morphisms (Cai et al., 2018). In particular, Liu et al. (2019) proposes a differentiable NAS framework - DARTS, converting the categorical operation selection problem into learning a continuous architecture mixing weight. They formulate a bi-level optimization objective, allowing the architecture search to be efficiently performed by a gradient-based optimizer.
+
+While current differentiable NAS methods achieve encouraging results, they still have shortcomings that hinder their real-world applications. Firstly, several works have cast doubt on the stability and generalization of these differentiable NAS methods (Chen & Hsieh, 2020; Zela et al., 2020a). They discover that directly optimizing the architecture mixing weight is prone to overfitting the validation set and often leads to distorted structures, e.g., searched architectures dominated by parameter-free operations. Secondly, there exist disparities between the search and evaluation phases, where proxy tasks are usually employed during search with smaller datasets or shallower and narrower networks, due to the large memory consumption of differentiable NAS.
+
+In this paper, we propose an effective approach that addresses the aforementioned shortcomings named Dirichlet Neural Architecture Search (DrNAS). Inspired by the fact that directly optimizing the architecture mixing weight is equivalent to performing point estimation (MLE/MAP) from a probabilistic perspective, we formulate the differentiable NAS as a distribution learning problem instead, which naturally induces stochasticity and encourages exploration. Making use of the probability simplex property of the Dirichlet samples, DrNAS models the architecture mixing weight as random variables sampled from a parameterized Dirichlet distribution. Optimizing the Dirichlet objective can thus be done efficiently in an end-to-end fashion, by employing the pathwise derivative estimators to compute the gradient of the distribution (Martin Jankowiak, 2018). A straightforward optimization, however, turns out to be problematic due to the uncontrolled variance of the Dirichlet, i.e., too much variance leads to training instability and too little variance suffers from insufficient exploration. In light of that, we apply an additional distance regularizer directly on the Dirichlet concentration parameter to strike a balance between the exploration and the exploitation. We further derive a theoretical bound showing that the constrained distributional objective promotes stability and generalization of architecture search by implicitly controlling the Hessian of the validation error.
+
+Furthermore, to enable a direct search on large-scale tasks, we propose a progressive learning scheme, eliminating the gap between the search and evaluation phases. Based on partial channel connection (Xu et al., 2020), we maintain a task-specific super-network of the same depth and number of channels as the evaluation phase throughout searching. To prevent loss of information and instability induced by partial connection, we divide the search phase into multiple stages and progressively increase the channel fraction via network transformation (Chen et al., 2016). Meanwhile, we prune the operation space according to the learnt distribution to maintain the memory efficiency.
+
+We conduct extensive experiments on different datasets and search spaces to demonstrate DrNAS’s effectiveness. Based on the DARTS search space (Liu et al., 2019), we achieve an average error rate of $2 . 4 6 \%$ on CIFAR-10, which ranks top amongst NAS methods. Furthermore, DrNAS achieves superior performance on large-scale tasks such as ImageNet. It obtains a top-1/5 error of $2 3 . 7 \% / 7 . 1 \%$ , surpassing the previous state-of-the-art $( 2 4 . 0 \% / 7 . 3 \% )$ under the mobile setting. On NAS-Bench201 (Dong & Yang, 2020), we also set new state-of-the-art results on all three datasets with low variance. Our code is available at https://github.com/xiangning-chen/DrNAS.
+
+# 2 THE PROPOSED APPROACH
+
+In this section, we first briefly review differentiable NAS setups and generalize the formulation to motivate distribution learning. We then layout our proposed DrNAS and describe its optimization in section 2.2. In section 2.3, we provide a generalization result by showing that our method implicitly regularizes the Hessian norm over the architecture parameter. The progressive architecture learning method that enables direct search is then described in section 2.4.
+
+# 2.1 PRELIMINARIES: DIFFERENTIABLE ARCHITECTURE SEARCH
+
+Cell-Based Search Space The cell-based search space is constructed by replications of normal and reduction cells (Zoph et al., 2018; Liu et al., 2019). A normal cell keeps the spatial resolution while a reduction cell halves it but doubles the number of channels. Every cell is represented by a DAG with $N$ nodes and $E$ edges, where every node represents a latent representation $\mathbf { x } ^ { i }$ and every edge $( i , j )$ is associated with an operations $o ^ { ( i , j ) }$ (e.g., max pooling or convolution) selected from a predefined candidate space $\mathcal { O }$ . The output of a node is a summation of all input flows, i.e., $\begin{array} { r } { \mathbf { x } ^ { j } = \sum _ { i < j } o ^ { ( i , j ) } ( \mathbf { x } ^ { i } ) } \end{array}$ and a concatenation of intermediate node outputs, i.e., $c o n c a t ( \mathbf { x } ^ { 2 } , . . . , \mathbf { x } ^ { N - 1 } )$ , composes the cell output, where the first two input nodes $\mathbf { x } ^ { 0 }$ and ${ \hat { \mathbf { x } } } ^ { 1 }$ are fixed to be the outputs of previous two cells.
+
+Gradient-Based Search via Continuous Relaxation To enable gradient-based optimization, Liu et al. (2019) apply a continuous relaxation to the discrete space. Concretely, the information passed from node $i$ to node $j$ is computed by a weighted sum of all operations alone the edge, forming a mixed-operation $\begin{array} { r } { \hat { o } ^ { ( i , j ) } ( x ) = \sum _ { o \in \mathcal { O } } \theta _ { o } ^ { ( i , j ) } o ( x ) } \end{array}$ . The operation mixing weight $\theta ^ { ( i , j ) }$ is defined over the probability simplex and its magnitude represents the strength of each operation. Therefore, the architecture search can be cast as selecting the operation associated with the highest mixing weight for each edge. To prevent abuse of terminology, we refer to $\theta$ as the architecture/operation mixing weight, and concentration parameter $\beta$ in DrNAS as the architecture parameter throughout the paper.
+
+Bilevel-Optimization with Simplex Constraints With continuous relaxation, the network weight $w$ and operation mixing weight $\theta$ can be jointly optimized by solving a constraint bi-level optimization
+
+problem:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } _ { v a l } ( w ^ { * } , \theta ) \mathrm { ~ s . t . ~ } w ^ { * } = \underset { w } { \arg \operatorname* { m i n } } \mathcal { L } _ { t r a i n } ( w , \theta ) , \sum _ { o = 1 } ^ { | \mathcal { O } | } \theta _ { o } ^ { ( i , j ) } = 1 , \forall ( i , j ) , i < j ,
+$$
+
+where the simplex constraint $\begin{array} { r } { \sum _ { o = 1 } ^ { | \mathcal { O } | } \theta _ { o } ^ { ( i , j ) } = 1 } \end{array}$ can be either solved explicitly via Lagrangian function (Li et al., 2020), or eliminated by substitution method (e.g., $\theta = S o f t m a x ( \alpha ) , \alpha \ \in$ $\mathcal { R } ^ { | \mathcal { O } | \times | E | } ,$ ) (Liu et al., 2019). In the next section we describe how this generalized formulation motivates our method.
+
+# 2.2 DIFFERENTIABLE ARCHITECTURE SEARCH AS DISTRIBUTION LEARNING
+
+Learning a Distribution over Operation Mixing Weight Previous differentiable architecture search methods view the operation mixing weight $\theta$ as learnable parameters that can be directly optimized (Liu et al., 2019; Xu et al., 2020; Li et al., 2020). This has been shown to cause $\theta$ to overfit the validation set and thus induce large generalization error (Zela et al., 2020a;b; Chen & Hsieh, 2020). We recognize that this treatment is equivalent to performing point estimation (e.g., MLE/MAP) of $\theta$ in probabilistic view, which is inherently prone to overfitting (Bishop, 2016; Gelman et al., 2004). Furthermore, directly optimizing $\theta$ lacks sufficient exploration in the search space, and thus cause the search algorithm to commit to suboptimal paths in the DAG that converges faster at the beginning but plateaus quickly (Shu et al., 2020).
+
+Based on these insights, we formulate the differentiable architecture search as a distribution learning problem. The operation mixing weight $\theta$ is treated as random variables sampled from a learnable distribution. Formally, let $q ( \theta | \beta )$ denote the distribution of $\theta$ parameterized by $\beta$ . The bi-level objective is then given by:
+
+$$
+\operatorname* { m i n } _ { \beta } E _ { q ( \theta | \beta ) } \big [ \mathcal { L } _ { v a l } ( w ^ { * } , \theta ) \big ] + \lambda d ( \beta , \hat { \beta } ) \mathrm { ~ s . t . ~ } w ^ { * } = \operatorname * { a r g m i n } _ { w } \mathcal { L } _ { t r a i n } ( w , \theta ) .
+$$
+
+where $d ( \cdot , \cdot )$ is a distance function. Since $\theta$ lies on the probability simplex, we select Dirichlet distribution to model its behavior, i.e., $q ( \theta | \beta ) \sim D i r ( \beta )$ , where $\beta$ represents the Dirichlet concentration parameter. Dirichlet distribution is a widely used distribution over the probability simplex (Joo et al., 2019; David M. Blei, 2003; Lee et al., 2020; Kessler et al., 2019), and it enjoys nice properties that enables gradient-based training (Martin Jankowiak, 2018).
+
+The concentration parameter $\beta$ controls the sampling behavior of Dirichlet distribution and is crucial in balancing exploration and exploitation during the search phase. Let $\beta _ { o }$ denote the concentration parameter assign to operation $o$ . When $\beta _ { o } \ll 1$ for most $o = 1 \sim | \mathcal { O } |$ , Dirichlet tends to produce sparse samples with high variance, reducing the training stability; when $\beta _ { o } \gg 1$ for most $o = 1 \sim | \mathcal { O } |$ , the samples will be dense with low variance, leading to insufficient exploration. Therefore, we add a penalty term in the objective (2) to regularize the distance between $\beta$ and the anchor $\hat { \beta } = 1$ , which corresponds to the symmetric Dirichlet. In section 2.3, we also derive a theoretical bound showing that our formulation additionally promotes stability and generalization of the architecture search by implicitly regularizing the Hessian of validation loss w.r.t. architecture parameters.
+
+Learning Dirichlet Parameters via Pathwise Derivative Estimator Optimizing objective (2) with gradient-based methods requires back-propagation through stochastic nodes of Dirichlet samples. The commonly used reparameterization trick does not apply to Dirichlet distribution, therefore we approximate the gradient of Dirichlet samples via pathwise derivative estimators (Martin Jankowiak, 2018)
+
+$$
+\frac { d \theta _ { i } } { d \beta _ { j } } = - \frac { \frac { \partial F _ { B e t a } } { \partial \beta _ { j } } \big ( \theta _ { j } \big | \beta _ { j } , \beta _ { t o t } - \beta _ { j } \big ) } { f _ { B e t a } \big ( \theta _ { j } \big | \beta _ { j } , \beta _ { t o t } - \beta _ { j } \big ) } \times \big ( \frac { \delta _ { i j } - \theta _ { i } } { 1 - \theta _ { j } } \big ) \quad i , j = 1 , . . . , \vert \mathcal { O } \vert ,
+$$
+
+where $F _ { B e t a }$ and $f _ { B e t a }$ denote the CDF and PDF of beta distribution respectively, $\delta _ { i j }$ is the indicator function, and $\beta _ { t o t }$ is the sum of concentrations. $F _ { B e t a }$ is the iregularised incomplete beta function, for which its gradient can be computed by simple numerical approximation. We refer to (Martin Jankowiak, 2018) for the complete derivations.
+
+Joint Optimization of Model Weight and Architecture Parameter With pathwise derivative estimator, the model weight $w$ and concentration $\beta$ can be jointly optimized with gradient descent. Concretely, we draw a sample $\theta \sim D i r ( \beta )$ for every forward pass, and the gradients can be obtained easily through backpropagation. Following DARTS (Liu et al., 2019), we approximate $w ^ { * }$ in the lower level objective of equation 2 with one step of gradient descent, and run alternative updates between $w ^ { * }$ and $\beta$ .
+
+Selecting the Best Architecture At the end of the search phase, a learnt distribution of operation mixing weight is obtained. We then select the best operation for each edge by the most likely operation in expectation:
+
+$$
+o ^ { ( i , j ) } = \arg \operatorname* { m a x } _ { o \in \mathcal { O } } E _ { q ( \theta _ { o } ^ { ( i , j ) } | \beta ^ { ( i , j ) } ) } \left[ \theta _ { o } ^ { ( i , j ) } \right] .
+$$
+
+In the Dirichlet case, the expectation term is simply the Dirichlet mean P o0 β(i,j)o0 . Note that under the distribution learning framework, we are able to sample a wide range of architectures from the learnt distribution. This property alone has many potentials. For example, in practical settings where both accuracy and latency are concerned, the learnt distribution can be used to find architectures under resource restrictions in a post search phase. We leave these extensions to future work.
+
+# 2.3 THE IMPLICIT REGULARIZATION ON HESSIAN
+
+It has been observed that the generalization error of differentiable NAS is highly related to the dominant eigenvalue of the Hessian of validation loss w.r.t. architecture parameter. Several recent works report that the large dominant eigenvalue of $\nabla _ { \theta } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w , \theta )$ in DARTS results in poor generalization performance (Zela et al., 2020a; Chen & Hsieh, 2020). Our objective (2) is the Lagrangian function of the following constraint objective:
+
+$$
+\operatorname* { m i n } _ { \beta } E _ { q ( \theta | \beta ) } \left[ \mathcal { L } _ { v a l } ( w ^ { * } , \theta ) \right] \mathrm { ~ s . t . ~ } w ^ { * } = \arg \operatorname* { m i n } _ { w } \mathcal { L } _ { t r a i n } ( w , \theta ) , \ d ( \beta , \hat { \beta } ) \leq \delta ,
+$$
+
+Here we derive an approximated lower bound based on (5), which demonstrates that our method implicitly controls this Hessian matrix.
+
+Proposition 1 Let $d ( { \boldsymbol { \beta } } , { \hat { \boldsymbol { \beta } } } ) = \| { \boldsymbol { \beta } } - { \hat { \boldsymbol { \beta } } } \| _ { 2 } \leq \delta$ and $\hat { \beta } = 1$ in the bi-level formulation (5). Let $\mu$ denote the mean under the Laplacian approximation of Dirichlet. If $\nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu )$ is Positive Semi-definite, the upper-level objective can be approximated bounded by:
+
+$$
+E _ { q ( \theta | \beta ) } ( \mathcal { L } _ { v a l } ( w , \theta ) ) \gtrsim \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \frac { 1 } { 2 } ( \frac { 1 } { 1 + \delta } ( 1 - \frac { 2 } { | \mathcal { O } | } ) + \frac { 1 } { | \mathcal { O } | } \frac { 1 } { 1 + \delta } ) t r \big ( \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \big )
+$$
+
+with:
+
+$$
+\tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) = \mathcal { L } _ { v a l } ( w ^ { * } , S o f t m a x ( \mu ) ) , \mu _ { o } = \log \beta _ { o } - \frac { 1 } { | \mathcal { O } | } \sum _ { o ^ { \prime } } \log \beta _ { o ^ { \prime } } , o = 1 , \ldots , | \mathcal { O } | .
+$$
+
+This proposition is driven by the Laplacian approximation to the Dirichlet distribution (MacKay, 1998; Akash Srivastava, 2017). The lower bound (6) indicates that minimizing the expected validation loss controls the trace norm of the Hessian matrix. Empirically, we observe that DrNAS always maintains the dominant eigenvalue of Hessian at a low level (Appendix A.4). The detailed proof can be found in Appendix A.1.
+
+# 2.4 PROGRESSIVE ARCHITECTURE LEARNING
+
+The GPU memory consumption of differentiable NAS methods grows linearly with the size of operation candidate space. Therefore, they usually use a easier proxy task such as training with a smaller dataset, or searching with fewer layers and number of channels (Cai et al., 2019). For instance, the architecture search is performed on 8 cells and 16 initial channels in DARTS (Liu et al., 2019). But during evaluation, the network has 20 cells and 36 initial channels. Such gap makes it hard to derive an optimal architecture for the target task (Cai et al., 2019).
+
+PC-DARTS (Xu et al., 2020) proposes a partial channel connection to reduce the memory overheads of differentiable NAS, where they only send a random subset of channels to the mixed-operation while directly bypassing the rest channels in a shortcut. However, their method causes loss of information and makes the selection of operation unstable since the sampled subsets may vary widely across iterations. This drawback is amplified when combining with the proposed method since we learn the architecture distribution from Dirichlet samples, which already injects certain stochasticity. As shown in Table 1, when directly applying partial channel connection with distribution learning, the test accuracy of the searched architecture decreases over $3 \%$ and $18 \%$ on CIFAR-10 and CIFAR-100 respectively if we send only 1/8 channels to the mixed-operation.
+
+To alleviate such information loss and instability problem while being memory-efficient, we propose a progressive learning scheme which gradually increases the fraction of channels that are forwarded to the mixed-operation and meanwhile prunes the operation space based on the learnt distribution. We split the search process into consecutive stages and construct a task-specific super-network with the same depth and number of channels as the evaluation phase at the initial stage. Then after each stage, we increase the partial channel fraction, which means that the super-network in the next stage will be wider, i.e., have more convolution channels, and in turn preserve more information. This is achieved by enlarging every convolution weight with a random mapping function similar to Net2Net (Chen et al., 2016). The mapping function $\bar { g } : \{ 1 , 2 , \dotsc , q \}  \{ \bar { 1 , 2 } , \dotsc , n \}$ with $q > n$ is defined as
+
+$$
+g ( j ) = { \left\{ \begin{array} { l l } { j } & { j \leq n } \\ { \operatorname { r a n d o m } \operatorname { s a m p l e } \operatorname { f r o m } \left\{ 1 , 2 , \dots , n \right\} } & { j > n } \end{array} \right. }
+$$
+
+To widen layer $l$ , we replace its convolution weight $\mathbf { W } ^ { ( l ) } \in \mathbb { R } ^ { O u t \times I n \times H \times W }$ with a new weight $\mathbf { U } ^ { ( l ) }$
+
+$$
+\mathbf { U } _ { o , i , h , w } ^ { ( l ) } = \mathbf { W } _ { g ( o ) , g ( i ) , h , w } ^ { ( l ) } ,
+$$
+
+where $O u t , I n , H , W$ denote the number of output and input channels, filter height and width respectively. Intuitively, we copy $\mathbf { W } ^ { ( l ) }$ directly into $\mathbf { U } ^ { ( l ) }$ and fulfill the rest part by choosing randomly as defined in $g$ . Unlike Net2Net, we do not divide $\mathbf { U } ^ { ( l ) }$ by a replication factor here because the information flow on each edge has the same scale no matter the partial fraction is. After widening the super-network, we reduce the operation space by pruning out less important operations according to the Dirichlet concentration parameter $\beta$ learnt from the previous stage, maintaining a consistent memory consumption. As illustrated in Table 1, the proposed progressive architecture learning scheme effectively discovers high accuracy architectures and retains a low GPU memory overhead.
+
+# 3 DISCUSSIONS AND RELATIONSHIP TO PRIOR WORK
+
+Early methods in NAS usually include a full training and evaluation procedure every iteration as the inner loop to guide the consecutive search (Zoph & Le, 2017; Zoph et al., 2018; Real et al., 2019). Consequently, their computational overheads are beyond acceptance for practical usage, especially on large-scale tasks.
+
+Differentiable NAS Recently, many works are proposed to improve the efficiency of NAS (Pham et al., 2018; Cai et al., 2018; Liu et al., 2019; Bender et al., 2018; Yao et al., 2020b;a; Mei et al., 2020). Amongst them, DARTS (Liu et al., 2019) proposes a differentiable NAS framework, which introduces a continuous architecture parameter that relaxes the discrete search space. Despite being efficient, DARTS only optimizes a single point on the simplex every search epoch, which has no guarantee to generalize well after the discretization during evaluation. So its stability and generalization have been widely challenged (Li & Talwalkar, 2019; Zela et al., 2020a; Chen & Hsieh, 2020; Wang et al., 2021). Following DARTS, SNAS (Xie et al., 2019) and GDAS (Dong & Yang, 2019)
+
+Table 1: Test accuracy of the derived architectures when searching on NASBench-201 with different partial channel fraction, where $1 / K$ channels are sent to the mixed-operation.   
+
+<table><tr><td>K</td><td>CIFAR-10 Test Accuracy (%)</td><td>GPU Memory (MB)</td></tr><tr><td>1</td><td>94.36±0.00</td><td>2437</td></tr><tr><td>2</td><td>93.49 ±0.28</td><td>1583</td></tr><tr><td>4</td><td>92.85 ± 0.35</td><td>1159 949</td></tr><tr><td>8 Ours</td><td>91.06 ± 0.00 94.36 ±0.00</td><td>949</td></tr><tr><td></td><td>CIFAR-100</td><td>GPU Memory</td></tr><tr><td>K</td><td>Test Accuracy (%)</td><td>(MB)</td></tr><tr><td>1 2</td><td>73.51± 0.00</td><td>2439 1583</td></tr><tr><td></td><td>68.48 ± 0.41</td><td>1161</td></tr><tr><td>4</td><td>66.68 ± 3.22</td><td>949</td></tr><tr><td>8 Ours</td><td>55.11 ± 13.78 73.51 ± 0.00</td><td>949</td></tr></table>
+
+leverage the gumbel-softmax trick to learn the exact architecture parameter. However, their reparameterization is motivated from reinforcement learning perspective, which is an approximation with softmax rather than an architecture distribution. Besides, their methods require tuning of temperature schedule (Yan et al., 2017; Caglar Gulcehre, 2017). GDAS linearly decreases the temperature from 10 to 1 while SNAS anneals it from 1 to 0.03. In comparison, the proposed method can automatically learn the architecture distribution without the requirement of handcrafted scheduling. BayesNAS (Zhou et al., 2019) applies Bayesian Learning in NAS. Specifically, they cast NAS as model compression problem and use Bayes Neural Network as the super-network, which is difficult to optimize and requires oversimplified approximation. While our method considers the stochasticity in architecture mixing weight, as it is directly related to the generalization of differentiable NAS algorithms (Zela et al., 2020a; Chen & Hsieh, 2020).
+
+Memory overhead When dealing with the large memory consumption of differentiable NAS, previous works mainly restrain the number of paths sampled during the search phase. For instance, ProxylessNAS (Cai et al., 2019) employs binary gates and samples two paths every search epoch. PARSEC (Casale et al., 2019) samples discrete architectures according to a categorical distribution to save memory. Similarly, GDAS (Dong & Yang, 2019) and DSNAS (Hu et al., 2020) both enforce a discrete constraint after the gumbel-softmax reparametrization. However, such discretization manifests premature convergence and cause search instability (Zela et al., 2020b; Zhang et al., 2020). Our experiments in section 4.3 also empirically demonstrate this phenomenon. As an alternative, PC-DARTS (Xu et al., 2020) proposes a partial channel connection, where only a portion of channels is sent to the mixed-operation. However, partial connection can cause loss of information as shown in section 2.4 and PC-DARTS searches on a shallower network with less channels, suffering the search and evaluation gap. Our solution, by progressively pruning the operation space and meanwhile widening the network, searches in a task-specific manner and achieves superior accuracy on challenging datasets like ImageNet $( + 2 . 8 \%$ over BayesNAS, $+ 2 . 3 \%$ over GDAS, $+ 2 . 3 \%$ over PARSEC, $+ 2 . 0 \%$ over DSNAS, $+ 1 . 2 \%$ over ProxylessNAS, and $+ 0 . 5 \%$ over PC-DARTS).
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate our proposed DrNAS on two search spaces: the CNN search space in DARTS (Liu et al., 2019) and NAS-Bench-201 (Dong & Yang, 2020). For DARTS space, we conduct experiments on both CIFAR-10 and ImageNet in section 4.1 and 4.2 respectively. For NAS-Bench201, we test all 3 supported datasets (CIFAR-10, CIFAR-100, ImageNet-16-120 (Chrabaszcz et al., 2017)) in section 4.3. Furthermore, we empirically study the dynamics of exploration and exploitation throughout the search process in section 4.4.
+
+# 4.1 RESULTS ON CIFAR-10
+
+Architecture Space For both search and evaluation phases, we stack 20 cells to compose the network and set the initial channel number as 36. We place the reduction cells at the 1/3 and 2/3 of the network and each cell consists of $N = 6$ nodes.
+
+Search Settings We equally divide the 50K training images into two parts, one is used for optimizing the network weights by momentum SGD and the other for learning the Dirichlet architecture distribution by an Adam optimizer. Since Dirichlet concentration $\beta$ must be positive, we apply the shifted exponential linear mapping $\beta = \mathrm { E L U } ( \eta ) + 1$ and optimize over $\eta$ instead. We use $l _ { 2 }$ norm to constrain the distance between $\eta$ and the anchor $\hat { \eta } = 0$ . The $\eta$ is initialized by standard Gaussian with scale 0.001, and $\lambda$ in (2) is set to 0.001. The ablation study in Appendix A.3 reveals the effectiveness of our anchor regularizer, and DrNAS is insensitive to a wide range of $\lambda$ . These settings are consistent for all experiments. For progressive architecture learning, the whole search process consists of 2 stages, each with 25 iterations. In the first stage, we set the partial channel parameter $K$ as 6 to fit the super-network into a single GTX 1080Ti GPU with 11GB memory, i.e., only 1/6 features are sampled on each edge. For the second stage, we prune half candidates and meanwhile widen the network twice, i.e., the operation space size reduces from 8 to 4 and $K$ becomes 3.
+
+Retrain Settings The evaluation phase uses the entire 50K training set to train the network from scratch for 600 epochs. The network weight is optimized by an SGD optimizer with a cosine annealing learning rate initialized as 0.025, a momentum of 0.9, and a weight decay of $3 \times 1 0 ^ { - 4 }$ . To allow a fair comparison with previous work, we also employ cutout regularization with length 16, drop-path (Zoph et al., 2018) with probability 0.3 and an auxiliary tower of weight 0.4.
+
+Table 2: Comparison with state-of-the-art image classifiers on CIFAR-10.   
+
+<table><tr><td>Architecture</td><td>Test Error (%)</td><td>Params (M)</td><td>Search Cost (GPU days)</td><td>Search Method</td></tr><tr><td>DenseNet-BC (Huang et al.,2017)*</td><td>3.46</td><td>25.6</td><td>1</td><td>manual</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>2.65</td><td>3.3</td><td>2000</td><td>RL</td></tr><tr><td>AmoebaNet-A (Real etal.,2019)</td><td>3.34±0.06</td><td>3.2</td><td>3150</td><td>evolution</td></tr><tr><td>AmoebaNet-B (Real etal.,2019)</td><td>2.55 ± 0.05</td><td>2.8</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al.,2018)*</td><td>3.41±0.09</td><td>3.2</td><td>225</td><td>SMBO</td></tr><tr><td>ENAS (Pham et al.,2018)</td><td>2.89</td><td>4.6</td><td>0.5</td><td>RL</td></tr><tr><td>DARTS (1st) (Liu et al.,2019)</td><td>3.00±0.14</td><td>3.3</td><td>0.4</td><td>gradient</td></tr><tr><td>DARTS (2nd) (Liu et al.,2019)</td><td>2.76 ±0.09</td><td>3.3</td><td>1.0</td><td>gradient</td></tr><tr><td>SNAS (moderate) (Xie et al.,2019)</td><td>2.85 ±0.02</td><td>2.8</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS(Dong &amp; Yang,2019)</td><td>2.93</td><td>3.4</td><td>0.3</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>2.81 ± 0.04</td><td>3.4</td><td>0.2</td><td>gradient</td></tr><tr><td>ProxylessNAS (Cai et al., 2019)†</td><td>2.08</td><td>5.7</td><td>4.0</td><td>gradient</td></tr><tr><td>PARSEC (Casale et al.,2019)</td><td>2.81 ± 0.03</td><td>3.7</td><td>1</td><td>gradient</td></tr><tr><td>P-DARTS (Chen et al.,2019)</td><td>2.50</td><td>3.4</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (Xu et al., 2020)</td><td>2.57 ±0.07</td><td>3.6</td><td>0.1</td><td>gradient</td></tr><tr><td>SDARTS-ADV (Chen &amp; Hsieh,2020)</td><td>2.61 ±0.02</td><td>3.3</td><td>1.3</td><td>gradient</td></tr><tr><td>GAEA+PC-DARTS (Li et al.,2020)</td><td>2.50± 0.06</td><td>3.7</td><td>0.1</td><td>gradient</td></tr><tr><td>DrNAS (without progressive learning)</td><td>2.54± 0.03</td><td>4.0</td><td>0.4</td><td>gradient</td></tr><tr><td>DrNAS</td><td>2.46 ±0.03</td><td>4.1</td><td>0.6</td><td>gradient</td></tr></table>
+
+? Obtained without cutout augmentation. † Obtained on a different space with PyramidNet (Han et al., 2017) as the backbone. ‡ Recorded on a single GTX 1080Ti GPU.
+
+Results Table 2 summarizes the performance of DrNAS compared with other popular NAS methods, and we also visualize the searched cells in Appendix A.2. DrNAS achieves an average test error of $2 . 4 6 \%$ , ranking top amongst recent NAS results. ProxylessNAS is the only method that achieves lower test error than us, but it searches on a different space with a much longer search time and has larger model size. We also perform experiments to assign proper credit to the two parts of our proposed algorithm, i.e., Dirichlet architecture distribution and progressive learning scheme. When searching on a proxy task with 8 stacked cells and 16 initial channels as the convention (Liu et al., 2019; Xu et al., 2020), we achieve a test error of $2 . 5 4 \%$ that surpasses most baselines. Our progressive learning algorithm eliminates the gap between the proxy and target tasks, which further reduces the test error. Consequently, both of the two parts contribute a lot to our performance gains.
+
+# 4.2 RESULTS ON IMAGENET
+
+Architecture Space The network architecture for ImageNet is slightly different from that for CIFAR-10 in that we stack 14 cells and set the initial channel number as 48. We also first downscale the spatial resolution from $2 2 4 \times 2 2 4$ to $2 8 \times 2 8$ with three convolution layers of stride 2 following previous works (Xu et al., 2020; Chen et al., 2019). The other settings are the same with section 4.1.
+
+Search Settings Following PC-DARTS (Xu et al., 2020), we randomly sample $10 \%$ and $2 . 5 \%$ images from the 1.3M training set to alternatively learn network weight and Dirichlet architecture distribution by a momentum SGD and an Adam optimizer respectively. We use 8 RTX 2080 Ti GPUs for both search and evaluation, and the setup of progressive pruning is the same with that on CIFAR-10, i.e., 2 stages with operation space size shrinking from 8 to 4, and the partial channel $K$ reduces from 6 to 3.
+
+Retrain Settings For architecture evaluation, we train the network for 250 epochs by an SGD optimizer with a momentum of 0.9, a weight decay of $3 \times 1 0 ^ { - 5 }$ , and a linearly decayed learning rate initialized as 0.5. We also use label smoothing and an auxiliary tower of weight 0.4 during training. The learning rate warm-up is employed for the first 5 epochs following previous works (Chen et al., 2019; Xu et al., 2020).
+
+Table 3: Comparison with state-of-the-art image classifiers on ImageNet in the mobile setting.   
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Test Error(%)</td><td rowspan="2">Params (M)</td><td rowspan="2">Search Cost (GPU days)</td><td rowspan="2">Search Method</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>Inception-vl (Szegedy et al., 2015)</td><td>30.1</td><td>10.1</td><td>6.6</td><td></td><td>manual</td></tr><tr><td>MobileNet (Howard et al.,2017)</td><td>29.4</td><td>10.5</td><td>4.2</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v1) (Zhang et al.,2018)</td><td>26.4</td><td>10.2</td><td>~5</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v2) (Ma et al.,2018)</td><td>25.1</td><td>-</td><td>~5</td><td>=</td><td>manual</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>26.0</td><td>8.4</td><td>5.3</td><td>2000</td><td>RL</td></tr><tr><td>AmoebaNet-C (Real etal.,2019)</td><td>24.3</td><td>7.6</td><td>6.4</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al., 2018)</td><td>25.8</td><td>8.1</td><td>5.1</td><td>225</td><td>SMBO</td></tr><tr><td>MnasNet-92 (Tan etal.,2019)</td><td>25.2</td><td>8.0</td><td>4.4</td><td>1</td><td>RL</td></tr><tr><td>DARTS (2nd) (Liu et al.,2019)</td><td>26.7</td><td>8.7</td><td>4.7</td><td>1.0</td><td>gradient</td></tr><tr><td>SNAS (mild) (Xie et al.,2019)</td><td>27.3</td><td>9.2</td><td>4.3</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>26.0</td><td>8.5</td><td>5.3</td><td>0.3</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>26.5</td><td>8.9</td><td>3.9</td><td>0.2</td><td>gradient</td></tr><tr><td>DSNAS (Hu et al., 2020)†</td><td>25.7</td><td>8.1</td><td>1</td><td>1</td><td>gradient</td></tr><tr><td>ProxylessNAS (GPU) (Cai et al.,2019)†</td><td>24.9</td><td>7.5</td><td>7.1</td><td>8.3</td><td>gradient</td></tr><tr><td>PARSEC (Casale et al.,2019)</td><td>26.0</td><td>8.4</td><td>5.6</td><td>1</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-1O) (Chen et al., 2019)</td><td>24.4</td><td>7.4</td><td>4.9</td><td>0.3</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-10O) (Chen et al.,2019)</td><td>24.7</td><td>7.5</td><td>5.1</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (CIFAR-10) (Xu et al., 2020)</td><td>25.1</td><td>7.8</td><td>5.3</td><td>0.1</td><td>gradient</td></tr><tr><td>PC-DARTS (ImageNet) (Xu et al., 2020)+</td><td>24.2</td><td>7.3</td><td>5.3</td><td>3.8</td><td>gradient</td></tr><tr><td>GAEA + PC-DARTS (Li et al., 2020)+</td><td>24.0</td><td>7.3</td><td>5.6</td><td>3.8</td><td>gradient</td></tr><tr><td>DrNAS (without progressive learning)†</td><td>24.2</td><td>7.3</td><td>5.2</td><td>3.9</td><td>gradient</td></tr><tr><td>DrNASt</td><td>23.7</td><td>7.1</td><td>5.7</td><td>4.6</td><td>gradient</td></tr></table>
+
+† The architecture is searched on ImageNet, otherwise it is searched on CIFAR-10 or CIFAR-100.
+
+Results As shown in Table 3, we achieve a top-1/5 test error of $2 3 . 7 \% / 7 . 1 \%$ , outperforming all compared baselines and achieving state-of-the-art performance in the ImageNet mobile setting. The searched cells are visualized in Appendix A.2. Similar to section 4.1, we also report the result achieved with 8 cells and 16 initial channels, which is a common setup for the proxy task on ImageNet (Xu et al., 2020). The obtained $2 4 . 2 \%$ top-1 accuracy is already highly competitive, which demonstrates the effectiveness of the architecture distribution learning on large-scale tasks. Then our progressive learning scheme further increases the top-1/5 accuracy for $0 . 5 \% / 0 . 2 \%$ . Therefore, learning in a task-specific manner is essential to discover better architectures.
+
+# 4.3 RESULTS ON NAS-BENCH-201
+
+Recently, some researchers doubt that the expert knowledge applied to the evaluation protocol plays an important role in the impressive results achieved by leading NAS methods (Yang et al., 2020; Li & Talwalkar, 2019). So to further verify the effectiveness of DrNAS, we perform experiments on NAS-Bench-201 (Dong & Yang, 2020), where architecture performance can be directly obtained by querying in the database. NAS-Bench-201 provides support for 3 dataset (CIFAR-10, CIFAR-100, ImageNet-16-120 (Chrabaszcz et al., 2017)) and has a unified cell-based search space containing 15,625 architectures. We refer to their paper (Dong & Yang, 2020) for details of the space. Our experiments are performed in a task-specific manner, i.e., the search and evaluation are based on the same dataset. The hyperparameters for all compared methods are set as their default and for DrNAS, we use the same search settings with section 4.1. We run every method 4 independent times with different random seeds and report the mean and standard deviation in Table 4.
+
+As shown, we achieve the best accuracy on all 3 datasets. On CIFAR-100, we even achieve the global optimal. Specifically, DrNAS outperforms DARTS, GDAS, DSNAS, PC-DARTS, and SNAS by $1 0 3 . 8 \%$ , $3 5 . 9 \%$ , $3 0 . 4 \%$ , $6 . 4 \%$ , and $4 . 3 \%$ on average. We notice that the two methods (GDAS and DSNAS) that enforce a discrete constraint, i.e., only sample a single path every search iteration, perform undesirable especially on CIFAR-100. In comparison, SNAS, employing a similar Gumbelsoftmax trick but without the discretization, performs much better. Consequently, a discrete constraint during search can reduce the GPU memory consumption but empirically suffers instability. In comparison, we develop the progressive learning scheme on top of the architecture distribution learning, enjoying both memory efficiency and strong search performance.
+
+Table 4: Comparison with state-of-the-art NAS methods on NAS-Bench-201.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td colspan="2">ImageNet-16-120</td></tr><tr><td>validation</td><td>test</td><td>validation</td><td>test</td><td>validation</td><td>test</td></tr><tr><td>ResNet (He et al.,2016)</td><td>90.83</td><td>93.97</td><td>70.42</td><td>70.86</td><td>44.53</td><td>43.63</td></tr><tr><td>Random (baseline)</td><td>90.93±0.36</td><td>93.70±0.36</td><td>70.60±1.37</td><td>70.65 ± 1.38</td><td>42.92 ± 2.00</td><td>42.96±2.15</td></tr><tr><td>RSPS (Li&amp; Talwalkar,2019)</td><td>84.16 ± 1.69</td><td>87.66 ± 1.69</td><td>45.78 ± 6.33</td><td>46.60 ± 6.57</td><td>31.09 ± 5.65</td><td>30.78 ±6.12</td></tr><tr><td>Reinforce (Zoph et al., 2018)</td><td>91.09 ± 0.37</td><td>93.85± 0.37</td><td>70.05 ± 1.67</td><td>70.17 ± 1.61</td><td>43.04 ± 2.18</td><td>43.16 ± 2.28</td></tr><tr><td>ENAS (Pham etal.,2018)</td><td>39.77±0.00</td><td>54.30±0.00</td><td>10.23±0.12</td><td>10.62 ± 0.27</td><td>16.43±0.00</td><td>16.32±0.00</td></tr><tr><td>DARTS(1st) (Liu et al.,2019)</td><td>39.77 ±0.00</td><td>54.30±0.00</td><td>38.57 ±0.00</td><td>38.97 ± 0.00</td><td>18.87 ± 0.00</td><td>18.41 ± 0.00</td></tr><tr><td>DARTS (2nd) (Liu et al., 2019)</td><td>39.77±0.00</td><td>54.30 ±0.00</td><td>38.57±0.00</td><td>38.97 ±0.00</td><td>18.87 ± 0.00</td><td>18.41 ± 0.00</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>90.01±0.46</td><td>93.23±0.23</td><td>24.05 ±8.12</td><td>24.20 ±8.08</td><td>40.66 ±0.00</td><td>41.02 ± 0.00</td></tr><tr><td>SNAS (Xie et al.,2019)</td><td>90.10 ± 1.04</td><td>92.77±0.83</td><td>69.69±2.39</td><td>69.34 ± 1.98</td><td>42.84 ± 1.79</td><td>43.16 ± 2.64</td></tr><tr><td>DSNAS (Hu et al.,2020)</td><td>89.66±0.29</td><td>93.08 ± 0.13</td><td>30.87±16.40</td><td>31.01 ± 16.38</td><td>40.61 ±0.09</td><td>41.07 ± 0.09</td></tr><tr><td>PC-DARTS(Xu et al.,2020)</td><td>89.96± 0.15</td><td>93.41 ± 0.30</td><td>67.12 ± 0.39</td><td>67.48± 0.89</td><td>40.83±0.08</td><td>41.31 ± 0.22</td></tr><tr><td>DrNAS</td><td>91.55 ± 0.00</td><td>94.36±0.00</td><td>73.49 ± 0.00</td><td>73.51 ± 0.00</td><td>46.37±0.00</td><td>46.34±0.00</td></tr><tr><td>optimal</td><td>91.61</td><td>94.37</td><td>73.49</td><td>73.51</td><td>46.77</td><td>47.31</td></tr></table>
+
+# 4.4 EMPIRICAL STUDY ON EXPLORATION V.S. EXPLOITATION
+
+We further conduct an empirical study on the dynamics of exploration and exploitation in the search phase of DrNAS on NAS-Bench-201. After every search epoch, We sample $1 0 0 \theta \mathrm { s }$ from the learned Dirichlet distribution and take the arg max to obtain 100 discrete architectures. We then plot the range of their accuracy along with the architecture selected by Dirichlet mean (solid line in Figure 1). Note that in our algorithm, we simply derive the architecture according to the Dirichlet mean as described in Section 2.2. As shown in Figure 1, the accuracy range of the sampled architectures starts very wide but narrows gradually during the search phase. It indicates that DrNAS learns to encourage exploration in the search space at the early stages and then gradually reduces it towards the end as the algorithm becomes more and more confident of the current choice. Moreover, the performance of our architectures can consistently match the best performance of the sampled architectures, indicating the effectiveness of DrNAS.
+
+![](images/42130c59f39e2b3211f0e4ab14b1f8da35d6e533b90256026a73b261df1e64e0.jpg)  
+Figure 1: Accuracy range (min-max) of the 100 sampled architectures. Note that the solid line is our derived architecture according to the Dirichlet mean as described in Section 2.2.
+
+# 5 CONCLUSION
+
+In this paper, we propose Dirichlet Neural Architecture Search (DrNAS). We formulate the differentiable NAS as a constraint distribution learning problem, which explicitly models the stochasticity in the architecture mixing weight and balances exploration and exploitation in the search space. The proposed method can be optimized efficiently via gradient-based algorithm, and possesses theoretical benefit to improve the generalization ability. Furthermore, we propose a progressive learning scheme to eliminate the search and evaluation gap. DrNAS consistently achieves strong performance across several image classification tasks, which reveals its potential to play a crucial role in future end-to-end deep learning platform.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by NSF under IIS-1901527, IIS-2008173, IIS-2048280 and by Army Research Laboratory under agreement number W911NF-20-2-0158.
+
+# REFERENCES
+
+Charles Sutton Akash Srivastava. Autoencoding variational inference for topic models. In International Conference on Learning Representations, 2017. URL https://arxiv.org/abs/ 1703.01488.
+
+Gabriel Bender, Pieter-Jan Kindermans, Barret Zoph, Vijay Vasudevan, and Quoc Le. Understanding and simplifying one-shot architecture search. 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. 550–559, Stockholmsmassan, Stockholm Sweden, 10–15 Jul ¨ 2018. PMLR. URL http://proceedings.mlr.press/v80/bender18a.html.
+
+Christopher Bishop. Pattern Recognition and Machine Learning. Springer New York, 2016.
+
+Yoshua Bengio Caglar Gulcehre, Sarath Chandar. Memory augmented neural networks with wormhole connections, 2017.
+
+Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Efficient architecture search by network transformation. In AAAI, 2018.
+
+Han Cai, Ligeng Zhu, and Song Han. ProxylessNAS: Direct neural architecture search on target task and hardware. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ HylVB3AqYm.
+
+Francesco Paolo Casale, Jonathan Gordon, and Nicolo Fusi. Probabilistic neural architecture search, 2019.
+
+Tianqi Chen, Ian Goodfellow, and Jonathon Shlens. Net2net: Accelerating learning via knowledge transfer. In International Conference on Learning Representations, 2016. URL http://arxiv. org/abs/1511.05641.
+
+Xiangning Chen and Cho-Jui Hsieh. Stabilizing differentiable architecture search via perturbationbased regularization. In Hal Daume III and Aarti Singh (eds.), ´ Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 1554–1565. PMLR, 13–18 Jul 2020. URL http://proceedings.mlr.press/v119/ chen20f.html.
+
+Xin Chen, Lingxi Xie, Jun Wu, and Qi Tian. Progressive differentiable architecture search: Bridging the depth gap between search and evaluation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1294–1303, 2019.
+
+Patryk Chrabaszcz, Ilya Loshchilov, and Frank Hutter. A downsampled variant of imagenet as an alternative to the cifar datasets, 2017.
+
+Jon D. McAuliffe David M. Blei, Alp Kucukelbir. Variational inference: A review for statisticians, 2016.
+
+Michael I. Jordan David M. Blei, Andrew Y. Ng. Latent dirichlet allocation. The Journal of Machine Learning Research, Mar 2003. ISSN 1532-4435. doi: 10.1162/jmlr.2003.3.4-5.993. URL http://dx.doi.org/10.1109/CVPR.2017.243.
+
+Xuanyi Dong and Yi Yang. Searching for a robust neural architecture in four gpu hours. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1761–1770, 2019.
+
+Xuanyi Dong and Yi Yang. Nas-bench-201: Extending the scope of reproducible neural architecture search. In International Conference on Learning Representations (ICLR), 2020. URL https: //openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ HJxyZkBKDr.
+
+Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin. Bayesian Data Analysis. Chapman and Hall/CRC, 2nd ed. edition, 2004.
+
+Dongyoon Han, Jiwhan Kim, and Junmo Kim. Deep pyramidal residual networks. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017. doi: 10.1109/cvpr. 2017.668. URL http://dx.doi.org/10.1109/CVPR.2017.668.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. pp. 770–778, 06 2016. doi: 10.1109/CVPR.2016.90.
+
+Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications, 2017.
+
+Shoukang Hu, Sirui Xie, Hehui Zheng, Chunxiao Liu, Jianping Shi, Xunying Liu, and Dahua Lin. Dsnas: Direct neural architecture search without parameter retraining, 2020.
+
+Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017. doi: 10.1109/cvpr.2017.243. URL http://dx.doi.org/10.1109/ CVPR.2017.243.
+
+Weonyoung Joo, Wonsung Lee, Sungrae Park, , and Il-Chul Moon. Dirichlet variational autoencoder, 2019. URL https://openreview.net/forum?id ${ . } = { }$ rkgsvoA9K7.
+
+Samuel Kessler, Vu Nguyen, Stefan Zohren, and Stephen Roberts. Hierarchical indian buffet neural networks for bayesian continual learning, 2019.
+
+Soochan Lee, Junsoo Ha, Dongsu Zhang, and Gunhee Kim. A neural dirichlet process mixture model for task-free continual learning,” in international conference on learning representations. In ICLR, 2020.
+
+Liam Li and Ameet Talwalkar. Random search and reproducibility for neural architecture search, 2019.
+
+Liam Li, Mikhail Khodak, Maria-Florina Balcan, and Ameet Talwalkar. Geometry-aware gradient algorithms for neural architecture search, 2020.
+
+Chenxi Liu, Barret Zoph, Maxim Neumann, Jonathon Shlens, Wei Hua, Li-Jia Li, Li Fei-Fei, Alan Yuille, Jonathan Huang, and Kevin Murphy. Progressive neural architecture search. Lecture Notes in Computer Science, pp. 19–35, 2018. ISSN 1611-3349. doi: 10.1007/978-3-030-01246-5 2. URL http://dx.doi.org/10.1007/978-3-030-01246-5_2.
+
+Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable architecture search. In International Conference on Learning Representations, 2019. URL https://openreview. net/forum?id ${ . } = { }$ S1eYHoC5FX.
+
+Ningning Ma, Xiangyu Zhang, Hai-Tao Zheng, and Jian Sun. Shufflenet v2: Practical guidelines for efficient cnn architecture design. In The European Conference on Computer Vision (ECCV), September 2018.
+
+David J. C. MacKay. Choice of basis for laplace approximation. Machine Language, October 1998. doi: 10.1023/A:1007558615313. URL https://link.springer.com/article/ 10.1023/A:1007558615313.
+
+Fritz Obermeyer Martin Jankowiak. Pathwise derivatives beyond the reparameterization trick. In International Conference on Machine Learning, 2018. URL https://arxiv.org/abs/ 1806.01851.
+
+Jieru Mei, Yingwei Li, Xiaochen Lian, Xiaojie Jin, Linjie Yang, Alan Yuille, and Jianchao Yang. Atomnas: Fine-grained end-to-end neural architecture search. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id= BylQSxHFwr.
+
+Hieu Pham, Melody Guan, Barret Zoph, Quoc Le, and Jeff Dean. Efficient neural architecture search via parameters sharing. 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. 4095–4104, Stockholmsmassan, Stockholm Sweden, 10–15 Jul 2018. PMLR. URL ¨ http://proceedings.mlr.press/v80/pham18a.html.
+
+Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V. Le. Regularized evolution for image classifier architecture search. Proceedings of the AAAI Conference on Artificial Intelligence, 33:4780–4789, Jul 2019. ISSN 2159-5399. doi: 10.1609/aaai.v33i01.33014780. URL http: //dx.doi.org/10.1609/aaai.v33i01.33014780.
+
+Yao Shu, Wei Wang, and Shaofeng Cai. Understanding architectures learnt by cell-based neural architecture search. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ BJxH22EKPS.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition (CVPR), 2015. URL http://arxiv.org/abs/1409.4842.
+
+Mingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, Mark Sandler, Andrew Howard, and Quoc V. Le. Mnasnet: Platform-aware neural architecture search for mobile. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
+
+Ruochen Wang, Minhao Cheng, Xiangning Chen, Xiaocheng Tang, and Cho-Jui Hsieh. Rethinking architecture selection in differentiable NAS. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id $=$ PKubaeJkw3.
+
+Sirui Xie, Hehui Zheng, Chunxiao Liu, and Liang Lin. SNAS: stochastic neural architecture search. In International Conference on Learning Representations, 2019. URL https://openreview. net/forum?id $=$ rylqooRqK7.
+
+Yuhui Xu, Lingxi Xie, Xiaopeng Zhang, Xin Chen, Guo-Jun Qi, Qi Tian, and Hongkai Xiong. PC-DARTS: Partial channel connections for memory-efficient architecture search. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum? id $=$ BJlS634tPr.
+
+Shiyang Yan, Jeremy S. Smith, Wenjin Lu, and Bailing Zhang. Hierarchical multi-scale attention networks for action recognition, 2017.
+
+Antoine Yang, Pedro M. Esperanc¸a, and Fabio M. Carlucci. Nas evaluation is frustratingly hard. In International Conference on Learning Representations, 2020. URL https://openreview. net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } = \underline { { \underline { { \mathbf { \delta \pi } } } } } .$ HygrdpVKvr.
+
+Quanming Yao, Xiangning Chen, James T. Kwok, Yong Li, and Cho-Jui Hsieh. Efficient neural interaction function search for collaborative filtering. In Proceedings of The Web Conference 2020, WWW ’20, pp. 1660–1670, New York, NY, USA, 2020a. Association for Computing Machinery. ISBN 9781450370233. doi: 10.1145/3366423.3380237. URL https://doi.org/10.1145/ 3366423.3380237.
+
+Quanming Yao, Ju Xu, Wei-Wei Tu, and Zhanxing Zhu. Efficient neural architecture search via proximal iterations. Proceedings of the AAAI Conference on Artificial Intelligence, 34(04):6664– 6671, Apr. 2020b. doi: 10.1609/aaai.v34i04.6143. URL https://ojs.aaai.org/index. php/AAAI/article/view/6143.
+
+Arber Zela, Thomas Elsken, Tonmoy Saikia, Yassine Marrakchi, Thomas Brox, and Frank Hutter. Understanding and robustifying differentiable architecture search. In International Conference on Learning Representations, 2020a. URL https://openreview.net/forum?id $=$ H1gDNyrKDS.
+
+Arber Zela, Julien Siems, and Frank Hutter. NAS-BENCH-1SHOT1: Benchmarking and dissecting one-shot neural architecture search. In International Conference on Learning Representations, 2020b. URL https://openreview.net/forum?id $=$ SJx9ngStPH.
+
+M. Zhang, H. Li, S. Pan, X. Chang, and S. Su. Overcoming multi-model forgetting in one-shot nas with diversity maximization. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 7806–7815, 2020.
+
+Xiangyu Zhang, Xinyu Zhou, Mengxiao Lin, and Jian Sun. Shufflenet: An extremely efficient convolutional neural network for mobile devices. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
+
+Hongpeng Zhou, Minghao Yang, Jun Wang, and Wei Pan. Bayesnas: A bayesian approach for neural architecture search. In ICML, pp. 7603–7613, 2019. URL http://proceedings. mlr.press/v97/zhou19e.html.
+
+Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. 2017. URL https://arxiv.org/abs/1611.01578.
+
+Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018. doi: 10.1109/cvpr.2018.00907. URL http://dx.doi.org/10. 1109/CVPR.2018.00907.
+
+# A APPENDIX
+
+# A.1 PROOF OF PROPOSITION 1
+
+Preliminaries: Before the development of Pathwise Derivative Estimator, Laplace Approximate with Softmax basis has been extensively used to approximate the Dirichlet Distribution (MacKay, 1998; Akash Srivastava, 2017). The approximated Dirichlet distribution is:
+
+$$
+p ( \theta ( \mathbf { h } ) | \beta ) = \frac { \Gamma ( \sum _ { o } \beta _ { o } ) } { \prod _ { o } \Gamma ( \beta _ { o } ) } \prod _ { o } \theta _ { o } ^ { \beta _ { o } } g ( \mathbf { 1 } ^ { T } \mathbf { h } )
+$$
+
+Where $\theta ( \mathbf { h } )$ is the softmax-transformed $\mathbf { h }$ , h follows multivariate normal distribution, and $g ( \cdot )$ is an arbitrary density to ensure integrability (Akash Srivastava, 2017). The mean $\mu$ and diagonal covariance matrix $\Sigma$ of $\mathbf { h }$ depends on the Dirichlet concentration parameter $\beta$ :
+
+$$
+\mu _ { o } = \log \beta _ { o } - \frac { 1 } { | \mathcal O | } \sum _ { o ^ { \prime } } \log \beta _ { o ^ { \prime } } \qquad \Sigma _ { o } = \frac { 1 } { \beta _ { o } } ( 1 - \frac { 2 } { | \mathcal O | } ) + \frac { 1 } { | \mathcal O | ^ { 2 } } \sum _ { o ^ { \prime } } \frac { 1 } { \beta _ { o ^ { \prime } } }
+$$
+
+It can be directly obtained from (10) that the Dirichlet mean $\begin{array} { r } { \frac { \beta _ { o } } { \sum _ { o ^ { \prime } } \beta _ { o ^ { \prime } } } = S o f t m a x ( \mu ) } \end{array}$ . Sampling from the approximated distribution can be down by first sampling from $\mathbf { h }$ and then applying Softmax function to obtain $\theta$ . We will leverage the fact that this approximation supports explicit reparameterization to derive our proof.
+
+Proof: Apply the above Laplace Approximation to Dirichlet distribution, the unconstrained upperlevel objective in (5) can then be written as:
+
+$$
+\begin{array} { r l } & { \quad E _ { \theta \sim D i r ( \beta ) } \big [ \mathcal { L } _ { v a l } ( w ^ { * } , \theta ) \big ] } \\ & { \approx E _ { \epsilon \sim \mathcal { N } ( 0 , \Sigma ) } \big [ \mathcal { L } _ { v a l } ( w ^ { * } , S o f t m a x ( \mu + \epsilon ) ) \big ] } \\ & { \equiv E _ { \epsilon \sim \mathcal { N } ( 0 , \Sigma ) } \big [ \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu + \epsilon ) \big ] } \\ & { \approx E _ { \epsilon \sim \mathcal { N } ( 0 , \Sigma ) } \big [ \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \epsilon ^ { T } \nabla _ { \mu } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \frac { 1 } { 2 } \epsilon ^ { T } \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \epsilon \big ] } \\ & { = \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \frac { 1 } { 2 } t r \big ( E _ { \epsilon \sim \mathcal { N } ( 0 , \Sigma ) } \big [ \epsilon \epsilon ^ { T } \big ] \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \big ) } \\ & { = \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \frac { 1 } { 2 } t r \big ( \Sigma \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \big ) } \end{array}
+$$
+
+In our full objective, we constrain the Euclidean distance between learnt Dirichlet concentration and fixed prior concentration $| | { \boldsymbol { \beta } } - \mathbf { 1 } | | _ { 2 } \leq \delta$ . The covariance matrix $\Sigma$ of approximated softmax Gaussian can be bounded as:
+
+$$
+\begin{array} { r } { \Sigma _ { o } = \displaystyle \frac { 1 } { \beta _ { o } } ( 1 - \frac { 2 } { | \mathcal { O } | } ) + \frac { 1 } { | \mathcal { O } | ^ { 2 } } \sum _ { o ^ { ' } } \frac { 1 } { \beta _ { o ^ { ' } } } } \\ { \geq \displaystyle \frac { 1 } { 1 + \delta } ( 1 - \frac { 2 } { | \mathcal { O } | } ) + \frac { 1 } { | \mathcal { O } | } \frac { 1 } { 1 + \delta } } \end{array}
+$$
+
+Then (11) becomes:
+
+$$
+\begin{array} { r l } & { \quad E _ { \theta \sim D i r ( \beta ) } \bigl [ \mathcal { L } _ { v a l } ( w ^ { * } , \theta ) \bigr ] } \\ & { \approx \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \cfrac { 1 } { 2 } t r \bigl ( \Sigma \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \bigr ) } \\ & { \geq \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) + \cfrac { 1 } { 2 } ( \cfrac { 1 } { 1 + \delta } ( 1 - \cfrac { 2 } { | \mathcal { O } | } ) + \cfrac { 1 } { | \mathcal { O } | } \cfrac { 1 } { 1 + \delta } ) t r \bigl ( \nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu ) \bigr ) } \end{array}
+$$
+
+The last line holds when $\nabla _ { \mu } ^ { 2 } \tilde { \mathcal { L } } _ { v a l } ( w ^ { * } , \mu )$ is positive semi-definite. In Appendix A.4 we provide an empirical justification for this implicit regularization effect of DrNAS.
+
+# A.2 SEARCHED ARCHITECTURES
+
+We visualize the searched normal and reduction cells in Figure 2 and 3, which is directly searched on CIFAR-10 and ImageNet respectively.
+
+![](images/d878e6f877230de00e6179098043604c3affb6fb7580503ae9bb930766151e11.jpg)  
+Figure 2: Normal and Reduction cells discovered by DrNAS on CIFAR-10.
+
+![](images/5cb59d84692b039cf61adfb54da76bb7cbab61347adc4374aec5c9850eac5f64.jpg)  
+Figure 3: Normal and Reduction cells discovered by DrNAS on ImageNet.
+
+A.3 ABLATION STUDY ON ANCHOR REGULARIZER PARAMETER λ
+
+Table 5 shows the accuracy of the searched architecture using different value of $\lambda$ while keeping all other settings the same. Using anchor regularizer? for a wide range of value can boost the accuracy and DrNAS performs quite stable under different λs.
+
+Table 5: Test accuracy of the searched architecture with different $\lambda s$ on NAS-Bench-201 (CIFAR-10). $\lambda = 1 e ^ { - 3 }$ is what we used for all of our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>入</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>5e-4</td><td rowspan=1 colspan=1>1e-3</td><td rowspan=1 colspan=1>5e-3</td><td rowspan=1 colspan=1>1e-2</td><td rowspan=1 colspan=1>1e-1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Accuracy</td><td rowspan=1 colspan=1>93.78</td><td rowspan=1 colspan=1>94.01</td><td rowspan=1 colspan=1>94.36</td><td rowspan=1 colspan=1>94.36</td><td rowspan=1 colspan=1>94.36</td><td rowspan=1 colspan=1>93.76</td><td rowspan=1 colspan=1>93.76</td></tr></table>
+
+# A.4 EMPIRICAL STUDY ON THE HESSIAN REGULARIZATION EFFECT
+
+We track the anytime Hessian norm on NAS-Bench-201 in Figure 4. The result is obtained by averaging from 4 independent runs. We observe that the largest eigenvalue expands about 10 times when searching by DARTS for 100 epochs. In comparison, DrNAS always maintains the Hessian norm at a low level, which is in agreement with our theoretical analysis in section 2.3. Figure 5 shows the regularization effect under various λs. As we can see, DrNAS can keep hessian norm at a low level for a wide range of $\lambda s$ , which is in accordance to the relatively stable performance in Table 5.
+
+Moreover, we compare DrNAS with DARTS and R-DARTS on 4 simplified space proposed in (Zela et al., 2020a) and record the endpoint dominant eigenvalue. The first space S1 contains 2 popular operators per edge based on DARTS search result. For S2, S3, and S4, the operation sets are $\{ 3 \times 3$ separable convolution, skip connection $\}$ , $\{ 3 \times 3$ separable convolution, skip connection, $z e r o \}$ , and $\{ 3 \times 3$ separable convolution, noise $\}$ respectively. As shown in Table 6, DrNAS consistently outperforms DARTS and R-DARTS. The endpoint eigenvalues for DrNAS are 0.0392, 0.0390, 0.0286, 0.0389 respectively. Figure 5 shows the Hessian norm trajectory under various $\lambda$ .
+
+# A.5 CONNECTION TO VARIATIONAL INFERENCE
+
+In this section, we draw a connection between DrNAS and Variational Inference (David M. Blei, 2016). We use $w , \theta .$ , and $\beta$ to denote the model weight, operation mixing weight, and Dirichlet concentration parameters respectively, following the main text. The true posterior distribution can
+
+![](images/2f8c53d08a634552563b632e326047f6a7a487915f4935d08dd668d5e05cc854.jpg)  
+Figure 4: Trajectory of the Hessian norm on NAS-Bench-201 when searching with CIFAR-10 (best viewed in color).
+
+![](images/558e9d893700c6ed82955d1a2ae46c1bd190cc7e061a2fd195833c6b5a5b17f1.jpg)  
+Figure 5: Trajectory of the Hessian norm under various $\lambda s$ on NAS-Bench-201 when searching with CIFAR-10 (best viewed in color).
+
+be written as $p ( \theta | w , D )$ , where $D = \{ x _ { n } , y _ { n } \} _ { n = 1 } ^ { N }$ is the dataset. Let $q ( \theta | \beta )$ denote the variational approximation of the true posterior; and assume that $q ( \theta | \beta )$ follows Dirichlet distribution. We follow Joo et al. (2019) to assume a symmetric Dirichlet distribution for the prior $p ( \theta )$ as well,
+
+Table 6: CIFAR-10 test error on 4 simplified spaces.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>s1</td><td rowspan=1 colspan=1>s2</td><td rowspan=1 colspan=1>s3</td><td rowspan=1 colspan=1>s4</td></tr><tr><td rowspan=1 colspan=1>DARTS</td><td rowspan=1 colspan=1>3.84</td><td rowspan=1 colspan=1>4.85</td><td rowspan=1 colspan=1>3.34</td><td rowspan=1 colspan=1>7.20</td></tr><tr><td rowspan=1 colspan=1>R-DARTS (DP)</td><td rowspan=1 colspan=1>3.11</td><td rowspan=1 colspan=1>3.48</td><td rowspan=1 colspan=1>2.93</td><td rowspan=1 colspan=1>3.58</td></tr><tr><td rowspan=1 colspan=1>R-DARTS (L2)</td><td rowspan=1 colspan=1>2.78</td><td rowspan=1 colspan=1>3.31</td><td rowspan=1 colspan=1>2.51</td><td rowspan=1 colspan=1>3.56</td></tr><tr><td rowspan=1 colspan=1>DrNAS</td><td rowspan=1 colspan=1>2.74</td><td rowspan=1 colspan=1>2.47</td><td rowspan=1 colspan=1>2.4</td><td rowspan=1 colspan=1>2.59</td></tr></table>
+
+i.e., $p ( \theta ) = D i r ( \mathbf { 1 } )$ . The goal is to minimize the KL divergence between the true posterior and the approximated form, i.e., $\begin{array} { r } { \operatorname* { m i n } _ { \beta } K L ( q ( \theta | \beta ) | | p ( \theta | w , D ) ) } \end{array}$ . It can be shown that this objective is equivalent to maximizing the evidence lower bound as below (David M. Blei, 2016):
+
+$$
+\mathcal { L } ( \beta ) = E _ { q ( \theta | \beta ) } { \big [ } \log p ( D | \theta , w ) { \big ] } - K L ( q ( \theta | \beta ) | | p ( \theta | w ) )
+$$
+
+The upper level objective of the bilevel optimization under variational inference framework is then given as:
+
+$$
+\operatorname* { m i n } _ { \beta } \ E _ { q ( \theta | \beta ) } \big [ - \log p ( D _ { v a l i d } | \theta , w ^ { * } ) \big ] + K L ( q ( \theta | \beta ) | | p ( \theta ) )
+$$
+
+Note that eq. (23) resembles eq. (2) if we use the negative log likelihood as the loss function and replace $d ( \cdot , \cdot )$ with KL divergence. In practice, we find that using a simple l2 distance regularization works well across datasets and search spaces.

parse/train/B1IDRdeCW/B1IDRdeCW.md

@@ -0,0 +1,270 @@
+# THE HIGH-DIMENSIONAL GEOMETRY OF BINARY NEURAL NETWORKS
+
+Alexander G. Anderson   
+Redwood Center for Theoretical Neuroscience   
+University of California, Berkeley   
+aga@berkeley.edu
+
+Cory P. Berg Redwood Center for Theoretical Neuroscience University of California, Berkeley cberg500@berkeley.edu
+
+# ABSTRACT
+
+Recent research has shown that one can train a neural network with binary weights and activations at train time by augmenting the weights with a high-precision continuous latent variable that accumulates small changes from stochastic gradient descent. However, there is a dearth of work to explain why one can effectively capture the features in data with binary weights and activations. Our main result is that the neural networks with binary weights and activations trained using the method of Courbariaux, Hubara et al. (2016) work because of the high-dimensional geometry of binary vectors. In particular, the ideal continuous vectors that extract out features in the intermediate representations of these BNNs are well-approximated by binary vectors in the sense that dot products are approximately preserved. Furthermore, the results and analysis used on BNNs are shown to generalize to neural networks with ternary weights and activations. Compared to previous research that demonstrated good classification performance with BNNs, our work explains why these BNNs work in terms of HD geometry. Our theory serves as a foundation for understanding not only BNNs but a variety of methods that seek to compress traditional neural networks. Furthermore, a better understanding of multilayer binary neural networks serves as a starting point for generalizing BNNs to other neural network architectures such as recurrent neural networks.
+
+# 1 INTRODUCTION
+
+The rapidly decreasing cost of computation has driven many successes in the field of deep learning in recent years. Consequently, researchers are now considering applications of deep learning in resource limited hardware such as neuromorphic chips, embedded devices and smart phones (Esser et al. (2016), Neftci et al. (2016), Andri et al. (2017)). A recent realization for both theoretical researchers and industry practitioners is that traditional neural networks can be compressed because they are highly over-parameterized. While there has been a large amount of experimental work dedicated to compressing neural networks (Sec. 2), we focus on the particular approach that replaces costly 32-bit floating point multiplications with cheap binary operations. Our analysis reveals a simple geometric picture based on the geometry of high dimensional binary vectors that allows us to understand the successes of the recent efforts to compress neural networks.
+
+Courbariaux et al. (2016) and Hubara et al. (2016) showed that one can efficiently train neural networks with binary weights and activations that have similar performance to their continuous counterparts. Such BNNs execute 7 times faster using a dedicated GPU kernel at test time. Furthermore, they argue that such BNNs require at least a factor of 32 fewer memory accesses at test time that should result in an even larger energy savings. There are two key ideas in their papers (Fig. 1). First, a continuous weight, $w ^ { c }$ , is associated with each binary weight, $w ^ { b }$ , that accumulates small changes from stochastic gradient descent. Second, the non-differentiable binarize function $( \theta ( x ) = 1$ if $x > 0$ and $- 1$ otherwise) is replaced with a continuous one during backpropagation. These modifications allow one to train neural networks that have binary weights and activations with stochastic gradient descent. While the work showed how to train such networks, the existence of neural networks with binary weights and activations needs to be reconciled with previous work that has sought to understand weight matrices as extracting out continuous features in data (e.g. Zeiler & Fergus (2014)). Summary of contributions:
+
+![](images/cb1b7425276e1d85e9981243987d7806fdb5720fbcbfa5db6227712c1266176b.jpg)  
+Figure 1: Review of the Courbariaux et al. (2016) BNN Training Algorithm: (a) A binary neural network is composed of binary convolution transformers (dashed green box). Each oval corresponds to a tensor and the derivative of the cost with respect to that tensor. Rectangles correspond to transformers that specify forward and backward propagation functions. Associated with each binary weight, $w ^ { b }$ , is a continuous weight, $w ^ { c }$ , that is used to accumulate gradients. $k$ denotes the kth layer of the network. (b) Each binarize transformer has a forward function and a backward function. The forward function simply binarizes the inputs. In the backward propagation step, one normally computes the derivative of the cost with respect to the input of a transformer via the Jacobian of the forward function and the derivative of the cost with respect to the output of that transformer $( \delta u \equiv d C / d u$ where $C$ is the cost function used to train the network). Since the binarize function is non-differentiable, the straight-through estimator (Bengio et al. (2013)), which is a smoothed version of the forward function, is used for the backward function .
+
+1. Angle Preservation Property: We demonstrate that binarization approximately preserves the direction of high-dimensional vectors. In particular, we show that the angle between a random vector (from a standard normal distribution) and its binarized version converges to arccos $\sqrt { 2 / \pi } \approx 3 7 ^ { \circ }$ as the dimension of the vector goes to infinity. This angle is an exceedingly small angle in high dimensions. Furthermore, we show that this property is present in the weight vectors of a network trained using the method of Courbariaux et al. (2016).
+
+2. Dot Product Proportionality Property: First, we empirically show that the weight-activation dot products, an important intermediate quantity a neural network, are approximately proportional under the binarization of the weight vectors. Next, we argue that if these weight activation dot products are proportional, then the continuous weights in the Courbariaux et al. (2016) method aren’t just a learning artifact. The continuous weights obtained from the BNN training algorithm (which decouples the forward and backward propagation steps) are an approximation of the weights one would learn if the network were trained with continuous weights and regular backpropagation.
+
+3. Generalized Binarization Transformation (GBT): We show that the computations done by the first layer of the network are fundamentally different than the computations being done in the rest of the network because correlations in the data result in high variance principal components that are not randomly oriented relative to the binarization. Thus we recommend an architecture that uses a continuous convolution for the first layer to embed the image in a high-dimensional binary space, after which it can be manipulated with cheap binary operations. Furthermore, we illustrate how a GBT (rotate, binarize, rotate back) is useful for embedding low dimensional data in a high-dimensional binary space.
+
+4. Generalization to Ternary Neural Networks: We show that the same analysis applies to ternary neural networks. In particular, the angle between a random vector from a standard normal distribution and the ternarized version of that vector predicts the empirical distribution of such angles in a network trained on CIFAR10. Furthermore, the dot product proportionality property is shown to hold for ternary neural networks.
+
+# 2 RELATED WORK
+
+Neural networks that achieve good performance on tasks such as IMAGENET object recognition are highly computationally intensive. For instance, AlexNet has 61 million parameters and executes 1.5 billion operations to classify one 224 by 224 image (30 thousand operations/pixel) (Rastegari et al. (2016)). Researchers have sought to reduce this computational cost for embedded applications using a number of different approaches.
+
+The first approach is to try and compress a pre-trained network. Kim et al. (2015) uses a Tucker decomposition of the kernel tensor and fine tunes the network afterwards. Han et al. (2015b) train a network, prune low magnitude connections, and retrain. Han et al. (2015a) extend their previous work to additionally include a weight sharing quantization step and Huffman coding of the weights. More recently, Han et al. (2017) train a dense network, sparsify it, and then retrain a dense network with the pruned weights initialized to zero. Second, researchers have sought to train networks using either low precision floating point numbers or fixed point numbers, which allow for cheaper multiplications (Courbariaux et al. (2014), Gupta et al. (2015), Judd et al. (2015), Gysel et al. (2016), Lin et al. (2016), Lai et al. (2017)).
+
+Third, one can train networks that have quantized weights and or activations. Bengio et al. (2013) looked at estimators for the gradient through a stochastic binary unit. Courbariaux et al. (2015) train networks with binary weights, and then later with binary weights and activations (Courbariaux et al. (2016)). Rastegari et al. (2016) replace a continuous weight matrix with a scalar times a binary matrix (and have a similar approximation for weight activation dot products). Kim & Smaragdis (2016) train a network with weights restricted in the range $- 1$ to 1 and then use a noisy backpropagation scheme train a network with binary weights and activations. Alemdar et al. (2016), Li et al. (2016) and Zhu et al. (2016) focus on networks with ternary weights. Further work seeks to quantize the weights and activations in neural networks to an arbitrary number of bits (Zhou et al. (2016), Hubara et al. (2016)). Zhou et al. (2017) use weights and activations that are zero or powers of two. Lin et al. (2015) and Zhou et al. (2016) quantize backpropagation in addition to the forward propagation.
+
+Beyond merely seeking to compress neural networks, the analysis of the internal representations of neural networks is useful for understanding how to to compress them. Agrawal et al. (2014) found that feature magnitudes in higher layers do not matter (e.g. binarizing features barely changes classification performance). Merolla et al. (2016) analyze the robustness of neural network representations to a collection of different distortions. Dosovitskiy & Brox (2016) observe that binarizing features in intermediate layers of a CNN and then using backpropagation to find an image with those features leads to relatively little distortion of the image compared to dropping out features. These papers naturally lead into our work where we are seeking to better understand the representations in neural networks based on the geometry of high-dimensional binary vectors.
+
+# 3 THEORY AND EXPERIMENTS
+
+We investigate the internal representations of neural networks with binary weights and activations. A binary neural network is trained on CIFAR-10 (same learning algorithm and architecture as in Courbariaux et al. (2016)). Experiments on MNIST were carried out using both fully connected and convolutional networks and produced similar results. The CIFAR-10 convolutional neural network has six layers of convolutions, all of which have a 3 by 3 spatial kernel. The number of feature maps in each layer are 128, 128, 256, 256, 512, 512. After the second, fourth, and sixth convolutions, there is a 2 by 2 max pooling operation. Then there are two fully connected layers with 1024 units each. Each layer has a batch norm layer in between. The experiments using ternary neural networks use the same network architecture. The dimensionality of the weight vectors in these networks (i.e. convolution converted to a matrix multiply) is the patch size $( = 3 * 3 = 9$ ) times the number of channels.
+
+# 3.1 PRESERVATION OF DIRECTION DURING BINARIZATION
+
+In this section, we analyze the angle distributions (i.e. geometry) of high-dimensional binary vectors. This is crucial for understanding binary neural networks because we can imagine that at each layer of a neural network, there are some ideal continuous weight vectors that extract out features. A binary neural network approximates these ideal continuous vectors with a binary vectors. In low dimensions, binarization strongly impacts the direction of a vector. However, we argue that binarization does not substantially change the direction of a high-dimensional continuous vector. It is often the case that the geometric properties of high-dimensional vectors are counter-intuitive. For instance, one key idea in the hyperdimensional computing theory of Kanerva (2009) is that two random, high-dimensional vectors of dimension $d$ whose entries are chosen uniformly from the set $\{ - 1 , 1 \}$ are approximately orthogonal. The result follows from the central limit theorem because the cosine angle between two√ such random vectors is normally distributed with $\mu = 0$ and $\sigma \sim 1 / \sqrt { d }$ . Then $\cos \theta \approx 0$ implies that $\theta \approx \frac { \pi } { 2 }$ . Building upon this work, we study the way in which binary vectors are distributed relative to continuous vectors. As binarizing a continuous vector gives the binary vector closest in angle to that continuous vector, we can get a sense of how binary vectors are distributed relative to continuous vectors in high dimensions by binarizing continuous vectors. The standard normal distribution, which serves as an informative null distribution because it is rotationally invariant, is used to generate random continuous vectors which are then binarized. This analysis gives a fundamental insight into understanding the recent success of binary neural networks. Binarizing a random continuous vector changes its direction by a small amount relative to the angle between two random vectors in moderately high dimensions (Fig. 2a). Binarization changes the direction of a vector by approximately $3 7 ^ { \circ }$ in high dimensions. This seems like a large change based on our low-dimensional intuition. Indeed, the angle between two randomly chosen vectors from a rotationally invariant distribution is uniform in two dimensions. However, two randomly chosen vectors are approximately orthogonal in high dimensions. Thus while it is common for two random vectors to have an angle less than $3 7 ^ { \circ }$ in low dimensions, it is exceedingly rare in high dimensions. Therefore $3 7 ^ { \circ }$ is a small angle in high dimensions.
+
+In order to test our theory of the binarization of random vectors chosen from a rotationally invariant distribution, we train a multilayer binary CNN on CIFAR10 (using the Courbariaux et al. (2016) method) and study the weight vectors1 of that network. Remarkably, there is a close correspondence between the experimental results and the theory for the angles between the binary and continuous weights (Fig. 2b). For each layer, the distribution of the angles between the binary and continuous weights is sharply peaked near the $d \to \infty$ expectation of arccos $\scriptstyle { \sqrt { 2 / \pi } }$ . We note that there is a small but systematic deviation from the theory towards larger angles for the higher layers of the network (Fig. 6). Ternary neural networks are considered in (SI Sec. 5.5) and yield a similar result.
+
+# 3.2 DOT PRODUCT PROPORTIONALITY AS A SUFFICIENT CONDITION FOR APPROXIMATING ANETWORK WITH CONTINUOUS WEIGHTS
+
+Given the previous discussion, an important question to ask is: are the so-called continuous weights a learning artifact without a clear correspondence to the binary weights? While we know that $w ^ { b } = \theta ( \overline { { w } } ^ { c } )$ , there are many continuous weights that map onto a particular binary weight vector. Which one is found when using the straight-through estimator to backpropagate through the binarize function? Remarkably, there is a clear answer to this question. In numerical experiments, we see that one gets the continuous weight vector such that the dot products of the activations with the prebinarization and post-binarization weights are highly correlated (Fig. 3). In equations, $a \cdot w ^ { b } \sim a \cdot w ^ { c }$ . We call this relation the Dot Product Proportionality (DPP) property. The proportionality constant, which is subsequently normalized away by a batch norm layer, depends on the magnitudes of the continuous and binary weight vectors and the cosine angle between the binary and continuous weight vectors. The theoretical consequences of the DPP property are explored in the rest of this section.
+
+We show that the modified gradient of the BNN training algorithm can be viewed as an estimator of the gradient that would be used to train the continuous weights in traditional backpropagation. This establishes the fundamental point that while the weights and activations are technically binary, they are operating as if the weights are continuous. For instance, one could imagine using an exhaustive search over all binary weights in the network. However, the additional structure in the problem associated with taking dot products makes the optimization simpler than that. Furthermore, we show that if the dot products of the activations with the pre-binarized and post-binarized weights are proportional then straight-through estimator gradient is proportional to the continuous weight network gradient. The key to the analysis is to focus on the transformers in the network whose forward and backward propagation functions are not related in the way that they would normally be related in typical gradient descent.
+
+![](images/769e8b20ccd61d1d0156f486458785077da1131dbe1c0038b64d4920a6a769e4.jpg)  
+Figure 2: Binarization of High-Dimensional Vectors Approximately Preserves their Direction in Theory and Practice: (a) Distribution of angles between two random vectors (blue), and between a vector and its binarized version (red), for a rotationally invariant distribution of dimension $d$ . The red distribution is peaked near the $d \to \infty$ limit of arccos $\sqrt { 2 / \pi } \approx 3 7 ^ { \circ }$ (SI, Sec. 1). While $3 7 ^ { \circ }$ may seem like a large angle, that angle is small as compared to the angle between two random vectors in moderately high dimensions (i.e. the blue and red curves are well-separated). (b) Angle distribution between continuous and binary weight vectors by layer for a binary CNN trained on CIFAR-10. For the higher layers, there is a close correspondence to the theory. There is a small, but systematic deviation towards large angles (SI, Fig. 6). $d$ is the dimension of the filters at each layer. (c) Standard deviations of the angle distributions from (b) by layer. We see a correspondence to the theoretical expectation that standard deviations of each of the angle distributions scales as $d ^ { - 0 . 5 }$ (SI, Sec. 1). (d) Histogram of the components of the continuous weights at each layer. The distribution is approximately Gaussian for all but the first layer. Furthermore, there is a high density of weights near zero, which is the threshold for the binarization function.
+
+Suppose that there is a neural network where two tensors, $u$ , and $v$ and the associated derivatives of the cost with respect to those tensors, $\delta u$ , and $\delta \boldsymbol { v }$ , are allocated. Suppose that the loss as a function of $v$ is $L ( x ) | _ { x = v }$ . Further, suppose that there are two potential forward propagation functions, $f$ , and $g$ . If the network is trained under normal conditions using $g$ as the forward propagation function, then the following computations are done:
+
+$$
+v  g ( u ) \qquad \delta v  L ^ { \prime } ( x = v = g ( u ) ) \qquad \delta u  \delta v \cdot g ^ { \prime } ( u )
+$$
+
+where $L ^ { \prime } ( x )$ denotes the derivative of $L$ with respect to the vector $x$ . In a modified backpropagation scheme, the following computations are done
+
+$$
+v  f ( u ) \qquad \delta v = L ^ { \prime } ( x = v = f ( u ) ) \qquad \delta u  \delta v \cdot g ^ { \prime } ( u )
+$$
+
+A sufficient condition for $\delta u$ to be the same in both cases is $L ^ { \prime } ( x = f ( u ) ) \sim L ^ { \prime } ( x = g ( u ) )$ where $a \sim b$ means that the vector $a$ is a scalar times the vector $b$ .
+
+Now this general observation is applied to the binarize transformer of Fig. 1. Here, $u$ is the continuous weight, $w ^ { c }$ , $f ( u )$ is the pointwise binarize function, $g ( u )$ is the identity function2, and $L$ is the loss of the network as a function of the weights in a particular layer. Given the network architecture, $L ( x ) = M ( a \cdot x )$ where $a$ are the activations corresponding to that layer and $M$ is the loss as a function of the weight-activation dot products. Then $L ^ { \prime } ( x ) = M ^ { \prime } ( a \cdot x ) \odot a$ where $\odot$ denotes a pointwise multiply. Thus the sufficient condition is $M ^ { \prime } ( a \cdot w ^ { b } ) \sim M ^ { \prime } ( a \cdot w ^ { c } )$ . Since the dot products are followed by a batch normalization, $M ( k { \vec { x } } ) = M ( { \vec { x } } )  M ^ { \prime } ( { \vec { x } } ) = k M ^ { \prime } ( k { \vec { x } } )$ . Therefore, it is sufficient that $\smash { \dot { a } \cdot w ^ { b } \sim a \cdot w ^ { c } }$ , which is the DPP property. When the DPP only approximately holds, the second derivative can be used to bound the error between the two gradients of the two learning procedures. In summary, the learning dynamics where $g$ is used for the forward and backward passes (i.e. training the network with continuous weights) is approximately equivalent to the modified learning dynamics $f$ on the forward pass, and $g$ on the backward pass) when we have the DPP property.
+
+While we demonstrated that the BNN learning dynamics approximate the dynamics that one would have by training a network with continuous weights using a mixture of empirical and theoretical arguments, the ideal result would be that the learning algorithm implies the DPP property. It should be noted that in the case of stochastic binarization where $E ( w ^ { b } ) \dot { = } w ^ { c }$ is chosen by definition, the DPP property is true by design. However, it is remarkable that the property still holds in the case of deterministic binarization, which is revealing of the fundamental nature of the representations used in neural networks.
+
+While the main focus of this section is the binarization of the weights, the arguments presented can also be applied to the binarize block that corresponds to the non-linearity of the network. The analogue of the DPP property for this binarize block is: $w ^ { b } \cdot a ^ { c } \sim w ^ { b } \cdot a ^ { b }$ where $a ^ { c }$ denotes the pre-binarized (post-batch norm) activations and $a ^ { b } = a$ denotes the binarized activations. This property is empirically verified to hold. For the sake of completeness, the dot product histogram corresponding to $w ^ { c } \cdot \bar { a } ^ { c } \sim w ^ { b } \cdot a ^ { b }$ is also computed, although it doesn’t directly correspond to removing one instance of a binarize transformer. This property is also empirically verified to hold (SI, Fig. 5).
+
+Impact on Classification: It is natural to ask to what extent the classification performance depends on the binarization of the weights. In experiments on CIFAR10, if the binarization of the weights on all of the convolutional layers is removed, the classification performance drops by only 3 percent relative to the original network. Looking at each layer individually, removing the weight binarization for the first layer accounts for this entire percentage, and removing the binarization of the weights for each other layer causes no degradation in performance. This result is evident by looking at the 2D dot product histograms in Fig 3. The off-diagonal quadrants show where switching the weights from binary to continuous changes the sign of the binarized weight-activation dot product. In all of the layers except the first layer, there are very few dot products in the off-diagonal quadrants. Thus we recommend the use of the dot product histograms for studying the performance of binary neural networks. Removing the binarization of the activations has a substantial impact on the classification performance because that removes the main non-linearity of the network.
+
+# 3.3 INPUT CORRELATIONS AND THE GENERALIZED BINARIZATION TRANSFORMATION
+
+Not surprisingly, some distributions are impacted more strongly by binarization than others. A binary neural network must adapt its internal representations in such a way to not be degraded too much by binarization at each layer. In this section we explore the idea that the principal components of the input to the binarization function should be randomly oriented relative to the binarization. While the network can adapt the higher level representations to satisfy this property, the part of the network that interfaces with the input doesn’t have that flexibility. We make the novel observation that the difficulties in training the first layer of the network are tied to the intrinsic correlations in the input data. In order to be more precise, we define the Generalized Binarization Transformation (GBT)
+
+$$
+\theta _ { R } ( x ) = R ^ { T } \theta ( R x )
+$$
+
+where $x$ is a column vector, $R$ is a fixed rotation matrix, and $\theta$ is the pointwise binarization function from before. The rows of $R$ are called the axes of binarization. If $R$ is the identity matrix, then $\theta _ { R } = \theta$ and the axes of binarization are the canonical basis vectors $( . . . , 0 , 1 , 0 , . . . ) .$ $R$ can either be chosen strategically or randomly.
+
+The GBT changes the distribution being binarized through a rotation. For appropriate choices of the rotation, $R$ , the directions of the input vectors, $x$ , are changed insignificantly by binarization. The angle between a vector and its binarized version is dependent on the dot product: ${ \bf \dot { x } } \cdot \theta _ { R } ( x )$ , which is equal to $x ^ { T } \theta _ { R } ( x ) = ( R x ) ^ { T } \theta ( R x ) = y \cdot \theta ( y )$ where $y = R x$ . As a concrete example of the benefits of the GBT, consider the case where $x \sim N ( 0 , \Sigma )$ and $\Sigma _ { i , j } = \delta _ { i , j } \exp ( 2 k i )$ for $k = 0 . 1$ (therefore $y \sim N ( 0 , R \Sigma R ^ { T } ) )$ . As the dimension goes to infinity, the angle between a vector drawn from this distribution and its binarized version approaches $\pi / 2$ . Thus binarization is destructive to vectors from this distribution. However, if the GBT is applied with a fixed random matrix3, the angle between the vector and its binarized version converges to $3 7 ^ { \circ }$ (Fig. 4). Thus a random rotation can compensate for the errors incurred from directly binarizing a non-isotropic Gaussian.
+
+![](images/dda78feb7ec99827c1081c16d52cd5c5917964a5de5826e4fcc43dfa4aa7f925.jpg)  
+Figure 3: Binarization Preserves Dot Products: Each subplot shows a 2D histogram of the dot products between the binarized weights and the activations (horizontal axis) and the dot products between the continuous weights and the activations (vertical axis) for different layers of a network trained on CIFAR10. Surprisingly, the dot products are highly correlated ( $r$ is the Pearson correlation coefficient). Thus replacing $\boldsymbol { w ^ { b } }$ with $w ^ { c }$ changes the overall constant in front of the dot products, while still preserving whether the dot product is zero or not zero. This overall constant is divided out by the subsequent batch norm layer. The shaded quadrants correspond to dot products where the sign changes when replacing the binary weights with the continuous weights. Notice that for all but the first layer, a very small fraction of the dot products lie in these off diagonal quadrants. The top left figure (labeled as Layer 1) corresponds to the input and the first convolution. Note that the correlation is weaker in the first layer.
+
+Moving into how this analysis applies to a binary neural network, the network weights must approximate the important directions in the activations using binary vectors. For instance, Gabor filters are intrinsic features in natural images and are often found in the first layer weights of neural networks trained on natural images (e.g. Olshausen et al. (1996); Krizhevsky et al. (2012)). While the network has flexibility in the higher layers, the first layer must interface directly with the input where the features are not necessarily randomly oriented. For instance, consider the 27 dimensional input to the first set of convolutions in our network: 3 color channels of a 3 by 3 patch of an image from CIFAR10 with the mean removed. 3 PCs capture 90 percent of the variance of this data and 4 PCs capture 94.5 percent of the variance. The first two PCs are spatially uniform colors. More generally, large images such as those in IMAGENET have the same issue. Translation invariance of the image covariance matrix implies that the principal components are the filters of the 2D Fourier transform. Scale invariance implies a $1 / f ^ { 2 }$ power spectrum, which results in the largest PCs corresponding to low frequencies (Field (1987)).
+
+Another manifestation of this issue can be seen in our trained networks. The first layer has a much smaller dot product correlation than the other layers (Fig. 3). To study this, we randomly permute the activations in order to generate a distribution with the same marginal statistics as the original data but independent joint statistics (a different permutation for each input image). Such a transformation gives a distribution with a correlation equal to the normalized dot product of the weight vectors (SI Sec. 3). The correlations for the higher layers decrease substantially but the correlation in the first layer increases (Fig. 4, right). For the first layer, the shuffling operation randomly permutes the pixels in the image. Thus we demonstrate that the binary weight vectors in the first layer are not well-aligned with the continuous weight vectors relative to the input data. Our theoretically grounded analysis is consistent with previous work. Han et al. (2015b) find that compressing the first set of convolutional weights of a particular layer by the same fraction has the highest impact on performance if done on the first layer. Zhou et al. (2016) find that accuracy degrades by about 0.5 to 1 percent on SHVN when quantizing the first layer weights. Thus it is recommended to rotate the input data before normalization or to use continuous weights for the first layer.
+
+![](images/4f8d8910287dc01352ec7948a857f9613ad9a594df342141736e9ae42f70e271.jpg)  
+Figure 4: Left: Random Rotation Improves Angle Preservation for a Non-Isotropic Gaussian. Random vectors are drawn from a Gaussian of dimension $d$ with a diagonal covariance matrix whose entries vary exponentially. As in Fig. 2, the red curve shows the angle between a random vector and its binarized version. Since the Gaussian is no longer isotropic, the red curve no longer peaks at $\theta = \operatorname { a r c c o s } { \sqrt { 2 / \pi } } .$ . However, if the binarization is replaced with a GBT with a fixed random matrix, the direction of the vector is again approximately preserved. Right: Permuting the activations shows that the correlations observed in Fig. 3 are not merely due to correlations between the binary and continuous weight vectors. The correlations are due to these weight vectors corresponding to high variance directions in the data.
+
+# 4 CONCLUSION
+
+Neural networks with binary weights and activations have similar performance to their continuous counterparts with substantially reduced execution time and power usage. We provide an experimentally verified theory for understanding how one can get away with such a massive reduction in precision based on the geometry of HD vectors. First, we show that binarization of high-dimensional vectors preserves their direction in the sense that the angle between a random vector and its binarized version is much smaller than the angle between two random vectors (Angle Preservation Property). Second, we take the perspective of the network and show that binarization approximately preserves weight-activation dot products (Dot Product Proportionality Property). More generally, when using a network compression technique, we recommend looking at the weight activation dot product histograms as a heuristic to help localize the layers that are most responsible for performance degradation. Third, we discuss the impacts of the low effective dimensionality of the data on the first layer of the network. We recommend either using continuous weights for the first layer or a Generalized Binarization Transformation. Such a transformation may be useful for architectures like LSTMs where the update for the hidden state declares a particular set of axes to be important (e.g. by taking the pointwise multiply of the forget gates with the cell state). Finally, we show that neural networks with ternary weights and activations can also be understood with our approach. More broadly speaking, our theory is useful for analyzing a variety of neural network compression techniques that transform the weights, activations or both to reduce the execution cost without degrading performance.
+
+# REFERENCES
+
+Pulkit Agrawal, Ross Girshick, and Jitendra Malik. Analyzing the performance of multilayer neural networks for object recognition. In European Conference on Computer Vision, pp. 329–344. Springer, 2014.
+
+Hande Alemdar, Nicholas Caldwell, Vincent Leroy, Adrien Prost-Boucle, and Frédéric Pétrot. Ternary neural networks for resource-efficient ai applications. arXiv preprint arXiv:1609.00222, 2016.
+
+Renzo Andri, Lukas Cavigelli, Davide Rossi, and Luca Benini. Yodann: An architecture for ultra-low power binary-weight cnn acceleration. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2017.
+
+Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013.
+
+Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Training deep neural networks with low precision multiplications. arXiv preprint arXiv:1412.7024, 2014.
+
+Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. In Advances in Neural Information Processing Systems, pp. 3123–3131, 2015.
+
+Matthieu Courbariaux, Itay Hubara, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks: Training neural networks with weights and activations constrained to $+ 1$ and $^ { - 1 }$ arXiv preprint arXiv:1602.02830, 2016.
+
+Alexey Dosovitskiy and Thomas Brox. Inverting visual representations with convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4829– 4837, 2016.
+
+Steven K Esser, Paul A Merolla, John V Arthur, Andrew S Cassidy, Rathinakumar Appuswamy, Alexander Andreopoulos, David J Berg, Jeffrey L McKinstry, Timothy Melano, Davis R Barch, et al. Convolutional networks for fast, energy-efficient neuromorphic computing. Proceedings of the National Academy of Sciences, pp. 201604850, 2016.
+
+David J Field. Relations between the statistics of natural images and the response properties of cortical cells. Josa a, 4(12):2379–2394, 1987.
+
+Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. In ICML, pp. 1737–1746, 2015.
+
+Philipp Gysel, Mohammad Motamedi, and Soheil Ghiasi. Hardware-oriented approximation of convolutional neural networks. arXiv preprint arXiv:1604.03168, 2016.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015a.
+
+Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In Advances in Neural Information Processing Systems, pp. 1135–1143, 2015b.
+
+Song Han, Jeff Pool, Sharan Narang, Huizi Mao, Enhao Gong, Shijian Tang, Erich Elsen, Peter Vajda, Manohar Paluri, John Tran, et al. Dsd: Dense-sparse-dense training for deep neural networks. arXiv prepring arXiv:1607.04381, 2017.
+
+Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Quantized neural networks: Training neural networks with low precision weights and activations. arXiv preprint arXiv:1609.07061, 2016.
+
+Patrick Judd, Jorge Albericio, Tayler Hetherington, Tor Aamodt, Natalie Enright Jerger, Raquel Urtasun, and Andreas Moshovos. Reduced-precision strategies for bounded memory in deep neural nets. arXiv preprint arXiv:1511.05236, 2015.
+
+Pentti Kanerva. Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors. Cognitive Computation, 1(2):139–159, 2009.
+
+Minje Kim and Paris Smaragdis. Bitwise neural networks. arXiv preprint arXiv:1601.06071, 2016.
+
+Yong-Deok Kim, Eunhyeok Park, Sungjoo Yoo, Taelim Choi, Lu Yang, and Dongjun Shin. Compression of deep convolutional neural networks for fast and low power mobile applications. arXiv preprint arXiv:1511.06530, 2015.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Liangzhen Lai, Naveen Suda, and Vikas Chandra. Deep convolutional neural network inference with floating-point weights and fixed-point activations. arXiv preprint arXiv:1703.03073, 2017.
+
+Fengfu Li, Bo Zhang, and Bin Liu. Ternary weight networks. arXiv preprint arXiv:1605.04711, 2016.
+
+Darryl Lin, Sachin Talathi, and Sreekanth Annapureddy. Fixed point quantization of deep convolutional networks. In International Conference on Machine Learning, pp. 2849–2858, 2016.
+
+Zhouhan Lin, Matthieu Courbariaux, Roland Memisevic, and Yoshua Bengio. Neural networks with few multiplications. arXiv preprint arXiv:1510.03009, 2015.
+
+Paul Merolla, Rathinakumar Appuswamy, John Arthur, Steve K Esser, and Dharmendra Modha. Deep neural networks are robust to weight binarization and other non-linear distortions. arXiv preprint arXiv:1606.01981, 2016.
+
+Emre Neftci, Charles Augustine, Somnath Paul, and Georgios Detorakis. Neuromorphic deep learning machines. arXiv preprint arXiv:1612.05596, 2016.
+
+Bruno A Olshausen et al. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583):607–609, 1996.
+
+Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. arXiv preprint arXiv:1603.05279, 2016.
+
+Gilbert W Stewart. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM Journal on Numerical Analysis, 17(3):403–409, 1980.
+
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014.
+
+Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. arXiv preprint arXiv:1702.03044, 2017.
+
+Shuchang Zhou, Yuxin Wu, Zekun Ni, Xinyu Zhou, He Wen, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. arXiv preprint arXiv:1606.06160, 2016.
+
+Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016.
+
+# 5 SUPPLEMENTARY INFORMATION
+
+# 5.1 EXPECTED ANGLES
+
+Random $n$ dimensional vectors are drawn from a rotationally invariant distribution. The angle between two random vectors and the angle between a vector and its binarized version are compared. A rotationally invariant distribution can be factorized into a pdf for the magnitude of the vector times a distribution on angles. In the expectations that we are calculating, the magnitude cancels out and there is only one rotationally invariant distribution on angles. Thus it suffices to compute these expectations using a Gaussian.
+
+# Lemmas:
+
+1. Consider a vector, $v$ , chosen from a standard normal distribution of dimension $n$ . Let $\begin{array} { r } { \rho = \frac { v _ { 1 } } { \sqrt { v _ { 1 } ^ { 2 } + \ldots + v _ { n } ^ { 2 } } } } \end{array}$ . Then $\rho$ is distributed according to: $\begin{array} { r } { g ( \rho ) = \frac { 1 } { \sqrt { \pi } } \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n - 1 ) / 2 ) } ( 1 - \rho ^ { 2 } ) ^ { \frac { n - 3 } { 2 } } } \end{array}$ where $\Gamma$ is the Gamma function. Proof: Begin by considering the integral $\begin{array} { r } { G ( \rho _ { 0 } ) = p ( \rho < \rho _ { 0 } ) = \int \prod _ { i } d v _ { i } p ( v ) I ( \rho ( v ) < \rho _ { 0 } ) } \end{array}$ where $I$ is an indicator function. The desired distribution comes from taking the derivative of this cumulative distribution g(ρ0) = ddρ0 . Consider the generalized spherical coordinate transformation: $v _ { 1 } = r \cos \phi _ { 1 }$ , $\begin{array} { r } { \dot { v } _ { 2 } = r \sin \phi _ { 1 } \cos \phi _ { 2 } . } \end{array}$ , $v _ { 3 } = r \sin \phi _ { 1 } \sin \phi _ { 2 } \cos \phi _ { 3 }$ , $\begin{array} { r } { \ldots \quad v _ { n - 1 } = r \sin \phi _ { 1 } \cdot \cdot \cdot \sin \phi _ { n - 2 } \cos \phi _ { n - 1 } , } \end{array}$ $v _ { n } ~ = ~ r \sin \phi _ { 1 } \cdot \cdot \cdot \sin \phi _ { n - 2 } \sin \phi _ { n - 1 }$ where $\phi _ { 1 } , \ldots , \phi _ { n - 2 } \in [ 0 , \pi ]$ and $\phi _ { n - 1 } \in [ 0 , 2 \pi ]$ . The generalized spherical volume element is $d v = r ^ { n - 1 } \sin ^ { n - 2 } ( \phi _ { 1 } ) \sin ^ { n - 3 } ( \phi _ { 2 } ) \cdot \cdot \cdot \sin ( \phi _ { n - 2 } ) d r d \phi _ { 1 } \cdot \cdot \cdot d \phi _ { n - 1 }$ . 4 Thus we can write out the integral $\begin{array} { r } { G ( \rho _ { 0 } ) = \int I ( \cos \phi _ { 1 } < \rho _ { 0 } ) \frac { e ^ { - r ^ { 2 } / 2 } } { \sqrt { ( 2 \pi ) ^ { n } } } r ^ { n - 1 } \sin ^ { n - 2 } ( \phi _ { 1 } ) \sin ^ { n - 3 } ( \phi _ { 2 } ) \cdot \cdot \cdot \sin ( \phi _ { n - 2 } ) d r d \phi _ { 1 } \cdot \cdot \cdot d \phi _ { n - 1 } } \end{array}$ The integral factorizes and all of the terms are independent of $\rho _ { 0 }$ except the integral over $\phi _ { 1 } \equiv \phi$ . Thus $\begin{array} { r } { G ( \rho _ { 0 } ) \sim \int _ { 0 } ^ { \pi } I ( \cos \phi < \rho _ { 0 } ) \sin ^ { \hat { n } - 2 } ( \phi ) d \phi } \end{array}$ . Using the substitution $\rho = \cos \phi$ (which is also consistent with the definition of $\rho$ above), $d \rho = - \sin ( \phi ) d \phi$ , $\sin \rho ~ = ~ ( 1 - \rho ^ { 2 } ) ^ { 0 . 5 }$ , so $\begin{array} { r } { G ( \rho _ { 0 } ) ~ \sim ~ \int _ { - 1 } ^ { 1 } I ( \rho ~ < ~ \rho _ { 0 } ) ( 1 - \rho ^ { 2 } ) ^ { ( n - 3 ) / 2 } d \rho } \end{array}$ . Taking the derivative with respect to $\rho _ { 0 }$ and using the fundamental theorem of calculus gives $g ( \rho ) \sim ( 1 - \rho ^ { 2 } ) ^ { ( \bar { n - 3 } ) / 2 }$ . The normalization constant is equal to a beta function that evaluates to the desired result (substitute $t = \rho ^ { 2 }$ ).
+
+Cases:
+
+• Distribution of angles between two random vectors. Since a Gaussian is a rotationally invariant distribution, we can say without loss of generality that one of the vectors is $( 1 , 0 , 0 , \ldots 0 )$ . Then the cosine angle between those two vectors is $\rho$ as defined above. While the exact distribution of $\rho$ is given by Lemma 1, we note that – $E ( \rho ) = 0$ due to the symmetry of the distribution. – $\begin{array} { r } { V a r ( \rho ) = E ( \rho ^ { 2 } ) = \frac { 1 } { n } } \end{array}$ because $\begin{array} { r } { 1 = E \left( \frac { \sum _ { i } x _ { i } ^ { 2 } } { \sum _ { j } x _ { j } ^ { 2 } } \right) = \sum _ { i } E \left( \frac { x _ { i } ^ { 2 } } { \sum _ { j } x _ { j } ^ { 2 } } \right) = n * E ( \rho ^ { 2 } ) } \end{array}$
+
+• Angles between a vector and the binarized version of that vector, η = v·θ(v)||v||·||θ(v)|| $\frac { \sum _ { i } \left| v _ { i } \right| } { \sqrt { \sum v _ { i } ^ { 2 } } * \sqrt { n } }$
+
+$$
+E ( \eta ) = \frac { \sqrt { n } } { \sqrt { \pi } } * \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n + 1 ) / 2 ) } \qquad \operatorname* { l i m } _ { n \to \infty } E ( \eta ) = \sqrt { \frac { 2 } { \pi } }
+$$
+
+First, we note $E ( \eta ) = \sqrt { n } E ( | \rho | )$ . Then $\begin{array} { r } { E ( | \rho | ) = \int _ { 0 } ^ { 1 } d \rho \rho \frac { 2 } { \sqrt { \pi } } \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n - 1 ) / 2 ) } ( 1 - \rho ^ { 2 } ) ^ { \frac { n - 3 } { 2 } } = } \end{array}$ $\frac { 2 } { \sqrt { \pi } } \ast \frac { 1 } { n - 1 } \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n - 1 ) / 2 ) }$ (substitute $u \ : = \ : \rho ^ { 2 }$ and use $\Gamma ( x + 1 ) = x \Gamma ( x )$ ). Lemma
+
+two gives the $n  \infty$ limit. $\begin{array} { r l r } { \frac { 2 } { \sqrt { \pi } } \frac { \sqrt { n } } { n - 1 } \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n - 1 ) / 2 ) } \ \approx \ \frac { 2 } { \sqrt { \pi } } \frac { \sqrt { n } } { n - 1 } \sqrt { \frac { n } { 2 } } \left[ 1 + \frac { 0 . 5 * 0 . 5 } { 2 ( n / 2 ) } \right] \ = } \end{array}$ q 2π 1 + 54 ∗ 1n 
 + O(1/n2)
+
+$$
+{ V a r } ( \eta ) = \frac { 1 } { n } \left( 1 - \frac { 1 } { \pi } \right) + { \cal O } ( 1 / n ^ { 2 } )
+$$
+
+Thus we have the normal scaling as in the central limit theorem of the large $n$ variance.   
+We can calculate this explicitly following the approach of 5.
+
+As $E ( \eta )$ has been calculated, it suffices to calculate $E ( \eta ^ { 2 } )$ . Expanding out $\eta ^ { 2 } , E ( \eta ^ { 2 } ) =$ $\begin{array} { r } { \frac { 1 } { n } + ( n - 1 ) * E \bigl ( \frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + \ldots v _ { n } ^ { 2 } } \bigr ) } \end{array}$ . Below we show that $\begin{array} { r } { E ( \frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + \ldots v _ { n } ^ { 2 } } ) = \frac { 2 } { \pi n } } \end{array}$ . Thus the variance is:
+
+$$
+{ \frac { 1 } { n } } * \left( 1 - { \frac { 2 } { \pi } } \right) + { \frac { 2 } { \pi } } - \left( { \frac { \sqrt { n } } { \sqrt { \pi } } } { \frac { \Gamma ( n / 2 ) } { \Gamma ( ( n + 1 ) / 2 ) } } \right) ^ { 2 }
+$$
+
+Using Lemma 2 to expand out the last term, we get $[ \frac { \sqrt { n } } { \sqrt { \pi } } ( n / 2 ) ^ { - 1 / 2 } ( 1 - 1 / ( 4 n ) +$ $O ( n ^ { - 2 } ) ) ] ^ { 2 } = \frac { _ { 2 } } { \pi } ( 1 - { 1 } / ( { 2 n } ) + O ( n ^ { - 2 } ) )$ . Plugging this in gives the desired result.
+
+Going back to the calculation of that expectation, change variables to $v _ { 1 } = r \cos \theta$ , $v _ { 2 } = r \sin { \theta }$ , $z ^ { 2 } = v _ { 3 } ^ { 2 } + \ldots + v _ { n } ^ { 2 }$ . The integration over the volume element $d v _ { 3 } \ldots d v _ { n }$ is rewritten as $d z d A _ { n - 3 }$ where $d A _ { n }$ denotes the surface element of a $n$ sphere. Since the integrand only depends on the magnitude, $z$ , $\textstyle { \int d A _ { n - 3 } = z ^ { n - 3 } * { \dot { S } } _ { n - 3 } }$ where $\begin{array} { r } { S _ { n } = \frac { 2 \pi ^ { ( n + 1 ) / 2 } } { \Gamma ( \frac { n + 1 } { 2 } ) } } \end{array}$ denotes the surface area of a unit $n$ -sphere. Then
+
+$$
+E \left( \frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + . . . v _ { n } ^ { 2 } } \right) = ( 2 \pi ) ^ { - n / 2 } S _ { n - 3 } \int _ { 0 } ^ { 2 \pi } d \theta | \cos \theta \sin \theta | * \int r d r z ^ { n - 3 } d z * \frac { r ^ { 2 } } { r ^ { 2 } + z ^ { 2 } } * e ^ { - ( z ^ { 2 } + r ^ { 2 } ) / 2 } .
+$$
+
+Then substitute $r = p \cos \phi$ , $z = p \sin \phi$ where $\phi \in [ 0 , \pi / 2 ]$
+
+$$
+= ( 2 \pi ) ^ { - n / 2 } \ast 2 S _ { n - 3 } \int _ { 0 } ^ { \pi / 2 } d \phi \cos \phi ^ { 3 } \ast \sin \phi ^ { n - 3 } \int _ { 0 } ^ { \infty } d p \ast p ^ { n - 1 } e ^ { - p ^ { 2 } / 2 }
+$$
+
+The first integral is $\frac { 2 } { n ( n - 2 ) }$ using $u = \sin ^ { 2 } \phi$ . The second integral is $2 ^ { ( n - 2 ) / 2 } \Gamma ( n / 2 )$ using $u = p ^ { 2 } / 2$ and the definition of the gamma function. Simplifying, the result is $\frac { 2 } { \pi * n }$
+
+Thus the angle between a vector and a binarized version of that vector converges to arccos ${ \sqrt { \frac { 2 } { \pi } } } \approx 3 7 ^ { \circ }$ , which is a very small angle in high dimensions.
+
+# 5.2 AN EXPLICIT EXAMPLE OF LEARNING DYNAMICS
+
+In this subsection, we look at the learning dynamics for the BNN training algorithm in a simple case and gain some insight about the learning algorithm. Consider the case of regression where the target output, $y$ , is predicted with a binary linear predictor with $x$ as the input. Using a squared error loss, ${ \cal L } \dot { = } ( y - \hat { y } ) \dot { ^ 2 } = ( y - w ^ { b } x ) ^ { 2 } = ( \bar { y } - \theta ( w ^ { c } \acute { } ) x ) ^ { 2 }$ . (In this notation, $x$ is a column vector.) Taking the derivative of this loss with respect to the continuous weights and using the rule for back propagating through the binarize function gives $\Delta w ^ { c } \sim - d L / d w ^ { c } = - d L / d w ^ { \check { b } } \cdot d w ^ { b } / d w ^ { c } = ( y ^ { \check { b } } - w ^ { \check { b } } x ) x ^ { \check { T } }$ . Finally, averaging over the training data gives
+
+$$
+\Delta w ^ { c } \sim C _ { y x } - \theta ( w ^ { c } ) \cdot C _ { x x } \qquad C _ { y x } = E [ y x ^ { T } ] \qquad C _ { x x } = E ( x x ^ { T } )
+$$
+
+It is worthwhile to compare this equation the corresponding equation from typical linear regression: $\Delta w ^ { c } \sim C _ { y x } - w ^ { c } \cdot C _ { x x }$ . For simplicity, consider the case where $C _ { x x }$ is the identity matrix. In this case, all of the components of $w$ become independent: $\delta w = \epsilon * ( \alpha - \theta ( w ) )$ where $\epsilon$ is the learning rate and $\alpha$ is the entry of $C _ { y x }$ corresponding to a particular element, $w$ . Compared to regular linear regression, it is clear that the stable point of these equations is when $w = \alpha$ . Since the weight is binarized, that equation cannot be satisfied. However, it can be shown $( \star )$ that in this special case of binary weight linear regression, $E ( \theta ( w ^ { c } ) ) = \alpha$ . Intuitively, if we consider a high dimensional vector and the fluctuations of each component are likely to be out of phase, then $\boldsymbol { w } ^ { b } \cdot \boldsymbol { x } \approx \boldsymbol { w } ^ { c } \cdot \boldsymbol { x }$ is going to be correct in expectation with a variance that scales as $\frac { 1 } { n }$ . During the actual learning process, we anneal the learning rate to a very small number, so the particular state of a fluctuating component of the vector is frozen in. Relatedly, the equation $C _ { y x } \approx w C _ { x x }$ is easier to satisfy in high dimensions, whereas in low dimensions, it is only satisfied in expectation.
+
+![](images/452741137351217b927a8a8324af28e914b59f5c7097c1f8fbf70109360b8f3e.jpg)  
+Figure 5: Activation Binarization Preserves Dot Products: Left: Each panel shows a 2D histogram of the dot products between the binarized weights and binarized activations (vertical axis) and post-batch norm (but pre-activation binarization) activations (horizontal axis). The binarization transformer does little to corrupt the dot products between weights and activations. Right: Dot products between the binary weights and binary activations (horizontal axis) compared against the dot products between the continuous weights and continuous activations (vertical axis). The dot products are not significantly impacted by removing binarization.
+
+Proof for $( \star )$ : Suppose that $| \alpha | \le 1$ . The basic idea of these dynamics is that steps of size proportional to $\epsilon$ are taken whose direction depends on whether $w > 0$ or $w < 0$ . In particular, if $w > 0$ , then the step is $- \epsilon \cdot | 1 - \alpha |$ and if $w < 0$ , the step is $\epsilon \cdot \left( \alpha + 1 \right)$ . It is evident that after a sufficient burn-in period, $| w | \leq \epsilon * \operatorname* { m a x } ( | 1 - \alpha | , 1 + \alpha ) \leq 2 \epsilon$ . Suppose $w > 0$ occurs with fraction $p$ and $w < 0$ occurs with fraction $1 - p$ . In order for $w$ to be in equilibrium, oscillating about zero, these steps balance out on average: $p ( 1 - \alpha ) = ( 1 - p ) ( 1 + \alpha )  p = ( 1 + \alpha ) / 2$ . Then the expected value of $\theta ( w )$ is $1 * p + ( - 1 ) * ( 1 - p ) = \alpha$ . When $| \alpha | > 1$ , the dynamics diverge because $\alpha - \theta ( w )$ will always have the same sign. This divergence demonstrates the importance of some normalization technique such as batch normalization or attempting to represent $w$ with a constant times a binary matrix.
+
+# 5.3 DOT PRODUCT CORRELATIONS AFTER ACTIVATION PERMUTATION
+
+Suppose that $A = w \cdot a$ and $\boldsymbol { B } = \boldsymbol { v } \cdot \boldsymbol { a }$ where $w , v$ are weight vectors and $a$ is the vector of activations. What is the correlation, $r$ , between $A$ and $B ?$ Assuming that $E ( a ) = 0$ , $E ( A ) = E ( B ) = 0$ . Then $\begin{array} { r } { E ( A B ) = \sum _ { i , j } w _ { i } v _ { j } E ( a _ { i } a _ { j } ) = w ^ { T } C v } \end{array}$ where $C _ { i , j } = E ( a _ { i } a _ { j } )$ . Setting $v = w$ and $w = v$ gives $E ( A ^ { 2 } ) = w ^ { T } C w$ and $v ^ { T } C v$ . Thus $\begin{array} { r } { r = \frac { w ^ { T } C v } { \sqrt { ( w ^ { T } C w ) ( v ^ { T } C v ) } } } \end{array}$ .
+
+In the case where the activations are randomly permuted, $C$ is proportional to the identity matrix, and thus the correlation between $A$ and $B$ is the cosine angle between $u$ and $v$ .
+
+# 5.4 ANGLE PLOT
+
+![](images/1acd23198be35f2ae8bf07828565f790fe198f42af03baa2da8695eb8ff226d2.jpg)  
+Figure 6: Angle distribution between continuous and binary weight vectors by layer for a binary CNN trained on CIFAR-10 (same plot as in Fig. 2b except zoomed in). Notice that there is a small but systematic deviation towards larger angles relative to the theoretical expectation (vertical dotted line). As the dimension of the vectors in the layer goes up, the distribution gets sharper. The theory√ predicts that the standard deviation of these distributions scales as $1 / \sqrt { d }$ . This relationship is shown to approximately hold in Fig. 2c.
+
+# 5.5 TERNARY NEURAL NETWORKS
+
+Moving beyond binarization, recent work has shown how to train neural networks where the activations are quantized to three (or more) values (e.g. Hubara et al. (2016)). Indeed, ternarization may be a more natural quantization method than binarization for neural network weights because one can express a positive association $( + 1 )$ , a negative association $( - 1 )$ , or no association (0) between two features in a neural network. We show that the analysis used on BNNs holds for ternary neural networks.
+
+The quantization function: t $\operatorname { x r } _ { a } ( x ) = 1$ if $x > a$ , 0 if $| x | < a$ , and $- 1$ if $x < - a$ is used in place of the binarize function with the same straight-through estimator for the gradient. Call $a$ the ternarization threshold. The exact value of $a$ is only important at initialization because the scaling constant of the batch normalization layer allows the network to adapt the standard deviation of the pre-nonlinearity activations to the value of $a$ . The network architecture from the previous experiments was used to classify images in CIFAR-10 and $a$ was chosen to be equal to 0.02. In practice, roughly 10 percent of the ternarized weights were zero (Fig. 7a) and 2 percent of the activations were zero. Thus the network learning process did not ignore the possibility of using zero weights. However, more work is needed to effectively use the zero value for the activations.
+
+The empirical distribution of angles between the continuous vectors and their ternarized counterparts is highly peaked at the value predicted by the theory (Fig. 7b). Random vectors are chosen from a standard normal distribution of dimension $d$ . As in the case of binarization, the ternarized version of a vector is close in angle to the original vector in high dimensions (Fig. 7c). These vectors are quantized using $\mathrm { t e r } _ { a }$ for different values of $a$ . The peak angle varies substantially as a function of $a$ (Fig. 7d). Note that for $a = 0$ , the ternarization function collapses into the binarization function. The empirical value of $a$ is the ratio of the empirical threshold to the empirical standard deviation of the continuous weights. Thus $a \approx 0 . 0 2 / 0 . 1 8 \approx 0 . 1 1$ for the higher layers. Remarkably, the theoretical prediction for the peak angle as a function of $a$ matches closely with the empirical result (Fig. 7b).
+
+Finally, the dot product proportionality property is also shown to hold for ternary neural networks (Fig. 7e). Thus the continuous weights found using the TNN training algorithm approximate the continuous weights that one would get if the network were trained with continuous weights and regular backpropagation.
+
+![](images/72bee4e85d8a824a5fa6b768abcb0acd56045eb210592b38724a042ad8feecdf.jpg)  
+Figure 7: Ternarization of High-Dimensional Vectors Preserves their Direction in Theory and Practice: (a) Histogram of the components of the continuous weights at each layer for a ternary CNN trained on CIFAR-10. The distribution is approximately Gaussian for all but the first layer. The vertical lines show the ternarization thresholds and approximately 10 percent of the weights are sent to zero. (b) Angle distribution between continuous and ternary weight vectors by layer. The vertical dotted line (at $\theta \approx 3 4 ^ { \circ }$ ) indicates the theoretical prediction given the empirical ternarization threshold $( \approx 0 . 1 1 )$ . $d$ is the dimension of the filters at each layer. (c) Distribution of angles between two random vectors (blue), and between a vector and its quantized version (red), for a rotationally invariant distribution of dimension $d$ . The ternarization threshold is chosen to match the trained network. As the red and blue curves have little overlap, ternarization causes a small change in angle in high dimensions. (d) Angle between a random vector and its ternarized version as a function of ternarization threshold for $d = 1 0 0 0$ . There is a large variation in the angle over different thresholds. (e) Ternarization preserves dot products. The 2D histogram shows the dot products between the ternarized weights and the activations (horizontal axis) and the dot products between the continuous weights and the activations (vertical axis) for layer 3 of the network. The dot products are highly correlated.

parse/train/B1IDRdeCW/B1IDRdeCW_content_list.json

@@ -0,0 +1,1462 @@
+[
+    {
+        "type": "text",
+        "text": "THE HIGH-DIMENSIONAL GEOMETRY OF BINARY NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            823,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexander G. Anderson   \nRedwood Center for Theoretical Neuroscience   \nUniversity of California, Berkeley   \naga@berkeley.edu ",
+        "bbox": [
+            183,
+            170,
+            488,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Cory P. Berg Redwood Center for Theoretical Neuroscience University of California, Berkeley cberg500@berkeley.edu ",
+        "bbox": [
+            506,
+            170,
+            813,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent research has shown that one can train a neural network with binary weights and activations at train time by augmenting the weights with a high-precision continuous latent variable that accumulates small changes from stochastic gradient descent. However, there is a dearth of work to explain why one can effectively capture the features in data with binary weights and activations. Our main result is that the neural networks with binary weights and activations trained using the method of Courbariaux, Hubara et al. (2016) work because of the high-dimensional geometry of binary vectors. In particular, the ideal continuous vectors that extract out features in the intermediate representations of these BNNs are well-approximated by binary vectors in the sense that dot products are approximately preserved. Furthermore, the results and analysis used on BNNs are shown to generalize to neural networks with ternary weights and activations. Compared to previous research that demonstrated good classification performance with BNNs, our work explains why these BNNs work in terms of HD geometry. Our theory serves as a foundation for understanding not only BNNs but a variety of methods that seek to compress traditional neural networks. Furthermore, a better understanding of multilayer binary neural networks serves as a starting point for generalizing BNNs to other neural network architectures such as recurrent neural networks. ",
+        "bbox": [
+            233,
+            291,
+            766,
+            541
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            566,
+            336,
+            582
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The rapidly decreasing cost of computation has driven many successes in the field of deep learning in recent years. Consequently, researchers are now considering applications of deep learning in resource limited hardware such as neuromorphic chips, embedded devices and smart phones (Esser et al. (2016), Neftci et al. (2016), Andri et al. (2017)). A recent realization for both theoretical researchers and industry practitioners is that traditional neural networks can be compressed because they are highly over-parameterized. While there has been a large amount of experimental work dedicated to compressing neural networks (Sec. 2), we focus on the particular approach that replaces costly 32-bit floating point multiplications with cheap binary operations. Our analysis reveals a simple geometric picture based on the geometry of high dimensional binary vectors that allows us to understand the successes of the recent efforts to compress neural networks. ",
+        "bbox": [
+            174,
+            597,
+            825,
+            736
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Courbariaux et al. (2016) and Hubara et al. (2016) showed that one can efficiently train neural networks with binary weights and activations that have similar performance to their continuous counterparts. Such BNNs execute 7 times faster using a dedicated GPU kernel at test time. Furthermore, they argue that such BNNs require at least a factor of 32 fewer memory accesses at test time that should result in an even larger energy savings. There are two key ideas in their papers (Fig. 1). First, a continuous weight, $w ^ { c }$ , is associated with each binary weight, $w ^ { b }$ , that accumulates small changes from stochastic gradient descent. Second, the non-differentiable binarize function $( \\theta ( x ) = 1$ if $x > 0$ and $- 1$ otherwise) is replaced with a continuous one during backpropagation. These modifications allow one to train neural networks that have binary weights and activations with stochastic gradient descent. While the work showed how to train such networks, the existence of neural networks with binary weights and activations needs to be reconciled with previous work that has sought to understand weight matrices as extracting out continuous features in data (e.g. Zeiler & Fergus (2014)). Summary of contributions: ",
+        "bbox": [
+            173,
+            743,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb1b7425276e1d85e9981243987d7806fdb5720fbcbfa5db6227712c1266176b.jpg",
+        "image_caption": [
+            "Figure 1: Review of the Courbariaux et al. (2016) BNN Training Algorithm: (a) A binary neural network is composed of binary convolution transformers (dashed green box). Each oval corresponds to a tensor and the derivative of the cost with respect to that tensor. Rectangles correspond to transformers that specify forward and backward propagation functions. Associated with each binary weight, $w ^ { b }$ , is a continuous weight, $w ^ { c }$ , that is used to accumulate gradients. $k$ denotes the kth layer of the network. (b) Each binarize transformer has a forward function and a backward function. The forward function simply binarizes the inputs. In the backward propagation step, one normally computes the derivative of the cost with respect to the input of a transformer via the Jacobian of the forward function and the derivative of the cost with respect to the output of that transformer $( \\delta u \\equiv d C / d u$ where $C$ is the cost function used to train the network). Since the binarize function is non-differentiable, the straight-through estimator (Bengio et al. (2013)), which is a smoothed version of the forward function, is used for the backward function . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            106,
+            787,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Angle Preservation Property: We demonstrate that binarization approximately preserves the direction of high-dimensional vectors. In particular, we show that the angle between a random vector (from a standard normal distribution) and its binarized version converges to arccos $\\sqrt { 2 / \\pi } \\approx 3 7 ^ { \\circ }$ as the dimension of the vector goes to infinity. This angle is an exceedingly small angle in high dimensions. Furthermore, we show that this property is present in the weight vectors of a network trained using the method of Courbariaux et al. (2016). ",
+        "bbox": [
+            214,
+            444,
+            825,
+            544
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Dot Product Proportionality Property: First, we empirically show that the weight-activation dot products, an important intermediate quantity a neural network, are approximately proportional under the binarization of the weight vectors. Next, we argue that if these weight activation dot products are proportional, then the continuous weights in the Courbariaux et al. (2016) method aren’t just a learning artifact. The continuous weights obtained from the BNN training algorithm (which decouples the forward and backward propagation steps) are an approximation of the weights one would learn if the network were trained with continuous weights and regular backpropagation. ",
+        "bbox": [
+            214,
+            547,
+            825,
+            660
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Generalized Binarization Transformation (GBT): We show that the computations done by the first layer of the network are fundamentally different than the computations being done in the rest of the network because correlations in the data result in high variance principal components that are not randomly oriented relative to the binarization. Thus we recommend an architecture that uses a continuous convolution for the first layer to embed the image in a high-dimensional binary space, after which it can be manipulated with cheap binary operations. Furthermore, we illustrate how a GBT (rotate, binarize, rotate back) is useful for embedding low dimensional data in a high-dimensional binary space. ",
+        "bbox": [
+            214,
+            664,
+            825,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "4. Generalization to Ternary Neural Networks: We show that the same analysis applies to ternary neural networks. In particular, the angle between a random vector from a standard normal distribution and the ternarized version of that vector predicts the empirical distribution of such angles in a network trained on CIFAR10. Furthermore, the dot product proportionality property is shown to hold for ternary neural networks. ",
+        "bbox": [
+            214,
+            780,
+            825,
+            849
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural networks that achieve good performance on tasks such as IMAGENET object recognition are highly computationally intensive. For instance, AlexNet has 61 million parameters and executes 1.5 billion operations to classify one 224 by 224 image (30 thousand operations/pixel) (Rastegari et al. (2016)). Researchers have sought to reduce this computational cost for embedded applications using a number of different approaches. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            203
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The first approach is to try and compress a pre-trained network. Kim et al. (2015) uses a Tucker decomposition of the kernel tensor and fine tunes the network afterwards. Han et al. (2015b) train a network, prune low magnitude connections, and retrain. Han et al. (2015a) extend their previous work to additionally include a weight sharing quantization step and Huffman coding of the weights. More recently, Han et al. (2017) train a dense network, sparsify it, and then retrain a dense network with the pruned weights initialized to zero. Second, researchers have sought to train networks using either low precision floating point numbers or fixed point numbers, which allow for cheaper multiplications (Courbariaux et al. (2014), Gupta et al. (2015), Judd et al. (2015), Gysel et al. (2016), Lin et al. (2016), Lai et al. (2017)). ",
+        "bbox": [
+            174,
+            210,
+            825,
+            335
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Third, one can train networks that have quantized weights and or activations. Bengio et al. (2013) looked at estimators for the gradient through a stochastic binary unit. Courbariaux et al. (2015) train networks with binary weights, and then later with binary weights and activations (Courbariaux et al. (2016)). Rastegari et al. (2016) replace a continuous weight matrix with a scalar times a binary matrix (and have a similar approximation for weight activation dot products). Kim & Smaragdis (2016) train a network with weights restricted in the range $- 1$ to 1 and then use a noisy backpropagation scheme train a network with binary weights and activations. Alemdar et al. (2016), Li et al. (2016) and Zhu et al. (2016) focus on networks with ternary weights. Further work seeks to quantize the weights and activations in neural networks to an arbitrary number of bits (Zhou et al. (2016), Hubara et al. (2016)). Zhou et al. (2017) use weights and activations that are zero or powers of two. Lin et al. (2015) and Zhou et al. (2016) quantize backpropagation in addition to the forward propagation. ",
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+        "type": "text",
+        "text": "Beyond merely seeking to compress neural networks, the analysis of the internal representations of neural networks is useful for understanding how to to compress them. Agrawal et al. (2014) found that feature magnitudes in higher layers do not matter (e.g. binarizing features barely changes classification performance). Merolla et al. (2016) analyze the robustness of neural network representations to a collection of different distortions. Dosovitskiy & Brox (2016) observe that binarizing features in intermediate layers of a CNN and then using backpropagation to find an image with those features leads to relatively little distortion of the image compared to dropping out features. These papers naturally lead into our work where we are seeking to better understand the representations in neural networks based on the geometry of high-dimensional binary vectors. ",
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+        "type": "text",
+        "text": "3 THEORY AND EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We investigate the internal representations of neural networks with binary weights and activations. A binary neural network is trained on CIFAR-10 (same learning algorithm and architecture as in Courbariaux et al. (2016)). Experiments on MNIST were carried out using both fully connected and convolutional networks and produced similar results. The CIFAR-10 convolutional neural network has six layers of convolutions, all of which have a 3 by 3 spatial kernel. The number of feature maps in each layer are 128, 128, 256, 256, 512, 512. After the second, fourth, and sixth convolutions, there is a 2 by 2 max pooling operation. Then there are two fully connected layers with 1024 units each. Each layer has a batch norm layer in between. The experiments using ternary neural networks use the same network architecture. The dimensionality of the weight vectors in these networks (i.e. convolution converted to a matrix multiply) is the patch size $( = 3 * 3 = 9$ ) times the number of channels. ",
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+        "type": "text",
+        "text": "3.1 PRESERVATION OF DIRECTION DURING BINARIZATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we analyze the angle distributions (i.e. geometry) of high-dimensional binary vectors. This is crucial for understanding binary neural networks because we can imagine that at each layer of a neural network, there are some ideal continuous weight vectors that extract out features. A binary neural network approximates these ideal continuous vectors with a binary vectors. In low dimensions, binarization strongly impacts the direction of a vector. However, we argue that binarization does not substantially change the direction of a high-dimensional continuous vector. It is often the case that the geometric properties of high-dimensional vectors are counter-intuitive. For instance, one key idea in the hyperdimensional computing theory of Kanerva (2009) is that two random, high-dimensional vectors of dimension $d$ whose entries are chosen uniformly from the set $\\{ - 1 , 1 \\}$ are approximately orthogonal. The result follows from the central limit theorem because the cosine angle between two√ such random vectors is normally distributed with $\\mu = 0$ and $\\sigma \\sim 1 / \\sqrt { d }$ . Then $\\cos \\theta \\approx 0$ implies that $\\theta \\approx \\frac { \\pi } { 2 }$ . Building upon this work, we study the way in which binary vectors are distributed relative to continuous vectors. As binarizing a continuous vector gives the binary vector closest in angle to that continuous vector, we can get a sense of how binary vectors are distributed relative to continuous vectors in high dimensions by binarizing continuous vectors. The standard normal distribution, which serves as an informative null distribution because it is rotationally invariant, is used to generate random continuous vectors which are then binarized. This analysis gives a fundamental insight into understanding the recent success of binary neural networks. Binarizing a random continuous vector changes its direction by a small amount relative to the angle between two random vectors in moderately high dimensions (Fig. 2a). Binarization changes the direction of a vector by approximately $3 7 ^ { \\circ }$ in high dimensions. This seems like a large change based on our low-dimensional intuition. Indeed, the angle between two randomly chosen vectors from a rotationally invariant distribution is uniform in two dimensions. However, two randomly chosen vectors are approximately orthogonal in high dimensions. Thus while it is common for two random vectors to have an angle less than $3 7 ^ { \\circ }$ in low dimensions, it is exceedingly rare in high dimensions. Therefore $3 7 ^ { \\circ }$ is a small angle in high dimensions. ",
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+        "text": "In order to test our theory of the binarization of random vectors chosen from a rotationally invariant distribution, we train a multilayer binary CNN on CIFAR10 (using the Courbariaux et al. (2016) method) and study the weight vectors1 of that network. Remarkably, there is a close correspondence between the experimental results and the theory for the angles between the binary and continuous weights (Fig. 2b). For each layer, the distribution of the angles between the binary and continuous weights is sharply peaked near the $d \\to \\infty$ expectation of arccos $\\scriptstyle { \\sqrt { 2 / \\pi } }$ . We note that there is a small but systematic deviation from the theory towards larger angles for the higher layers of the network (Fig. 6). Ternary neural networks are considered in (SI Sec. 5.5) and yield a similar result. ",
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+        "type": "text",
+        "text": "3.2 DOT PRODUCT PROPORTIONALITY AS A SUFFICIENT CONDITION FOR APPROXIMATING ANETWORK WITH CONTINUOUS WEIGHTS",
+        "text_level": 1,
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+        "text": "Given the previous discussion, an important question to ask is: are the so-called continuous weights a learning artifact without a clear correspondence to the binary weights? While we know that $w ^ { b } = \\theta ( \\overline { { w } } ^ { c } )$ , there are many continuous weights that map onto a particular binary weight vector. Which one is found when using the straight-through estimator to backpropagate through the binarize function? Remarkably, there is a clear answer to this question. In numerical experiments, we see that one gets the continuous weight vector such that the dot products of the activations with the prebinarization and post-binarization weights are highly correlated (Fig. 3). In equations, $a \\cdot w ^ { b } \\sim a \\cdot w ^ { c }$ . We call this relation the Dot Product Proportionality (DPP) property. The proportionality constant, which is subsequently normalized away by a batch norm layer, depends on the magnitudes of the continuous and binary weight vectors and the cosine angle between the binary and continuous weight vectors. The theoretical consequences of the DPP property are explored in the rest of this section. ",
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+        "text": "We show that the modified gradient of the BNN training algorithm can be viewed as an estimator of the gradient that would be used to train the continuous weights in traditional backpropagation. This establishes the fundamental point that while the weights and activations are technically binary, they are operating as if the weights are continuous. For instance, one could imagine using an exhaustive search over all binary weights in the network. However, the additional structure in the problem associated with taking dot products makes the optimization simpler than that. Furthermore, we show that if the dot products of the activations with the pre-binarized and post-binarized weights are proportional then straight-through estimator gradient is proportional to the continuous weight network gradient. The key to the analysis is to focus on the transformers in the network whose forward and backward propagation functions are not related in the way that they would normally be related in typical gradient descent. ",
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+        "image_caption": [
+            "Figure 2: Binarization of High-Dimensional Vectors Approximately Preserves their Direction in Theory and Practice: (a) Distribution of angles between two random vectors (blue), and between a vector and its binarized version (red), for a rotationally invariant distribution of dimension $d$ . The red distribution is peaked near the $d \\to \\infty$ limit of arccos $\\sqrt { 2 / \\pi } \\approx 3 7 ^ { \\circ }$ (SI, Sec. 1). While $3 7 ^ { \\circ }$ may seem like a large angle, that angle is small as compared to the angle between two random vectors in moderately high dimensions (i.e. the blue and red curves are well-separated). (b) Angle distribution between continuous and binary weight vectors by layer for a binary CNN trained on CIFAR-10. For the higher layers, there is a close correspondence to the theory. There is a small, but systematic deviation towards large angles (SI, Fig. 6). $d$ is the dimension of the filters at each layer. (c) Standard deviations of the angle distributions from (b) by layer. We see a correspondence to the theoretical expectation that standard deviations of each of the angle distributions scales as $d ^ { - 0 . 5 }$ (SI, Sec. 1). (d) Histogram of the components of the continuous weights at each layer. The distribution is approximately Gaussian for all but the first layer. Furthermore, there is a high density of weights near zero, which is the threshold for the binarization function. "
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+        "text": "Suppose that there is a neural network where two tensors, $u$ , and $v$ and the associated derivatives of the cost with respect to those tensors, $\\delta u$ , and $\\delta \\boldsymbol { v }$ , are allocated. Suppose that the loss as a function of $v$ is $L ( x ) | _ { x = v }$ . Further, suppose that there are two potential forward propagation functions, $f$ , and $g$ . If the network is trained under normal conditions using $g$ as the forward propagation function, then the following computations are done: ",
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+        "text": "$$\nv  g ( u ) \\qquad \\delta v  L ^ { \\prime } ( x = v = g ( u ) ) \\qquad \\delta u  \\delta v \\cdot g ^ { \\prime } ( u )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $L ^ { \\prime } ( x )$ denotes the derivative of $L$ with respect to the vector $x$ . In a modified backpropagation scheme, the following computations are done ",
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+        "text": "$$\nv  f ( u ) \\qquad \\delta v = L ^ { \\prime } ( x = v = f ( u ) ) \\qquad \\delta u  \\delta v \\cdot g ^ { \\prime } ( u )\n$$",
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+        "type": "text",
+        "text": "A sufficient condition for $\\delta u$ to be the same in both cases is $L ^ { \\prime } ( x = f ( u ) ) \\sim L ^ { \\prime } ( x = g ( u ) )$ where $a \\sim b$ means that the vector $a$ is a scalar times the vector $b$ . ",
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+        "text": "Now this general observation is applied to the binarize transformer of Fig. 1. Here, $u$ is the continuous weight, $w ^ { c }$ , $f ( u )$ is the pointwise binarize function, $g ( u )$ is the identity function2, and $L$ is the loss of the network as a function of the weights in a particular layer. Given the network architecture, $L ( x ) = M ( a \\cdot x )$ where $a$ are the activations corresponding to that layer and $M$ is the loss as a function of the weight-activation dot products. Then $L ^ { \\prime } ( x ) = M ^ { \\prime } ( a \\cdot x ) \\odot a$ where $\\odot$ denotes a pointwise multiply. Thus the sufficient condition is $M ^ { \\prime } ( a \\cdot w ^ { b } ) \\sim M ^ { \\prime } ( a \\cdot w ^ { c } )$ . Since the dot products are followed by a batch normalization, $M ( k { \\vec { x } } ) = M ( { \\vec { x } } )  M ^ { \\prime } ( { \\vec { x } } ) = k M ^ { \\prime } ( k { \\vec { x } } )$ . Therefore, it is sufficient that $\\smash { \\dot { a } \\cdot w ^ { b } \\sim a \\cdot w ^ { c } }$ , which is the DPP property. When the DPP only approximately holds, the second derivative can be used to bound the error between the two gradients of the two learning procedures. In summary, the learning dynamics where $g$ is used for the forward and backward passes (i.e. training the network with continuous weights) is approximately equivalent to the modified learning dynamics $f$ on the forward pass, and $g$ on the backward pass) when we have the DPP property. ",
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+        "type": "text",
+        "text": "While we demonstrated that the BNN learning dynamics approximate the dynamics that one would have by training a network with continuous weights using a mixture of empirical and theoretical arguments, the ideal result would be that the learning algorithm implies the DPP property. It should be noted that in the case of stochastic binarization where $E ( w ^ { b } ) \\dot { = } w ^ { c }$ is chosen by definition, the DPP property is true by design. However, it is remarkable that the property still holds in the case of deterministic binarization, which is revealing of the fundamental nature of the representations used in neural networks. ",
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+        "type": "text",
+        "text": "While the main focus of this section is the binarization of the weights, the arguments presented can also be applied to the binarize block that corresponds to the non-linearity of the network. The analogue of the DPP property for this binarize block is: $w ^ { b } \\cdot a ^ { c } \\sim w ^ { b } \\cdot a ^ { b }$ where $a ^ { c }$ denotes the pre-binarized (post-batch norm) activations and $a ^ { b } = a$ denotes the binarized activations. This property is empirically verified to hold. For the sake of completeness, the dot product histogram corresponding to $w ^ { c } \\cdot \\bar { a } ^ { c } \\sim w ^ { b } \\cdot a ^ { b }$ is also computed, although it doesn’t directly correspond to removing one instance of a binarize transformer. This property is also empirically verified to hold (SI, Fig. 5). ",
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+        "type": "text",
+        "text": "Impact on Classification: It is natural to ask to what extent the classification performance depends on the binarization of the weights. In experiments on CIFAR10, if the binarization of the weights on all of the convolutional layers is removed, the classification performance drops by only 3 percent relative to the original network. Looking at each layer individually, removing the weight binarization for the first layer accounts for this entire percentage, and removing the binarization of the weights for each other layer causes no degradation in performance. This result is evident by looking at the 2D dot product histograms in Fig 3. The off-diagonal quadrants show where switching the weights from binary to continuous changes the sign of the binarized weight-activation dot product. In all of the layers except the first layer, there are very few dot products in the off-diagonal quadrants. Thus we recommend the use of the dot product histograms for studying the performance of binary neural networks. Removing the binarization of the activations has a substantial impact on the classification performance because that removes the main non-linearity of the network. ",
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+        "type": "text",
+        "text": "3.3 INPUT CORRELATIONS AND THE GENERALIZED BINARIZATION TRANSFORMATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Not surprisingly, some distributions are impacted more strongly by binarization than others. A binary neural network must adapt its internal representations in such a way to not be degraded too much by binarization at each layer. In this section we explore the idea that the principal components of the input to the binarization function should be randomly oriented relative to the binarization. While the network can adapt the higher level representations to satisfy this property, the part of the network that interfaces with the input doesn’t have that flexibility. We make the novel observation that the difficulties in training the first layer of the network are tied to the intrinsic correlations in the input data. In order to be more precise, we define the Generalized Binarization Transformation (GBT) ",
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+        "text": "$$\n\\theta _ { R } ( x ) = R ^ { T } \\theta ( R x )\n$$",
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+        "type": "text",
+        "text": "where $x$ is a column vector, $R$ is a fixed rotation matrix, and $\\theta$ is the pointwise binarization function from before. The rows of $R$ are called the axes of binarization. If $R$ is the identity matrix, then $\\theta _ { R } = \\theta$ and the axes of binarization are the canonical basis vectors $( . . . , 0 , 1 , 0 , . . . ) .$ $R$ can either be chosen strategically or randomly. ",
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+        "type": "text",
+        "text": "The GBT changes the distribution being binarized through a rotation. For appropriate choices of the rotation, $R$ , the directions of the input vectors, $x$ , are changed insignificantly by binarization. The angle between a vector and its binarized version is dependent on the dot product: ${ \\bf \\dot { x } } \\cdot \\theta _ { R } ( x )$ , which is equal to $x ^ { T } \\theta _ { R } ( x ) = ( R x ) ^ { T } \\theta ( R x ) = y \\cdot \\theta ( y )$ where $y = R x$ . As a concrete example of the benefits of the GBT, consider the case where $x \\sim N ( 0 , \\Sigma )$ and $\\Sigma _ { i , j } = \\delta _ { i , j } \\exp ( 2 k i )$ for $k = 0 . 1$ (therefore $y \\sim N ( 0 , R \\Sigma R ^ { T } ) )$ . As the dimension goes to infinity, the angle between a vector drawn from this distribution and its binarized version approaches $\\pi / 2$ . Thus binarization is destructive to vectors from this distribution. However, if the GBT is applied with a fixed random matrix3, the angle between the vector and its binarized version converges to $3 7 ^ { \\circ }$ (Fig. 4). Thus a random rotation can compensate for the errors incurred from directly binarizing a non-isotropic Gaussian. ",
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+        "img_path": "images/dda78feb7ec99827c1081c16d52cd5c5917964a5de5826e4fcc43dfa4aa7f925.jpg",
+        "image_caption": [
+            "Figure 3: Binarization Preserves Dot Products: Each subplot shows a 2D histogram of the dot products between the binarized weights and the activations (horizontal axis) and the dot products between the continuous weights and the activations (vertical axis) for different layers of a network trained on CIFAR10. Surprisingly, the dot products are highly correlated ( $r$ is the Pearson correlation coefficient). Thus replacing $\\boldsymbol { w ^ { b } }$ with $w ^ { c }$ changes the overall constant in front of the dot products, while still preserving whether the dot product is zero or not zero. This overall constant is divided out by the subsequent batch norm layer. The shaded quadrants correspond to dot products where the sign changes when replacing the binary weights with the continuous weights. Notice that for all but the first layer, a very small fraction of the dot products lie in these off diagonal quadrants. The top left figure (labeled as Layer 1) corresponds to the input and the first convolution. Note that the correlation is weaker in the first layer. "
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+        "type": "text",
+        "text": "Moving into how this analysis applies to a binary neural network, the network weights must approximate the important directions in the activations using binary vectors. For instance, Gabor filters are intrinsic features in natural images and are often found in the first layer weights of neural networks trained on natural images (e.g. Olshausen et al. (1996); Krizhevsky et al. (2012)). While the network has flexibility in the higher layers, the first layer must interface directly with the input where the features are not necessarily randomly oriented. For instance, consider the 27 dimensional input to the first set of convolutions in our network: 3 color channels of a 3 by 3 patch of an image from CIFAR10 with the mean removed. 3 PCs capture 90 percent of the variance of this data and 4 PCs capture 94.5 percent of the variance. The first two PCs are spatially uniform colors. More generally, large images such as those in IMAGENET have the same issue. Translation invariance of the image covariance matrix implies that the principal components are the filters of the 2D Fourier transform. Scale invariance implies a $1 / f ^ { 2 }$ power spectrum, which results in the largest PCs corresponding to low frequencies (Field (1987)). ",
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+    {
+        "type": "text",
+        "text": "Another manifestation of this issue can be seen in our trained networks. The first layer has a much smaller dot product correlation than the other layers (Fig. 3). To study this, we randomly permute the activations in order to generate a distribution with the same marginal statistics as the original data but independent joint statistics (a different permutation for each input image). Such a transformation gives a distribution with a correlation equal to the normalized dot product of the weight vectors (SI Sec. 3). The correlations for the higher layers decrease substantially but the correlation in the first layer increases (Fig. 4, right). For the first layer, the shuffling operation randomly permutes the pixels in the image. Thus we demonstrate that the binary weight vectors in the first layer are not well-aligned with the continuous weight vectors relative to the input data. Our theoretically grounded analysis is consistent with previous work. Han et al. (2015b) find that compressing the first set of convolutional weights of a particular layer by the same fraction has the highest impact on performance if done on the first layer. Zhou et al. (2016) find that accuracy degrades by about 0.5 to 1 percent on SHVN when quantizing the first layer weights. Thus it is recommended to rotate the input data before normalization or to use continuous weights for the first layer. ",
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+        "image_caption": [
+            "Figure 4: Left: Random Rotation Improves Angle Preservation for a Non-Isotropic Gaussian. Random vectors are drawn from a Gaussian of dimension $d$ with a diagonal covariance matrix whose entries vary exponentially. As in Fig. 2, the red curve shows the angle between a random vector and its binarized version. Since the Gaussian is no longer isotropic, the red curve no longer peaks at $\\theta = \\operatorname { a r c c o s } { \\sqrt { 2 / \\pi } } .$ . However, if the binarization is replaced with a GBT with a fixed random matrix, the direction of the vector is again approximately preserved. Right: Permuting the activations shows that the correlations observed in Fig. 3 are not merely due to correlations between the binary and continuous weight vectors. The correlations are due to these weight vectors corresponding to high variance directions in the data. "
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+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Neural networks with binary weights and activations have similar performance to their continuous counterparts with substantially reduced execution time and power usage. We provide an experimentally verified theory for understanding how one can get away with such a massive reduction in precision based on the geometry of HD vectors. First, we show that binarization of high-dimensional vectors preserves their direction in the sense that the angle between a random vector and its binarized version is much smaller than the angle between two random vectors (Angle Preservation Property). Second, we take the perspective of the network and show that binarization approximately preserves weight-activation dot products (Dot Product Proportionality Property). More generally, when using a network compression technique, we recommend looking at the weight activation dot product histograms as a heuristic to help localize the layers that are most responsible for performance degradation. Third, we discuss the impacts of the low effective dimensionality of the data on the first layer of the network. We recommend either using continuous weights for the first layer or a Generalized Binarization Transformation. Such a transformation may be useful for architectures like LSTMs where the update for the hidden state declares a particular set of axes to be important (e.g. by taking the pointwise multiply of the forget gates with the cell state). Finally, we show that neural networks with ternary weights and activations can also be understood with our approach. More broadly speaking, our theory is useful for analyzing a variety of neural network compression techniques that transform the weights, activations or both to reduce the execution cost without degrading performance. ",
+        "bbox": [
+            174,
+            579,
+            825,
+            842
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pulkit Agrawal, Ross Girshick, and Jitendra Malik. Analyzing the performance of multilayer neural networks for object recognition. In European Conference on Computer Vision, pp. 329–344. Springer, 2014. ",
+        "bbox": [
+            174,
+            126,
+            826,
+            167
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hande Alemdar, Nicholas Caldwell, Vincent Leroy, Adrien Prost-Boucle, and Frédéric Pétrot. Ternary neural networks for resource-efficient ai applications. arXiv preprint arXiv:1609.00222, 2016. ",
+        "bbox": [
+            169,
+            176,
+            823,
+            207
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Renzo Andri, Lukas Cavigelli, Davide Rossi, and Luca Benini. Yodann: An architecture for ultra-low power binary-weight cnn acceleration. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2017. ",
+        "bbox": [
+            173,
+            214,
+            825,
+            257
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013. ",
+        "bbox": [
+            171,
+            266,
+            823,
+            296
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Training deep neural networks with low precision multiplications. arXiv preprint arXiv:1412.7024, 2014. ",
+        "bbox": [
+            171,
+            304,
+            823,
+            334
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. In Advances in Neural Information Processing Systems, pp. 3123–3131, 2015. ",
+        "bbox": [
+            174,
+            342,
+            825,
+            385
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Courbariaux, Itay Hubara, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks: Training neural networks with weights and activations constrained to $+ 1$ and $^ { - 1 }$ arXiv preprint arXiv:1602.02830, 2016. ",
+        "bbox": [
+            173,
+            393,
+            825,
+            436
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alexey Dosovitskiy and Thomas Brox. Inverting visual representations with convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4829– 4837, 2016. ",
+        "bbox": [
+            173,
+            444,
+            826,
+            488
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Steven K Esser, Paul A Merolla, John V Arthur, Andrew S Cassidy, Rathinakumar Appuswamy, Alexander Andreopoulos, David J Berg, Jeffrey L McKinstry, Timothy Melano, Davis R Barch, et al. Convolutional networks for fast, energy-efficient neuromorphic computing. Proceedings of the National Academy of Sciences, pp. 201604850, 2016. ",
+        "bbox": [
+            173,
+            496,
+            826,
+            554
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David J Field. Relations between the statistics of natural images and the response properties of cortical cells. Josa a, 4(12):2379–2394, 1987. ",
+        "bbox": [
+            174,
+            561,
+            825,
+            592
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. In ICML, pp. 1737–1746, 2015. ",
+        "bbox": [
+            173,
+            599,
+            823,
+            630
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Philipp Gysel, Mohammad Motamedi, and Soheil Ghiasi. Hardware-oriented approximation of convolutional neural networks. arXiv preprint arXiv:1604.03168, 2016. ",
+        "bbox": [
+            173,
+            637,
+            826,
+            667
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015a. ",
+        "bbox": [
+            173,
+            675,
+            826,
+            705
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In Advances in Neural Information Processing Systems, pp. 1135–1143, 2015b. ",
+        "bbox": [
+            173,
+            713,
+            825,
+            755
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Jeff Pool, Sharan Narang, Huizi Mao, Enhao Gong, Shijian Tang, Erich Elsen, Peter Vajda, Manohar Paluri, John Tran, et al. Dsd: Dense-sparse-dense training for deep neural networks. arXiv prepring arXiv:1607.04381, 2017. ",
+        "bbox": [
+            174,
+            765,
+            825,
+            808
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Quantized neural networks: Training neural networks with low precision weights and activations. arXiv preprint arXiv:1609.07061, 2016. ",
+        "bbox": [
+            173,
+            815,
+            823,
+            859
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Patrick Judd, Jorge Albericio, Tayler Hetherington, Tor Aamodt, Natalie Enright Jerger, Raquel Urtasun, and Andreas Moshovos. Reduced-precision strategies for bounded memory in deep neural nets. arXiv preprint arXiv:1511.05236, 2015. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pentti Kanerva. Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors. Cognitive Computation, 1(2):139–159, 2009. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Minje Kim and Paris Smaragdis. Bitwise neural networks. arXiv preprint arXiv:1601.06071, 2016. ",
+        "bbox": [
+            174,
+            141,
+            823,
+            156
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yong-Deok Kim, Eunhyeok Park, Sungjoo Yoo, Taelim Choi, Lu Yang, and Dongjun Shin. Compression of deep convolutional neural networks for fast and low power mobile applications. arXiv preprint arXiv:1511.06530, 2015. ",
+        "bbox": [
+            176,
+            165,
+            823,
+            208
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. ",
+        "bbox": [
+            176,
+            215,
+            825,
+            258
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Liangzhen Lai, Naveen Suda, and Vikas Chandra. Deep convolutional neural network inference with floating-point weights and fixed-point activations. arXiv preprint arXiv:1703.03073, 2017. ",
+        "bbox": [
+            173,
+            268,
+            825,
+            297
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fengfu Li, Bo Zhang, and Bin Liu. Ternary weight networks. arXiv preprint arXiv:1605.04711, 2016. ",
+        "bbox": [
+            173,
+            305,
+            825,
+            334
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Darryl Lin, Sachin Talathi, and Sreekanth Annapureddy. Fixed point quantization of deep convolutional networks. In International Conference on Machine Learning, pp. 2849–2858, 2016. ",
+        "bbox": [
+            173,
+            343,
+            825,
+            372
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhouhan Lin, Matthieu Courbariaux, Roland Memisevic, and Yoshua Bengio. Neural networks with few multiplications. arXiv preprint arXiv:1510.03009, 2015. ",
+        "bbox": [
+            171,
+            381,
+            823,
+            410
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Paul Merolla, Rathinakumar Appuswamy, John Arthur, Steve K Esser, and Dharmendra Modha. Deep neural networks are robust to weight binarization and other non-linear distortions. arXiv preprint arXiv:1606.01981, 2016. ",
+        "bbox": [
+            173,
+            419,
+            825,
+            462
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Emre Neftci, Charles Augustine, Somnath Paul, and Georgios Detorakis. Neuromorphic deep learning machines. arXiv preprint arXiv:1612.05596, 2016. ",
+        "bbox": [
+            174,
+            469,
+            821,
+            500
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bruno A Olshausen et al. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583):607–609, 1996. ",
+        "bbox": [
+            173,
+            508,
+            823,
+            537
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. arXiv preprint arXiv:1603.05279, 2016. ",
+        "bbox": [
+            173,
+            545,
+            823,
+            575
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gilbert W Stewart. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM Journal on Numerical Analysis, 17(3):403–409, 1980. ",
+        "bbox": [
+            174,
+            583,
+            823,
+            613
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014. ",
+        "bbox": [
+            173,
+            621,
+            823,
+            651
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. arXiv preprint arXiv:1702.03044, 2017. ",
+        "bbox": [
+            173,
+            659,
+            823,
+            689
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Shuchang Zhou, Yuxin Wu, Zekun Ni, Xinyu Zhou, He Wen, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. arXiv preprint arXiv:1606.06160, 2016. ",
+        "bbox": [
+            173,
+            696,
+            826,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016. ",
+        "bbox": [
+            171,
+            748,
+            825,
+            777
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 SUPPLEMENTARY INFORMATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            475,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.1 EXPECTED ANGLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            352,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Random $n$ dimensional vectors are drawn from a rotationally invariant distribution. The angle between two random vectors and the angle between a vector and its binarized version are compared. A rotationally invariant distribution can be factorized into a pdf for the magnitude of the vector times a distribution on angles. In the expectations that we are calculating, the magnitude cancels out and there is only one rotationally invariant distribution on angles. Thus it suffices to compute these expectations using a Gaussian. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lemmas: ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            251,
+            236,
+            263
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. Consider a vector, $v$ , chosen from a standard normal distribution of dimension $n$ . Let $\\begin{array} { r } { \\rho = \\frac { v _ { 1 } } { \\sqrt { v _ { 1 } ^ { 2 } + \\ldots + v _ { n } ^ { 2 } } } } \\end{array}$ . Then $\\rho$ is distributed according to: $\\begin{array} { r } { g ( \\rho ) = \\frac { 1 } { \\sqrt { \\pi } } \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n - 1 ) / 2 ) } ( 1 - \\rho ^ { 2 } ) ^ { \\frac { n - 3 } { 2 } } } \\end{array}$ where $\\Gamma$ is the Gamma function. Proof: Begin by considering the integral $\\begin{array} { r } { G ( \\rho _ { 0 } ) = p ( \\rho < \\rho _ { 0 } ) = \\int \\prod _ { i } d v _ { i } p ( v ) I ( \\rho ( v ) < \\rho _ { 0 } ) } \\end{array}$ where $I$ is an indicator function. The desired distribution comes from taking the derivative of this cumulative distribution g(ρ0) = ddρ0 . Consider the generalized spherical coordinate transformation: $v _ { 1 } = r \\cos \\phi _ { 1 }$ , $\\begin{array} { r } { \\dot { v } _ { 2 } = r \\sin \\phi _ { 1 } \\cos \\phi _ { 2 } . } \\end{array}$ , $v _ { 3 } = r \\sin \\phi _ { 1 } \\sin \\phi _ { 2 } \\cos \\phi _ { 3 }$ , $\\begin{array} { r } { \\ldots \\quad v _ { n - 1 } = r \\sin \\phi _ { 1 } \\cdot \\cdot \\cdot \\sin \\phi _ { n - 2 } \\cos \\phi _ { n - 1 } , } \\end{array}$ $v _ { n } ~ = ~ r \\sin \\phi _ { 1 } \\cdot \\cdot \\cdot \\sin \\phi _ { n - 2 } \\sin \\phi _ { n - 1 }$ where $\\phi _ { 1 } , \\ldots , \\phi _ { n - 2 } \\in [ 0 , \\pi ]$ and $\\phi _ { n - 1 } \\in [ 0 , 2 \\pi ]$ . The generalized spherical volume element is $d v = r ^ { n - 1 } \\sin ^ { n - 2 } ( \\phi _ { 1 } ) \\sin ^ { n - 3 } ( \\phi _ { 2 } ) \\cdot \\cdot \\cdot \\sin ( \\phi _ { n - 2 } ) d r d \\phi _ { 1 } \\cdot \\cdot \\cdot d \\phi _ { n - 1 }$ . 4 Thus we can write out the integral $\\begin{array} { r } { G ( \\rho _ { 0 } ) = \\int I ( \\cos \\phi _ { 1 } < \\rho _ { 0 } ) \\frac { e ^ { - r ^ { 2 } / 2 } } { \\sqrt { ( 2 \\pi ) ^ { n } } } r ^ { n - 1 } \\sin ^ { n - 2 } ( \\phi _ { 1 } ) \\sin ^ { n - 3 } ( \\phi _ { 2 } ) \\cdot \\cdot \\cdot \\sin ( \\phi _ { n - 2 } ) d r d \\phi _ { 1 } \\cdot \\cdot \\cdot d \\phi _ { n - 1 } } \\end{array}$ The integral factorizes and all of the terms are independent of $\\rho _ { 0 }$ except the integral over $\\phi _ { 1 } \\equiv \\phi$ . Thus $\\begin{array} { r } { G ( \\rho _ { 0 } ) \\sim \\int _ { 0 } ^ { \\pi } I ( \\cos \\phi < \\rho _ { 0 } ) \\sin ^ { \\hat { n } - 2 } ( \\phi ) d \\phi } \\end{array}$ . Using the substitution $\\rho = \\cos \\phi$ (which is also consistent with the definition of $\\rho$ above), $d \\rho = - \\sin ( \\phi ) d \\phi$ , $\\sin \\rho ~ = ~ ( 1 - \\rho ^ { 2 } ) ^ { 0 . 5 }$ , so $\\begin{array} { r } { G ( \\rho _ { 0 } ) ~ \\sim ~ \\int _ { - 1 } ^ { 1 } I ( \\rho ~ < ~ \\rho _ { 0 } ) ( 1 - \\rho ^ { 2 } ) ^ { ( n - 3 ) / 2 } d \\rho } \\end{array}$ . Taking the derivative with respect to $\\rho _ { 0 }$ and using the fundamental theorem of calculus gives $g ( \\rho ) \\sim ( 1 - \\rho ^ { 2 } ) ^ { ( \\bar { n - 3 } ) / 2 }$ . The normalization constant is equal to a beta function that evaluates to the desired result (substitute $t = \\rho ^ { 2 }$ ). ",
+        "bbox": [
+            214,
+            275,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cases: ",
+        "bbox": [
+            173,
+            617,
+            218,
+            631
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• Distribution of angles between two random vectors. Since a Gaussian is a rotationally invariant distribution, we can say without loss of generality that one of the vectors is $( 1 , 0 , 0 , \\ldots 0 )$ . Then the cosine angle between those two vectors is $\\rho$ as defined above. While the exact distribution of $\\rho$ is given by Lemma 1, we note that – $E ( \\rho ) = 0$ due to the symmetry of the distribution. – $\\begin{array} { r } { V a r ( \\rho ) = E ( \\rho ^ { 2 } ) = \\frac { 1 } { n } } \\end{array}$ because $\\begin{array} { r } { 1 = E \\left( \\frac { \\sum _ { i } x _ { i } ^ { 2 } } { \\sum _ { j } x _ { j } ^ { 2 } } \\right) = \\sum _ { i } E \\left( \\frac { x _ { i } ^ { 2 } } { \\sum _ { j } x _ { j } ^ { 2 } } \\right) = n * E ( \\rho ^ { 2 } ) } \\end{array}$ ",
+        "bbox": [
+            214,
+            642,
+            831,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• Angles between a vector and the binarized version of that vector, η = v·θ(v)||v||·||θ(v)|| $\\frac { \\sum _ { i } \\left| v _ { i } \\right| } { \\sqrt { \\sum v _ { i } ^ { 2 } } * \\sqrt { n } }$ ",
+        "bbox": [
+            215,
+            756,
+            825,
+            805
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d59000e59599616a98b171635ac99013c0150aa8f115ff083ae8a1fd130a1b6a.jpg",
+        "text": "$$\nE ( \\eta ) = \\frac { \\sqrt { n } } { \\sqrt { \\pi } } * \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n + 1 ) / 2 ) } \\qquad \\operatorname* { l i m } _ { n \\to \\infty } E ( \\eta ) = \\sqrt { \\frac { 2 } { \\pi } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            810,
+            723,
+            847
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "First, we note $E ( \\eta ) = \\sqrt { n } E ( | \\rho | )$ . Then $\\begin{array} { r } { E ( | \\rho | ) = \\int _ { 0 } ^ { 1 } d \\rho \\rho \\frac { 2 } { \\sqrt { \\pi } } \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n - 1 ) / 2 ) } ( 1 - \\rho ^ { 2 } ) ^ { \\frac { n - 3 } { 2 } } = } \\end{array}$ $\\frac { 2 } { \\sqrt { \\pi } } \\ast \\frac { 1 } { n - 1 } \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n - 1 ) / 2 ) }$ (substitute $u \\ : = \\ : \\rho ^ { 2 }$ and use $\\Gamma ( x + 1 ) = x \\Gamma ( x )$ ). Lemma ",
+        "bbox": [
+            263,
+            852,
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+            896
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "two gives the $n  \\infty$ limit. $\\begin{array} { r l r } { \\frac { 2 } { \\sqrt { \\pi } } \\frac { \\sqrt { n } } { n - 1 } \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n - 1 ) / 2 ) } \\ \\approx \\ \\frac { 2 } { \\sqrt { \\pi } } \\frac { \\sqrt { n } } { n - 1 } \\sqrt { \\frac { n } { 2 } } \\left[ 1 + \\frac { 0 . 5 * 0 . 5 } { 2 ( n / 2 ) } \\right] \\ = } \\end{array}$ q 2π \u00001 + 54 ∗ 1n \u0001 + O(1/n2) ",
+        "bbox": [
+            258,
+            101,
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+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe8570bc0e42b9d54cd4a5cfbd02e7a9ba30dfc1f9ff5f589068aae2c7bfc3bf.jpg",
+        "text": "$$\n{ V a r } ( \\eta ) = \\frac { 1 } { n } \\left( 1 - \\frac { 1 } { \\pi } \\right) + { \\cal O } ( 1 / n ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            156,
+            663,
+            191
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Thus we have the normal scaling as in the central limit theorem of the large $n$ variance.   \nWe can calculate this explicitly following the approach of 5. ",
+        "bbox": [
+            264,
+            193,
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+            222
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As $E ( \\eta )$ has been calculated, it suffices to calculate $E ( \\eta ^ { 2 } )$ . Expanding out $\\eta ^ { 2 } , E ( \\eta ^ { 2 } ) =$ $\\begin{array} { r } { \\frac { 1 } { n } + ( n - 1 ) * E \\bigl ( \\frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + \\ldots v _ { n } ^ { 2 } } \\bigr ) } \\end{array}$ . Below we show that $\\begin{array} { r } { E ( \\frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + \\ldots v _ { n } ^ { 2 } } ) = \\frac { 2 } { \\pi n } } \\end{array}$ . Thus the variance is: ",
+        "bbox": [
+            264,
+            222,
+            825,
+            270
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27c25af3e4aaee3aa0cf438b24930e58e3acb5d283e2e1973031d49dc8efabb8.jpg",
+        "text": "$$\n{ \\frac { 1 } { n } } * \\left( 1 - { \\frac { 2 } { \\pi } } \\right) + { \\frac { 2 } { \\pi } } - \\left( { \\frac { \\sqrt { n } } { \\sqrt { \\pi } } } { \\frac { \\Gamma ( n / 2 ) } { \\Gamma ( ( n + 1 ) / 2 ) } } \\right) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            281,
+            694,
+            318
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Using Lemma 2 to expand out the last term, we get $[ \\frac { \\sqrt { n } } { \\sqrt { \\pi } } ( n / 2 ) ^ { - 1 / 2 } ( 1 - 1 / ( 4 n ) +$ $O ( n ^ { - 2 } ) ) ] ^ { 2 } = \\frac { _ { 2 } } { \\pi } ( 1 - { 1 } / ( { 2 n } ) + O ( n ^ { - 2 } ) )$ . Plugging this in gives the desired result. ",
+        "bbox": [
+            264,
+            323,
+            826,
+            359
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Going back to the calculation of that expectation, change variables to $v _ { 1 } = r \\cos \\theta$ , $v _ { 2 } = r \\sin { \\theta }$ , $z ^ { 2 } = v _ { 3 } ^ { 2 } + \\ldots + v _ { n } ^ { 2 }$ . The integration over the volume element $d v _ { 3 } \\ldots d v _ { n }$ is rewritten as $d z d A _ { n - 3 }$ where $d A _ { n }$ denotes the surface element of a $n$ sphere. Since the integrand only depends on the magnitude, $z$ , $\\textstyle { \\int d A _ { n - 3 } = z ^ { n - 3 } * { \\dot { S } } _ { n - 3 } }$ where $\\begin{array} { r } { S _ { n } = \\frac { 2 \\pi ^ { ( n + 1 ) / 2 } } { \\Gamma ( \\frac { n + 1 } { 2 } ) } } \\end{array}$ denotes the surface area of a unit $n$ -sphere. Then ",
+        "bbox": [
+            264,
+            359,
+            826,
+            440
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e6317a88e73e808052e6c94816aa02262f397cce32d868b40ade1514ab0d308.jpg",
+        "text": "$$\nE \\left( \\frac { | v _ { 1 } v _ { 2 } | } { v _ { 1 } ^ { 2 } + . . . v _ { n } ^ { 2 } } \\right) = ( 2 \\pi ) ^ { - n / 2 } S _ { n - 3 } \\int _ { 0 } ^ { 2 \\pi } d \\theta | \\cos \\theta \\sin \\theta | * \\int r d r z ^ { n - 3 } d z * \\frac { r ^ { 2 } } { r ^ { 2 } + z ^ { 2 } } * e ^ { - ( z ^ { 2 } + r ^ { 2 } ) / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            452,
+            888,
+            488
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Then substitute $r = p \\cos \\phi$ , $z = p \\sin \\phi$ where $\\phi \\in [ 0 , \\pi / 2 ]$ ",
+        "bbox": [
+            264,
+            494,
+            656,
+            511
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b92a8f204aebd66e20a92b50c4574ae827f16296d6b814a3beb292b4ceaeed59.jpg",
+        "text": "$$\n= ( 2 \\pi ) ^ { - n / 2 } \\ast 2 S _ { n - 3 } \\int _ { 0 } ^ { \\pi / 2 } d \\phi \\cos \\phi ^ { 3 } \\ast \\sin \\phi ^ { n - 3 } \\int _ { 0 } ^ { \\infty } d p \\ast p ^ { n - 1 } e ^ { - p ^ { 2 } / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            520,
+            779,
+            556
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The first integral is $\\frac { 2 } { n ( n - 2 ) }$ using $u = \\sin ^ { 2 } \\phi$ . The second integral is $2 ^ { ( n - 2 ) / 2 } \\Gamma ( n / 2 )$ using $u = p ^ { 2 } / 2$ and the definition of the gamma function. Simplifying, the result is $\\frac { 2 } { \\pi * n }$ ",
+        "bbox": [
+            264,
+            560,
+            825,
+            611
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Thus the angle between a vector and a binarized version of that vector converges to arccos ${ \\sqrt { \\frac { 2 } { \\pi } } } \\approx 3 7 ^ { \\circ }$ , which is a very small angle in high dimensions. ",
+        "bbox": [
+            176,
+            622,
+            825,
+            660
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5.2 AN EXPLICIT EXAMPLE OF LEARNING DYNAMICS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            675,
+            565,
+            690
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we look at the learning dynamics for the BNN training algorithm in a simple case and gain some insight about the learning algorithm. Consider the case of regression where the target output, $y$ , is predicted with a binary linear predictor with $x$ as the input. Using a squared error loss, ${ \\cal L } \\dot { = } ( y - \\hat { y } ) \\dot { ^ 2 } = ( y - w ^ { b } x ) ^ { 2 } = ( \\bar { y } - \\theta ( w ^ { c } \\acute { } ) x ) ^ { 2 }$ . (In this notation, $x$ is a column vector.) Taking the derivative of this loss with respect to the continuous weights and using the rule for back propagating through the binarize function gives $\\Delta w ^ { c } \\sim - d L / d w ^ { c } = - d L / d w ^ { \\check { b } } \\cdot d w ^ { b } / d w ^ { c } = ( y ^ { \\check { b } } - w ^ { \\check { b } } x ) x ^ { \\check { T } }$ . Finally, averaging over the training data gives ",
+        "bbox": [
+            173,
+            700,
+            826,
+            800
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75fde38297295a099c15f15aed083e979de557024ed883f22fc98381b47d8217.jpg",
+        "text": "$$\n\\Delta w ^ { c } \\sim C _ { y x } - \\theta ( w ^ { c } ) \\cdot C _ { x x } \\qquad C _ { y x } = E [ y x ^ { T } ] \\qquad C _ { x x } = E ( x x ^ { T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            805,
+            723,
+            824
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It is worthwhile to compare this equation the corresponding equation from typical linear regression: $\\Delta w ^ { c } \\sim C _ { y x } - w ^ { c } \\cdot C _ { x x }$ . For simplicity, consider the case where $C _ { x x }$ is the identity matrix. In this case, all of the components of $w$ become independent: $\\delta w = \\epsilon * ( \\alpha - \\theta ( w ) )$ where $\\epsilon$ is the learning rate and $\\alpha$ is the entry of $C _ { y x }$ corresponding to a particular element, $w$ . Compared to regular linear regression, it is clear that the stable point of these equations is when $w = \\alpha$ . Since the weight is binarized, that equation cannot be satisfied. However, it can be shown $( \\star )$ that in this special case of binary weight linear regression, $E ( \\theta ( w ^ { c } ) ) = \\alpha$ . Intuitively, if we consider a high dimensional vector and the fluctuations of each component are likely to be out of phase, then $\\boldsymbol { w } ^ { b } \\cdot \\boldsymbol { x } \\approx \\boldsymbol { w } ^ { c } \\cdot \\boldsymbol { x }$ is going to be correct in expectation with a variance that scales as $\\frac { 1 } { n }$ . During the actual learning process, we anneal the learning rate to a very small number, so the particular state of a fluctuating component of the vector is frozen in. Relatedly, the equation $C _ { y x } \\approx w C _ { x x }$ is easier to satisfy in high dimensions, whereas in low dimensions, it is only satisfied in expectation. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/452741137351217b927a8a8324af28e914b59f5c7097c1f8fbf70109360b8f3e.jpg",
+        "image_caption": [
+            "Figure 5: Activation Binarization Preserves Dot Products: Left: Each panel shows a 2D histogram of the dot products between the binarized weights and binarized activations (vertical axis) and post-batch norm (but pre-activation binarization) activations (horizontal axis). The binarization transformer does little to corrupt the dot products between weights and activations. Right: Dot products between the binary weights and binary activations (horizontal axis) compared against the dot products between the continuous weights and continuous activations (vertical axis). The dot products are not significantly impacted by removing binarization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            373,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof for $( \\star )$ : Suppose that $| \\alpha | \\le 1$ . The basic idea of these dynamics is that steps of size proportional to $\\epsilon$ are taken whose direction depends on whether $w > 0$ or $w < 0$ . In particular, if $w > 0$ , then the step is $- \\epsilon \\cdot | 1 - \\alpha |$ and if $w < 0$ , the step is $\\epsilon \\cdot \\left( \\alpha + 1 \\right)$ . It is evident that after a sufficient burn-in period, $| w | \\leq \\epsilon * \\operatorname* { m a x } ( | 1 - \\alpha | , 1 + \\alpha ) \\leq 2 \\epsilon$ . Suppose $w > 0$ occurs with fraction $p$ and $w < 0$ occurs with fraction $1 - p$ . In order for $w$ to be in equilibrium, oscillating about zero, these steps balance out on average: $p ( 1 - \\alpha ) = ( 1 - p ) ( 1 + \\alpha )  p = ( 1 + \\alpha ) / 2$ . Then the expected value of $\\theta ( w )$ is $1 * p + ( - 1 ) * ( 1 - p ) = \\alpha$ . When $| \\alpha | > 1$ , the dynamics diverge because $\\alpha - \\theta ( w )$ will always have the same sign. This divergence demonstrates the importance of some normalization technique such as batch normalization or attempting to represent $w$ with a constant times a binary matrix. ",
+        "bbox": [
+            173,
+            478,
+            825,
+            604
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5.3 DOT PRODUCT CORRELATIONS AFTER ACTIVATION PERMUTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            621,
+            679,
+            636
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Suppose that $A = w \\cdot a$ and $\\boldsymbol { B } = \\boldsymbol { v } \\cdot \\boldsymbol { a }$ where $w , v$ are weight vectors and $a$ is the vector of activations. What is the correlation, $r$ , between $A$ and $B ?$ Assuming that $E ( a ) = 0$ , $E ( A ) = E ( B ) = 0$ . Then $\\begin{array} { r } { E ( A B ) = \\sum _ { i , j } w _ { i } v _ { j } E ( a _ { i } a _ { j } ) = w ^ { T } C v } \\end{array}$ where $C _ { i , j } = E ( a _ { i } a _ { j } )$ . Setting $v = w$ and $w = v$ gives $E ( A ^ { 2 } ) = w ^ { T } C w$ and $v ^ { T } C v$ . Thus $\\begin{array} { r } { r = \\frac { w ^ { T } C v } { \\sqrt { ( w ^ { T } C w ) ( v ^ { T } C v ) } } } \\end{array}$ . ",
+        "bbox": [
+            173,
+            647,
+            826,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the case where the activations are randomly permuted, $C$ is proportional to the identity matrix, and thus the correlation between $A$ and $B$ is the cosine angle between $u$ and $v$ . ",
+        "bbox": [
+            171,
+            731,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5.4 ANGLE PLOT",
+        "text_level": 1,
+        "bbox": [
+            174,
+            775,
+            307,
+            790
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1acd23198be35f2ae8bf07828565f790fe198f42af03baa2da8695eb8ff226d2.jpg",
+        "image_caption": [
+            "Figure 6: Angle distribution between continuous and binary weight vectors by layer for a binary CNN trained on CIFAR-10 (same plot as in Fig. 2b except zoomed in). Notice that there is a small but systematic deviation towards larger angles relative to the theoretical expectation (vertical dotted line). As the dimension of the vectors in the layer goes up, the distribution gets sharper. The theory√ predicts that the standard deviation of these distributions scales as $1 / \\sqrt { d }$ . This relationship is shown to approximately hold in Fig. 2c. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            116,
+            656,
+            265
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5.5 TERNARY NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
+            431,
+            431
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Moving beyond binarization, recent work has shown how to train neural networks where the activations are quantized to three (or more) values (e.g. Hubara et al. (2016)). Indeed, ternarization may be a more natural quantization method than binarization for neural network weights because one can express a positive association $( + 1 )$ , a negative association $( - 1 )$ , or no association (0) between two features in a neural network. We show that the analysis used on BNNs holds for ternary neural networks. ",
+        "bbox": [
+            173,
+            443,
+            825,
+            526
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The quantization function: t $\\operatorname { x r } _ { a } ( x ) = 1$ if $x > a$ , 0 if $| x | < a$ , and $- 1$ if $x < - a$ is used in place of the binarize function with the same straight-through estimator for the gradient. Call $a$ the ternarization threshold. The exact value of $a$ is only important at initialization because the scaling constant of the batch normalization layer allows the network to adapt the standard deviation of the pre-nonlinearity activations to the value of $a$ . The network architecture from the previous experiments was used to classify images in CIFAR-10 and $a$ was chosen to be equal to 0.02. In practice, roughly 10 percent of the ternarized weights were zero (Fig. 7a) and 2 percent of the activations were zero. Thus the network learning process did not ignore the possibility of using zero weights. However, more work is needed to effectively use the zero value for the activations. ",
+        "bbox": [
+            174,
+            532,
+            825,
+            659
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The empirical distribution of angles between the continuous vectors and their ternarized counterparts is highly peaked at the value predicted by the theory (Fig. 7b). Random vectors are chosen from a standard normal distribution of dimension $d$ . As in the case of binarization, the ternarized version of a vector is close in angle to the original vector in high dimensions (Fig. 7c). These vectors are quantized using $\\mathrm { t e r } _ { a }$ for different values of $a$ . The peak angle varies substantially as a function of $a$ (Fig. 7d). Note that for $a = 0$ , the ternarization function collapses into the binarization function. The empirical value of $a$ is the ratio of the empirical threshold to the empirical standard deviation of the continuous weights. Thus $a \\approx 0 . 0 2 / 0 . 1 8 \\approx 0 . 1 1$ for the higher layers. Remarkably, the theoretical prediction for the peak angle as a function of $a$ matches closely with the empirical result (Fig. 7b). ",
+        "bbox": [
+            173,
+            665,
+            825,
+            791
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Finally, the dot product proportionality property is also shown to hold for ternary neural networks (Fig. 7e). Thus the continuous weights found using the TNN training algorithm approximate the continuous weights that one would get if the network were trained with continuous weights and regular backpropagation. ",
+        "bbox": [
+            173,
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/72bee4e85d8a824a5fa6b768abcb0acd56045eb210592b38724a042ad8feecdf.jpg",
+        "image_caption": [
+            "Figure 7: Ternarization of High-Dimensional Vectors Preserves their Direction in Theory and Practice: (a) Histogram of the components of the continuous weights at each layer for a ternary CNN trained on CIFAR-10. The distribution is approximately Gaussian for all but the first layer. The vertical lines show the ternarization thresholds and approximately 10 percent of the weights are sent to zero. (b) Angle distribution between continuous and ternary weight vectors by layer. The vertical dotted line (at $\\theta \\approx 3 4 ^ { \\circ }$ ) indicates the theoretical prediction given the empirical ternarization threshold $( \\approx 0 . 1 1 )$ . $d$ is the dimension of the filters at each layer. (c) Distribution of angles between two random vectors (blue), and between a vector and its quantized version (red), for a rotationally invariant distribution of dimension $d$ . The ternarization threshold is chosen to match the trained network. As the red and blue curves have little overlap, ternarization causes a small change in angle in high dimensions. (d) Angle between a random vector and its ternarized version as a function of ternarization threshold for $d = 1 0 0 0$ . There is a large variation in the angle over different thresholds. (e) Ternarization preserves dot products. The 2D histogram shows the dot products between the ternarized weights and the activations (horizontal axis) and the dot products between the continuous weights and the activations (vertical axis) for layer 3 of the network. The dot products are highly correlated. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    }
+]

parse/train/B1xtFpVtvB/B1xtFpVtvB_content_list.json

@@ -0,0 +1,1416 @@
+[
+    {
+        "type": "text",
+        "text": "IMPROVING THE GENERALIZATION OF VISUAL NAVIGATION POLICIES USING INVARIANCE REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            138
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            162,
+            398,
+            190
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            227,
+            544,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training agents to operate in one environment often yields overfitted models that are unable to generalize to the changes in that environment. However, due to the numerous variations that can occur in the real-world, the agent is often required to be robust in order to be useful. This has not been the case for agents trained with reinforcement learning $( R L )$ algorithms. In this paper, we investigate the overfitting of RL agents to the training environments in visual navigation tasks. Our experiments show that deep RL agents can overfit even when trained on multiple environments simultaneously. We propose a regularization method which combines RL with supervised learning methods by adding a term to the RL objective that would encourage the invariance of a policy to variations in the observations that ought not to affect the action taken. The results of this method, called invariance regularization, show an improvement in the generalization of policies to environments not seen during training. ",
+        "bbox": [
+            233,
+            262,
+            766,
+            443
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            478,
+            336,
+            494
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning control policies from high-dimensional sensory input has been gaining more traction lately due to the popularity of deep reinforcement learning (DRL) Mnih et al. (2015); Levine et al. (2015); Zhang et al. (2018b); Rakelly et al. (2019), which enables learning the perception and control modules simultaneously. However, most of the work done in RL chooses to evaluate the learned policies in the same environment in which training occurred Cobbe et al. (2018). ",
+        "bbox": [
+            174,
+            512,
+            825,
+            582
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Using the same environments to train and test agents does not give any insight into the generalization abilities of the learned policy. There could be a number of changes in the environment at test time that would degrade the agent’s performance. Variations could appear in the visual aspects that determine the agent’s observation, the physical structure that determines the agent’s state and even some aspects that are related to the agent’s goal (Figure 1). For example, different observations of the same room are encountered at different times of the day (different lighting conditions). New obstacles could be present. Levels of a game could be different, yet playing a few levels should often be enough to figure out how to play the rest. Such variations might result in a new environment where the control model that defined the training environment has changed. A robust policy should generalize from its experience and perform the same skills in the presence of these variations. ",
+        "bbox": [
+            174,
+            589,
+            825,
+            728
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DRL agents have been notorious for overfitting to their training environments Cobbe et al. (2018). An agent could have drastically different performance on testing environments even if it manages to maximize the reward during training Zhang et al. (2018a). Supervised learning algorithms have been shown to have some generalization guarantees when adding proper regularization Mohri et al. (2018). However, these guarantees are weakened in reinforcement learning algorithms where the source of the data is not i.i.d.. In order to make use of the progress of DRL algorithms in practice we need policies that are robust to possible changes in the sensory inputs, surrounding structure and even some aspects of the task. ",
+        "bbox": [
+            174,
+            736,
+            825,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we study the notion of generalization that is appropriate for visual navigation control policies that are learned with DRL. We present: (1) a study of the generalization of visual control policies to certain changes in the underlying dynamical system; (2) an alternative training method that combines DRL with supervised learning, thus using DRL to learn a controller while leveraging the generalization properties of supervised learning. In our experiments we use the VizDoom platform ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kempka et al. (2016) which is easily customizable and enables the generation of numerous variants of a given environment. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            152,
+            339,
+            167
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Visual navigation for mobile robots combines the domains of vision and control. Navigation can be described as finding a suitable and safe path between a starting state and a goal state Bonin-Font et al. (2008). Classical approaches split the problem into a sequence of sub-tasks, such as map construction, localization, planning and path following Bonin-Font et al. (2008). However, each sub-task requires some handengineering that is specific to the environment and task which makes it hard to adapt it to different scenarios without performing some tuning. Deep learning approaches enable the use of highly non-linear classifiers that can adapt their inner representations to learn to robustly solve complicated tasks Goodfellow et al. (2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we use reinforcement learning algorithms coupled with deep learning approaches to solve the task of navigating an agent towards ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/801395498f4cba41a2833c4f0a1153b6efefe4d8cb2b2abf170446f28cc0b916.jpg",
+        "image_caption": [
+            "Figure 1: The figure shows how environments may differ in their visual aspects, like textures of the surfaces. The textures provide a differentiator for each environment, where without them the environments would have shared the same state space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "a goal object using only its visual observations as input. The field of view of the agent is limited, i.e., it does not observe the full environment, and we do not provide an explicit map of the environment to that agent. ",
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+    {
+        "type": "text",
+        "text": "2.1 PROBLEM STATEMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We model the problem of visual navigation as a partially observed Markov decision process (POMDP) (Spaan, 2012). A POMDP is given by a tuple ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d41d174d47237fedf2c3853c72c1c004a9d8a9c0fdf95d776e5dd266c7079d64.jpg",
+        "text": "$$\n\\mathcal { P } : = \\langle \\mathcal { S } , \\mathcal { A } , \\Omega , R , T , O , P _ { 0 } \\rangle ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $s$ is the set of states, $\\mathcal { A }$ is the set of actions and $\\Omega$ is the set of observations, all which are assumed to be finite sets. The reward function is $R : S \\times A \\to \\mathbb { R }$ . The conditional transition probability mass function is $T : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ , with the interpretation that $T ( s , a , s ^ { \\prime } ) =$ $\\bar { p } ( s _ { t + 1 } = \\cdot | s _ { t } = s , a _ { t } = a )$ is the probability that the next state is $s ^ { \\prime }$ given that the current state is $s$ and that action $a$ is taken. The conditional observation probability mass function is $O : S \\times A \\times \\Omega  [ 0 , 1 ]$ , with the interpretation that $O ( s , a , o ) = p \\bar { ( } o _ { t } = o | s _ { t } = s , a _ { t - 1 } = a )$ is the probability of observing $o$ in state $s$ when the last action taken was $a$ , and we allow for a special observation probability $\\bar { O ( s , o ) } = p ( o _ { 0 } = o | s _ { 0 } = s )$ when in the initial state $s$ and no action has yet been taken. Finally, $P _ { 0 }$ is the initial state probability mass function, so that $P _ { 0 } ( s ) = p ( s _ { 0 } = s )$ is the probability that the initial state is $s$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In DRL, we work with a parameterized policy $\\pi _ { \\boldsymbol { \\theta } } ( h , a ) \\ : = \\ : p _ { \\boldsymbol { \\theta } } ( a _ { t } \\ : = \\ : a | h _ { t } \\ : = \\ : h )$ with parameters $\\theta ~ \\in ~ \\Theta$ , giving the probability of taking action $a$ given observation-action history $h _ { t } \\ : =$ $\\left( o _ { 0 } , a _ { 0 } , o _ { 1 } , a _ { 1 } , \\ldots , a _ { t - 1 } , o _ { t } \\right)$ . The objective is to adjust the parameters $\\theta$ to attain a high value for the discounted reward ",
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+    {
+        "type": "equation",
+        "img_path": "images/61f99ddd380d8a39808432603a6e67286ca1bd3805ee2f9eece794249dee7ad0.jpg",
+        "text": "$$\nJ _ { \\mathcal { P } } ( \\theta ) : = \\mathbb { E } _ { \\mathcal { P } } ^ { \\pi _ { \\theta } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( s _ { t } , a _ { t } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "with discount factor $\\gamma ~ \\in ~ [ 0 , 1 )$ . The expectation is over state-observation-action sequences $( s _ { t } , o _ { t } , a _ { t } ) _ { t = 0 } ^ { \\infty }$ where the initial state $s _ { 0 }$ is drawn from $P _ { 0 }$ and other elements of such a sequence are drawn from $T , O$ and $\\pi _ { \\theta }$ respectively (Sutton & Barto, 1998). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Many methods for attempting to approximate optimal policies have been proposed. For instance, policy gradient methods perform gradient ascent on estimates of the expected discounted reward. In ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "this work we use the proximal policy optimization (PPO) algorithm, which arguably shows relatively robust performance on a wide range of different tasks (Schulman et al., 2017). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 FORMALIZING GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As in classification, we wish to learn from a finite training set but still perform well on previouslyunseen examples from a test set. To formalize this, we have a distribution $\\mathcal { D }$ over POMDPs, representing multiple environments or tasks, and we sample $n ^ { \\mathrm { t r a i n } }$ POMDPs from this distribution $\\mathcal { P } _ { 1 } , \\mathcal { P } _ { 2 } , \\ldots , \\mathcal { P } _ { n ^ { \\mathrm { t a i n } } }$ . In the context of navigation, these POMDPs might differ in terms of their observation distributions, perhaps representing views of the same environment at different times of day or year, in terms of their transition distributions, perhaps representing maps with different geometries, or in terms of their reward distributions, perhaps corresponding to the specification of different goal states. Given this sample, we then learn a policy $\\pi _ { \\theta }$ from a finite collection of state-observation-action sequences from these POMDPs. In order to have a meaningful common policy across these POMDPs, we require that they have common state, action and observation spaces $s , A$ and $\\Omega$ . By analogy with the notion of generalization risk in classification (Mohri et al., 2018), we say that policy $\\pi _ { \\theta }$ generalizes well if it attains a high value for the expectation of the discounted reward over the full distribution of POMDPs, which we call the discounted generalization reward, so that $\\mathbb { E } _ { \\mathcal { P } \\sim \\mathcal { D } } J _ { \\mathcal { P } } ( \\boldsymbol { \\theta } )$ is high in some sense.This is our own terminology as we did not find a semantically-equivalent term in the literature. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "It is not hard to see that the discounted generalization reward is actually the discounted reward $J _ { \\mathcal { P } ^ { D } } ( \\theta )$ of a single larger POMDP $\\mathcal { P } ^ { \\mathcal { D } }$ , whose state space may however no longer be finite. To see this, let us associate a unique identifier $i ( \\mathcal { P } )$ with any POMDP sampled from $\\mathcal { D }$ and let ${ \\boldsymbol { \\mathcal { T } } } ^ { \\mathcal { D } }$ be the set of all such unique identifiers. In the large POMDP $\\mathcal { P } ^ { \\mathcal { D } }$ , the state space is the Cartesian product $\\boldsymbol { S } \\times \\boldsymbol { \\mathcal { T } } ^ { \\mathcal { D } }$ of the original states and these unique identifiers, but the action and observation spaces are just $\\mathcal { A }$ and $\\Omega$ . The initial state distribution is obtained by first sampling a POMDP $\\mathcal { P } \\sim \\mathcal { D }$ and then sampling $s _ { \\mathrm { e n v } } \\sim P _ { 0 } ^ { \\mathcal { P } }$ from that POMDP’s initial state distribution. The initial state in the large POMDP is then the concatenation $( s _ { \\mathrm { e n v } } , i ( \\mathcal { P } ) )$ . Thus one might succinctly state the problem of generalization in POMDPs as follows: given a distribution $\\mathcal { D }$ over POMDPs with common state, action and observation spaces and access to a sample of state-observation-action sequences from a sample of POMDPs drawn from $\\mathcal { D }$ , choose a policy $\\pi _ { \\theta }$ that obtains a high value for the discounted reward $J _ { \\mathcal { P } ^ { D } } ( \\theta )$ . ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Training in synthetic environments enables the simulation of huge amounts of experience in a span of a few hours. Simulations are convenient to use when training reinforcement learning agents that are often highly sample inefficient Sutton & Barto (1998). There is, frequently, a gap between the synthetic world and the real-world, mainly due to the manner in which the simulators depict the real-world dynamics and visual appearances. Often, these simulated worlds capture the richness and noise of the real-world with low-fidelity Tobin et al. (2017). Many have tried to propose transfer learning techniques to bridge the reality gap in order to still make use of fast simulators for training Taylor & Stone (2009). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One popular method to bridge the reality gap is by randomizing some aspects of the training environment Sadeghi & Levine (2016). This domain randomization technique has been shown to be successful for the transfer of grasping policies from simulated training environments to the real-world Tobin et al. (2017). However, the learned models resulting from that work are not control policies, but perception modules. Previous work has showed some success in transferring the perception module learned in simulation to the real world, but not the controller. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Cobbe et al. (2018) conduct a large scale study on generalization using a new environment, that resembles an arcade game, which they call CoinRun. They experiment by training on different background images and different level structures. They test with different regularization strategies and network architectures finding that the RL agent has a surprising tendency to overfit even to large training sets. Zhang et al. (2018a) reach a similar conclusion, when learning in grid-world environments, and state that the agents have a tendency to memorize levels of the training set. Unlike Cobbe et al. (2018), however, they argue that the methods that inject stochasticity into the dynamics of the system to prevent memorization, such as sticky actions Machado et al. (2017) and random initializations Hausknecht & Stone (2015), often do not help. In our work we are interested in generalization when navigating under partial observability unlike the fully observable CoinRun or grid-world environments. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Domain adaptation methods have also been used for simulated to real transfer. They allow models trained on a source domain to generalize to a target domain. Bousmalis et al. (2017) train a generative model to adapt the synthetic images of the simulator to appear like the real environment. It was shown to successfully transfer a grasping policy trained in simulation to the real world. However, they do not discuss whether the policy generalizes when variations happen in the target domain. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Another aspect of generalization is the transfer of learned skills to solve different tasks. In other words, generalization to the goal of the trained agent $g$ . Achieving different tasks would require the agent to have the ability to maximize different reward functions. Schaul et al. (2015) consider working with value functions that contain the goal $g$ as part of the agent’s state. They call them universal value functions. The reward will then become a function of a state-action-goal tuple $( s , a , g )$ instead of a classical state-action pair. In the paper, the authors present universal value function approximators (UVFA). A method that attempts to learn a universal value function estimate $V _ { \\theta } ( s , g )$ They show that UVFA’s can generalize for unseen state-goal pairs in grid-world setup. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Deep reinforcement learning has been used to train control policies. These DRL based methods generally propose to learn motor control commands from raw camera images, thus mapping pixels to commands that control the robot’s motors Levine et al. (2015). DRL algorithms have been used for various navigation tasks such as goal conditioned navigation Mirowski et al. (2016); Zhu et al. (2016) and mapless navigation Mirowski et al. (2018). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 GENERALIZATION IN VISUAL CONTROL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Control policies learned from high-dimensional visual input are often brittle and lack the robustness to operate in novel situations. Our main contribution is to propose a regularization term that can be added to the RL objective to improve the robustness of the learned policy to variations in the observations, presented in Section 4.2. However, to motivate the necessity of our proposed method we study domain randomization in Section 4.1; one of the current main practices that aims at learning a policy that generalizes well. ",
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+    {
+        "type": "text",
+        "text": "4.1 DOMAIN RANDOMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Domain randomization is typically used to train policies that can generalize to variations and noise in the observations. It is done by training on several POMDP’s that share the same $s , A , \\Omega$ spaces, however they could in their observation distribution. The motivation behind domain randomization is that it is assumed to be an effective technique to provide a policy that is invariant to the changes that would appear in the observations. We explore the problem of navigating the agent towards a goal object with random noise added to the agent’s observations. If the agent is able to perform the task in an environment defined by a POMDP $\\mathcal { P } _ { 1 }$ then it should still be able to perform the task in another POMDP $\\mathcal { P } _ { 2 }$ , if certain features $f$ of the environment that are specific to successfully achieving the task exist and are invariant to these variations, i.e., $f ( \\mathcal { P } _ { 1 } ) = f ( \\mathcal { P } _ { 2 } )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Section 5.1, we study domain randomization when added to RL training and the ability of resulting policies to generalize in unseen POMDPs. We want to investigate if the policy does in fact overfit to the training POMDPs and whether we mitigate that overfitting by training the policies on multiple POMDPs. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.2 INVARIANCE REGULARIZATION (PROPOSED METHOD) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the previous sections, we discussed how overfitting to the training environment can be a big problem in RL. Furthermore, we should be careful not to jump to the conclusion that training on different environments will ensure policies that generalize well to new environments. It is merely an assumption that has been shown to empirically hold up when used in a supervised learning context. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, we show in this work that this assumption might not hold for reinforcement learning techniques. This is compatible with the findings in Cobbe et al. (2018) and Zhang et al. (2018a). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We reason that in order to generalize well, the training objective should include a term that encourages policy generalization. Therefore, putting the weight of the problem of generalizing explicitly in the objective function itself. Formally, a function $h$ of variable $x$ is invariant to a transformation $\\phi$ of $x$ if $h ( x ) = h ( \\phi ( x ) )$ . We can deduce the same definition for the invariance of a policy $\\pi$ to changes in the observation given by some transformation $\\tau$ , $\\pi ( o ) = \\pi ( { \\mathcal T } ( o ) )$ . We add this regularization penalty term to the RL objective as shown in Equation (1): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e5bbca4591ad1f33a9702d06ddd7e68b4d1bc8c0ba3ed05079480c661da2369.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\quad \\quad L _ { p p o } ( { \\cal O } ; \\pi _ { \\theta } ) - \\frac { \\lambda } { N M } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { M } d ( \\pi _ { \\theta } ( o _ { i } ) , \\pi _ { \\theta } ( \\mathcal { T } _ { j } ( o _ { i } ) ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $L _ { p p o }$ is the PPO objective Schulman et al. (2017), $\\theta$ is the set of parameters that define the policy $\\pi _ { \\theta }$ , $d$ is a distance function between the two conditional distributions, and $\\lambda$ is a weighting coefficient of the penalty. $O = \\{ o _ { 1 } , o _ { 2 } , . . . o _ { N } \\}$ is a sequence of $N$ observations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\tau$ is a transformation of the observations. Given an observation $o$ and a transformation on that observation $\\tau$ where the transformation still holds the semantic context of the underlying state, but with added visual variations. We can think of the difference between observing a room with observation $o$ and observing the same room with observation $\\mathcal { T } ( o )$ as the color of the wall for example. Therefore, let us say that we observe $o$ in POMDP $\\mathcal { P }$ and observe $\\mathcal { T } ( o )$ in POMDP $\\mathcal { P } ^ { \\mathcal { T } }$ then ${ \\dot { \\boldsymbol { f } } } ( { \\mathcal { P } } ) = f ( { \\mathcal { P } } ^ { \\tau } )$ , where $f ( \\mathcal P )$ is the set of invariant features of the environment defined by the POMDP $\\mathcal { P }$ . We further discuss the nature of $\\tau$ in the experiments section. $M$ is the number of transformations of each observations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The penalty $d$ in Equation 1 resembles adding a constraint on the PPO objective, where the new objective dictates that the policy should simultaneously obtain a high reward while behaving similarly for the observations $o$ and $\\mathcal { T } ( o )$ . The idea is similar, in spirit, to trust region policy optimization Schulman et al. (2015) where a penalty term, resembling that which would result from imposing a trustregion constraint, is added to ensure monotonic improvement of the average return with each policy update. We call this method in Equation 1 invariance regularization (IR) since the regularization term indicates the invariance of the learned policy to a transformation of given observations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose two ways to solve the RL problem in Equation 1. The first is to directly optimize the full objective by adding the penalty to the original PPO loss. The second method splits the training process to two stages of training RL first and then performing a supervised learning step to minimize $d ( \\pi ( o ) , \\pi ( T ( o ) ) )$ which presents an elegant form that combines reinforcement learning with supervised learning, more details of the second method is available in Appendix A. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the next section, we will discuss experiments using both methods. Before that, we will describe a study on the effectiveness of domain randomization as a mean to reducing overfitting in DRL agents. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we present the results of two experiments. The first is about training RL with domain randomization. We discuss the ability of the learned policies to generalize to unseen environments when trained on variations of the training environment. The next part presents the results obtained when using the invariance regularization (IR) method, proposed in Section 4.2, with domain randomization and shows that it improves the success rate considerably. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We performed these experiments because we are interested in the following questions: (1) Does training on environments with random variations (as domain randomization suggests) learn a representation of the invariant $f$ with which the policy can generalize to other environments that share the same invariant features? (2) Can we find a training algorithm that would empirically guarantee finding these invariant features $f$ ? ",
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+    {
+        "type": "text",
+        "text": "5.1 DOMAIN RANDOMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We leverage the customizability of VizDoom maps Kempka et al. (2016) with hundreds of unique textures to generate train/test scenarios. The agent is required to reach an object in order to get a ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Num training envs:</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>500</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=5>PPO</td></tr><tr><td rowspan=1 colspan=1>RGB</td><td rowspan=1 colspan=1>0.21 ± 0.04</td><td rowspan=1 colspan=1>0.17± 0.04</td><td rowspan=1 colspan=1>0.35 ± 0.13</td><td rowspan=1 colspan=1>0.35 ± 0.16</td><td rowspan=1 colspan=1>0.34 ± 0.14</td></tr><tr><td rowspan=1 colspan=1>RGB-D</td><td rowspan=1 colspan=1>0.05±0.04</td><td rowspan=1 colspan=1>0.89 ±0.05</td><td rowspan=1 colspan=1>0.90±0.05</td><td rowspan=1 colspan=1>0.61 ±0.37</td><td rowspan=1 colspan=1>0.77 ± 0.33</td></tr><tr><td rowspan=1 colspan=1>Grayscale</td><td rowspan=1 colspan=1>0.36 ± 0.04</td><td rowspan=1 colspan=1>0.33 ± 0.13</td><td rowspan=1 colspan=1>0.37± 0.04</td><td rowspan=1 colspan=1>0.47±0.14</td><td rowspan=1 colspan=1>0.41 ± 0.22</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=5>PPO-IR (split)</td></tr><tr><td rowspan=1 colspan=1>RGB</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0.64 ± 0.05</td><td rowspan=1 colspan=1>0.69 ±0.03</td><td rowspan=1 colspan=1>0.72 ± 0.016</td><td rowspan=1 colspan=1>0.75 ± 0.02</td></tr><tr><td rowspan=1 colspan=1>RGB-D</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.85 ± 0.02</td><td rowspan=1 colspan=1>0.90±0.05</td><td rowspan=1 colspan=1>0.94±0.01</td><td rowspan=1 colspan=1>0.95± 0.02</td></tr><tr><td rowspan=1 colspan=1>Grayscale</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.69±0.01</td><td rowspan=1 colspan=1>0.76±0.02</td><td rowspan=1 colspan=1>0.75 ±0.02</td><td rowspan=1 colspan=1>0.76±0.02</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=5>PPO-IR (full objective)</td></tr><tr><td rowspan=1 colspan=1>RGB</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.79± 0.05</td><td rowspan=1 colspan=1>0.79 ±0.03</td><td rowspan=1 colspan=1>0.81 ±0.03</td><td rowspan=1 colspan=1>0.81±0.02</td></tr><tr><td rowspan=1 colspan=1>RGB-D</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.98±0.01</td><td rowspan=1 colspan=1>0.97 ± 0.01</td><td rowspan=1 colspan=1>0.99 ± 0.01</td><td rowspan=1 colspan=1>0.99± 0.01</td></tr><tr><td rowspan=1 colspan=1>Grayscale</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.79± 0.03</td><td rowspan=1 colspan=1>0.79±0.01</td><td rowspan=1 colspan=1>0.79±0.02</td><td rowspan=1 colspan=1>0.80±0.02</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Average success rate and standard deviation of agents, that are trained on a different number of randomly environments, when tested on 50 test environments whose textures are not seen during training. The bold values represent the algorithm that resulted in the best average success rate according to an amount of training environments and an input type. We see that our method brings stability to the average results and improves generalization even when no depth is added. ",
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+        "text": "reward. We train an actor-critic style agent Konda & Tsitsiklis (1999) to solve the task. The network consists of three convolutional layers and 2 fully connected layers, followed by the policy and value function estimator layers. The policy output is a four-dimensional fully-connected layer, where the four dimensions corresponds to four actions; move forward, turn right, turn left and do nothing. The ouput of the policy layer is a log-probability of each action. The value layer is a single unit that predicts the value function. This network architecture was proposed by Mnih et al. (2015). ReLUs are used as the non-linear operations in all layers Nair & Hinton (2010). As mentioned, we optimize the PPO objective Schulman et al. (2017) with a binary reward function $( + 1$ if goal is reached, 0 otherwise) and a discount factor $\\gamma = 0 . 9 9$ . ",
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+    {
+        "type": "text",
+        "text": "We generate the variations of the training environment by changing the textures on the surfaces using the numerous textures provided by VizDoom Kempka et al. (2016). We train agents on a subset of 1, 10, 50, 100 and 500 rooms from the generated environments and test on 50 rooms with textures from a hold-out set which are not seen in training. We detail this experimental setup in Appendix B. We experiment with different types of visual input; RGB, RGB-D and Grayscale. The number of training iterations is fixed at $5 \\times 1 0 ^ { 6 }$ to ensure repeatability of our experiment. The results are therefore potentially pessimistic, and in future work we would like to choose the number of iterations for each network independently so as to maximize generalization performance. The agent and the goal object are initialized at random positions in the environment at the start of each episode. ",
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+        "type": "text",
+        "text": "The role of depth. Adding a depth channel to the observation plays a significant role in generalization. Depth is invariant to many changes in the visible spectrum of the observations. This might lead the training agent to partly find an invariance in observations in its implicit perception model, which in this case can be as simple as focusing on the depth channel only. Therefore, it was not surprising to see, in Table 1, that the depth agents (RGB-D) generalize better than the agents without any depth information. ",
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+        "type": "text",
+        "text": "Table 1 shows the success rate of the PPO models with respect to the number of training environments used and the input type (RGB, RGB-D). The results are averaged over 5 seeds, a standard practice in the RL literature today. We notice the superior performance of the agent with depth than the agent without depth. The fact that the RGB agent is not able to generalize well even when exposed to numerous environments tells us that it might not be learning the invariance relating the environments. On the other hand, the RGB-D agents perform well on the testing environments even when the agents are only exposed to 10 random training environments. ",
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+        "type": "text",
+        "text": "Looking at the RGB and RGB-D experiments, the agents trained on 100 and 500 environments generalize worse on average than the ones trained on 10 and 50, which indicates that some agents might be overfitting. This is inspite of the fact that these agents are able to maximize the reward in the training set regardless of the set size. Looking at the max statistic of these results (not shown in this paper) the 100 and 500 experiments outperform the rest. However, the 100 and 500 experiments have a higher variance in the success rates of different seeds than the 10 and 50 experiments. High variance in the test results of the 100/500 RGB-D experiments shows that some seeds are able to achieve a near perfect score on the testing environment and others completely fail, thus there is then no empirical guarantee that RL agents will generalize when exposed to numerous environments. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The average success rate for the RGB input without the depth shows that domain randomization alone might not be an effective method to adapt the policy to variations in the observations, at least not in the context of RL. In fact, it shows little progress, e.g., the RGB agent exposed to one environment achieves around a $20 \\%$ success on the testing environments and the agents exposed to $5 0 +$ environments achieve less than $40 \\%$ success. These results are consistent when running with a grayscale channel (see Table 1). ",
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+    {
+        "type": "text",
+        "text": "While training by randomizing the environment did show some success in making supervised learning models generalize better, it fails to do so in RL policies. It is clear from these results, that adding random variations and relying solely on the RL objective is not enough to ensure generalization. Much of the success of domain randomization in previous works Tobin et al. (2017) was reported using supervised learning. Also, the generalization abilities of machine learning algorithms have been linked to supervised learning setups. Therefore, it would make sense to adapt supervised learning techniques to regularize the models trained with DRL. ",
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+    {
+        "type": "text",
+        "text": "5.2 INVARIANCE REGULARIZATION EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we will discuss the results obtained from training the agent using the method proposed in Section 4.2. As mentioned in Section 4.2, we propose two methods of using the proposed IR penalty. The first is to add to the PPO objective as in Equation 1, this method is referred to as (full objective) in the results. The second, which is referred to as (split) in the results, is to split the objective into two parts; RL step and a supervised learning step (more details available in Appendix A). The value of $\\lambda$ in Equation 1 used in all IR (full objective) experiments is 1.0. ",
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+    {
+        "type": "text",
+        "text": "As for the nature of transformation $\\tau$ of the observations, we tested with the same randomly textured environments from VizDoom, that were used in the previous section, in order to be able to make fair comparisons with the pure RL and domain randomization agents. Regarding the distance penalty term $d$ in Equation 1, we did preliminary experiments with the KL divergence, $L 1$ , $L 2$ and cross-entropy losses and the KL divergence returned the best results. Table 1 shows the results for combining PPO with the IR penalty using the two proposed implementations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observing the split method’s results, we see that the proposed training procedure returns stable success rates that are improves as more environments are added. The split version was able to outperform vanilla PPO and substantially improve the generalization especially in the cases of RGB/Grayscale inputs. Training with the full objective, however, returned the best results that outperform vanilla PPO with domain randomization and the split version of the IR algorithm. Similar to the split version, training on the full objective shows stable performance for the different inputs across different seeds. ",
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+    {
+        "type": "text",
+        "text": "The results, in Table 1, also show that the trained models, on the full objective, are achieving test success rate, with only 10 training environments, that is close if not identical to the agents trained on 50, 100 and 500 environments. These results suggests that training with the full objective version of the IR algorithms does not require a large number of environments to learn the invariant features. Notice the average testing success rate is similar across the different number of training environments since the model learns the invariant features from only 10 environments and adding more environments that share the same invariant features will not make a difference. We can verify that hypothesis when looking at the RGB-D testing results in the full objective part. All agents achieve a near perfect score which we attribute to the availability of an invariant feature map in the input (the depth channel) which only the agents trained with the full objective are able to catch. ",
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+    {
+        "type": "text",
+        "text": "5.2.1 COMPARISONS WITH OTHER REGULARIZATION TECHNIQUES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Regularization has been shown to help in the generalization of supervised learning models Srivastava et al. (2014). Using regularization in supervised learning often improves the performance of the trained models on test sets. However, regularization has not been frequently used in DRL setups, possibly due to the previously-common poor practise of testing and training in the same environment so there is no generalization gap Cobbe et al. (2018). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/18eddee47a70d9a9819ece91270c1aab5c304c516180c32953e656ebb986745f.jpg",
+        "image_caption": [
+            "Figure 2: Different regularization methods tested on the RGB input. (Left): The average success rate show that some of these methods achieve similar results to ours in some instances. (Right): The lower SPL of the other regularization methods relative to ours indicates some randomness in their learned policies "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "We compare our method with some regularization techniques that are frequently used; dropout, batchnorm and $L 2$ . The first experiment has a dropout layer added after each convolutional layer Srivastava et al. (2014), the second has a batchnorm layer added after every convolutional layer Ioffe & Szegedy (2015) and the last uses $L 2$ regularization. We choose the dropout probability to be 0.1 and the $L 2$ weight to be $1 0 ^ { - 4 }$ , the same values that were proposed by Cobbe et al. (2018). As in the previous setup, we train five models (different seeds) for each technique and evaluate on 50 environments whose textures are sampled from a hold-out set. We report the experiments done with RGB input only as it poses a harder problem and a larger gap than RGB-D. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 (left) shows the average success rate over 5 seeds for the four methods. We see that our proposed method is the only one that is steadily improving when more environments are added. The batchnorm models performed worst while dropout and $L 2$ achieved similar success rates to the split version of our method given 50 and 500 training environments. However, the entropy of the learned policies is substantially higher when dropout and $L 2$ are added to the model. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We hypothesize that the high entropy policies are able to generalize by acting randomly in some instances and this makes them more robust in certain situations. We show the success weighted shortest path length $( S P L )$ in Figure 2 (right). A random behavior that displays robustness (has a high success probability) would return a relatively lower SPL due to the fact that this random behavior will probably not take the shortest possible path to the goal. Details of the formulation of SPL are available in Appendix C. Figure 2 (right) shows that the dropout and L2 agents have a lower SPL than the IR agents indicating that these policies with higher entropy are inefficient. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We present a study of the generalization capabilities of visual navigation agents trained with deep reinforcement learning algorithms. We formalize what it means to generalize in the context of a POMDP. We find that the tendency of RL agent to overfit even when exposed to large training sets is quite visible. We show that using domain randomization with RL, without adding invariant features to the input such as the depth maps, is not enough to generalize. In the second part, we proposed Invariance Regularization (IR), a method that attempts to regularize the RL model with a supervised learning loss. It improves the generalization success and displays stable performance across different seeds. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we focused our experimentation on generalization to changes in the input observation. However, it is also interesting to generalize the learned skills to different architectural designs of the environment, just as one one wishes to generalize to different levels of the game as proposed in the retro competition Nichol et al. (2018). Another avenue of future work is to explore the appropriate transformation function $\\tau$ of the observations.One might consider an adaptive form of $\\tau$ learned with data augmentation Cubuk et al. (2018) or adversarial examples Goodfellow et al. (2015). ",
+        "bbox": [
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+            840,
+            825,
+            924
+        ],
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            285,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Angel X. Chang, Devendra Singh Chaplot, Alexey Dosovitskiy, Saurabh Gupta, Vladlen Koltun, Jana Kosecka, Jitendra Malik, Roozbeh Mottaghi, Manolis Savva, and Amir Roshan Zamir. On evaluation of embodied navigation agents. CoRR, abs/1807.06757, 2018. URL http://arxiv.org/abs/1807.06757. ",
+        "bbox": [
+            173,
+            126,
+            826,
+            183
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Francisco Bonin-Font, Alberto Ortiz, and Gabriel Oliver. Visual navigation for mobile robots: A survey. Journal of Intelligent and Robotic Systems, 53:263–296, 11 2008. doi: 10.1007/ s10846-008-9235-4. ",
+        "bbox": [
+            176,
+            189,
+            825,
+            233
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Konstantinos Bousmalis, Alex Irpan, Paul Wohlhart, Yunfei Bai, Matthew Kelcey, Mrinal Kalakrishnan, Laura Downs, Julian Ibarz, Peter Pastor, Kurt Konolige, Sergey Levine, and Vincent Vanhoucke. Using simulation and domain adaptation to improve efficiency of deep robotic grasping. CoRR, abs/1709.07857, 2017. URL http://arxiv.org/abs/1709.07857. ",
+        "bbox": [
+            173,
+            241,
+            826,
+            297
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Karl Cobbe, Oleg Klimov, Christopher Hesse, Taehoon Kim, and John Schulman. Quantifying generalization in reinforcement learning. CoRR, abs/1812.02341, 2018. URL http://arxiv. org/abs/1812.02341. ",
+        "bbox": [
+            174,
+            306,
+            825,
+            348
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ekin Dogus Cubuk, Barret Zoph, Dandelion Mané, Vijay Vasudevan, and Quoc V. Le. Autoaugment: Learning augmentation policies from data. CoRR, abs/1805.09501, 2018. URL http://arxiv. org/abs/1805.09501. ",
+        "bbox": [
+            173,
+            357,
+            826,
+            400
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http: //www.deeplearningbook.org. ",
+        "bbox": [
+            173,
+            409,
+            823,
+            438
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ian J. Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. CoRR, abs/1412.6572, 2015. ",
+        "bbox": [
+            173,
+            445,
+            823,
+            474
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthew Hausknecht and Peter Stone. The impact of determinism on learning Atari 2600 games. In AAAI Workshop on Learning for General Competency in Video Games, January 2015. ",
+        "bbox": [
+            174,
+            482,
+            825,
+            512
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. CoRR, abs/1502.03167, 2015. URL http://arxiv.org/ abs/1502.03167. ",
+        "bbox": [
+            173,
+            518,
+            825,
+            563
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Michał Kempka, Marek Wydmuch, Grzegorz Runc, Jakub Toczek, and Wojciech Jaskowski. ViZ- ´ Doom: A Doom-based AI research platform for visual reinforcement learning. In IEEE Conference on Computational Intelligence and Games, pp. 341–348, Santorini, Greece, Sep 2016. IEEE. URL http://arxiv.org/abs/1605.02097. ",
+        "bbox": [
+            173,
+            570,
+            826,
+            627
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Vijay R. Konda and John N. Tsitsiklis. Actor-critic algorithms. In Advances in Neural Information Processing Systems 12, pp. 1008–1014, 1999. URL http://papers.nips.cc/paper/ 1786-actor-critic-algorithms. ",
+        "bbox": [
+            174,
+            636,
+            826,
+            679
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep visuomotor policies. CoRR, abs/1504.00702, 2015. URL http://arxiv.org/abs/1504. 00702. ",
+        "bbox": [
+            173,
+            686,
+            825,
+            728
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Marlos C. Machado, Marc G. Bellemare, Erik Talvitie, Joel Veness, Matthew J. Hausknecht, and Michael Bowling. Revisiting the arcade learning environment: Evaluation protocols and open problems for general agents. CoRR, abs/1709.06009, 2017. URL http://arxiv.org/abs/ 1709.06009. ",
+        "bbox": [
+            174,
+            737,
+            826,
+            794
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Piotr Mirowski, Razvan Pascanu, Fabio Viola, Hubert Soyer, Andrew J. Ballard, Andrea Banino, Misha Denil, Ross Goroshin, Laurent Sifre, Koray Kavukcuoglu, Dharshan Kumaran, and Raia Hadsell. Learning to navigate in complex environments. CoRR, abs/1611.03673, 2016. URL http://arxiv.org/abs/1611.03673. ",
+        "bbox": [
+            174,
+            803,
+            825,
+            859
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Piotr Mirowski, Matthew Koichi Grimes, Mateusz Malinowski, Karl Moritz Hermann, Keith Anderson, Denis Teplyashin, Karen Simonyan, Koray Kavukcuoglu, Andrew Zisserman, and Raia Hadsell. Learning to navigate in cities without a map. CoRR, abs/1804.00168, 2018. URL http://arxiv.org/abs/1804.00168. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin A. Riedmiller, Andreas Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518:529–533, 2015. ",
+        "bbox": [
+            174,
+            103,
+            826,
+            174
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT Press, second edition, 2018. ",
+        "bbox": [
+            173,
+            181,
+            823,
+            210
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Vinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference on International Conference on Machine Learning, ICML’10, pp. 807–814, USA, 2010. Omnipress. ISBN 978-1-60558-907-7. URL http://dl.acm.org/citation.cfm?id $\\equiv$ 3104322.3104425. ",
+        "bbox": [
+            173,
+            219,
+            825,
+            276
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Nichol, Vicki Pfau, Christopher Hesse, Oleg Klimov, and John Schulman. Gotta learn fast: A new benchmark for generalization in RL. CoRR, abs/1804.03720, 2018. URL http://arxiv. org/abs/1804.03720. ",
+        "bbox": [
+            173,
+            284,
+            825,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kate Rakelly, Aurick Zhou, Deirdre Quillen, Chelsea Finn, and Sergey Levine. Efficient off-policy meta-reinforcement learning via probabilistic context variables. CoRR, abs/1903.08254, 2019. URL http://arxiv.org/abs/1903.08254. ",
+        "bbox": [
+            174,
+            334,
+            825,
+            377
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fereshteh Sadeghi and Sergey Levine. Real single-image flight without a single real image. CoRR, abs/1611.04201, 2016. URL http://arxiv.org/abs/1611.04201. ",
+        "bbox": [
+            173,
+            386,
+            825,
+            415
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tom Schaul, Dan Horgan, Karol Gregor, and David Silver. Universal value function approximators. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML’15, pp. 1312–1320. JMLR.org, 2015. URL http://dl.acm. org/citation.cfm?id=3045118.3045258. ",
+        "bbox": [
+            173,
+            422,
+            828,
+            479
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Sergey Levine, Philipp Moritz, Michael I. Jordan, and Pieter Abbeel. Trust region policy optimization. CoRR, abs/1502.05477, 2015. URL http://arxiv.org/abs/1502. 05477. ",
+        "bbox": [
+            173,
+            487,
+            825,
+            530
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017. URL http://arxiv.org/abs/ 1707.06347. ",
+        "bbox": [
+            171,
+            537,
+            825,
+            580
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "M.T.J. Spaan. Partially observable Markov decision processes. In Reinforcement Learning, pp. 387–414. Springer, 2012. ",
+        "bbox": [
+            169,
+            589,
+            825,
+            618
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15:1929–1958, 2014. URL http://jmlr.org/papers/v15/ srivastava14a.html. ",
+        "bbox": [
+            173,
+            626,
+            826,
+            683
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Richard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, Cambridge, MA, USA, 1st edition, 1998. ISBN 0262193981. ",
+        "bbox": [
+            171,
+            691,
+            826,
+            720
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Matthew E. Taylor and Peter Stone. Transfer learning for reinforcement learning domains: A survey. J. Mach. Learn. Res., 10:1633–1685, December 2009. ISSN 1532-4435. URL http: //dl.acm.org/citation.cfm?id=1577069.1755839. ",
+        "bbox": [
+            176,
+            728,
+            825,
+            771
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Joshua Tobin, Rachel Fong, Alex Ray, Jonas Schneider, Wojciech Zaremba, and Pieter Abbeel. Domain randomization for transferring deep neural networks from simulation to the real world. CoRR, abs/1703.06907, 2017. URL http://arxiv.org/abs/1703.06907. ",
+        "bbox": [
+            176,
+            779,
+            825,
+            821
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang, Oriol Vinyals, Rémi Munos, and Samy Bengio. A study on overfitting in deep reinforcement learning. CoRR, abs/1804.06893, 2018a. URL http://arxiv.org/abs/ 1804.06893. ",
+        "bbox": [
+            173,
+            830,
+            825,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Marvin Zhang, Sharad Vikram, Laura Smith, Pieter Abbeel, Matthew J. Johnson, and Sergey Levine. SOLAR: deep structured latent representations for model-based reinforcement learning. CoRR, abs/1808.09105, 2018b. URL http://arxiv.org/abs/1808.09105. ",
+        "bbox": [
+            174,
+            882,
+            826,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yuke Zhu, Roozbeh Mottaghi, Eric Kolve, Joseph J. Lim, Abhinav Gupta, Li Fei-Fei, and Ali Farhadi. Target-driven visual navigation in indoor scenes using deep reinforcement learning. CoRR, abs/1609.05143, 2016. URL http://arxiv.org/abs/1609.05143. ",
+        "bbox": [
+            179,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            263,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A IR SPLIT VERSION ",
+        "bbox": [
+            176,
+            133,
+            336,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 RL with iterative supervision ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            166,
+            454,
+            181
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Initialize $k _ { 1 } , k _ { 2 } , \\theta _ { 0 } , \\mathcal { T } _ { i = \\{ 1 . . . , N \\} } , e n v$   \nwhile not converged do for $i = 1 \\ldots , k _ { 1 }$ do // Train $\\pi _ { \\theta }$ on env on the RL objective $\\theta _ { i } \\gets \\operatorname* { m a x } _ { \\theta } L _ { p p o } ( o ^ { e n v } ; \\pi _ { \\theta _ { i - 1 } } )$ end for for $j = 1 \\ldots , k _ { 2 }$ do // Train $\\pi$ on env and $\\tau ( e n v )$ Sample $\\{ o _ { t } ^ { e n v } , \\pi _ { \\theta _ { k _ { 1 } } } ( o _ { t } ^ { e n v } ) \\}$ Generate $\\{ o _ { t } ^ { \\mathcal { T } _ { i } ( e n v ) } , \\pi _ { \\theta _ { k _ { 1 } } } ( o _ { t } ^ { \\mathcal { T } _ { i } ( e n v ) } ) \\} ^ { i = 1 \\dots N }$ $\\begin{array} { r } { \\theta _ { j }  \\operatorname* { m i n } _ { \\theta } d ( \\pi _ { \\theta _ { k _ { 1 } } } ( o _ { e n v } ) | | \\pi _ { \\theta _ { j - 1 } } ( \\mathcal T _ { i } ( o _ { e n v } ) ) } \\end{array}$ end for   \nend while   \nreturn $\\pi _ { \\theta }$ ",
+        "bbox": [
+            178,
+            185,
+            508,
+            392
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The first part consists of training RL on the observations of the original training environment, while the second part can be seen as a supervised learning objective on the transformed observations, as shown in Algorithm 1. ",
+        "bbox": [
+            176,
+            417,
+            823,
+            459
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The first step trains RL on one environment and then use the actions that the trained policy would have taken in that environment to tune the model with supervised learning on the textured environments. In the reported experiments using the split version, the model is trained with one iteration of the algorithm. Therefore, the training process has two stages, train RL then train with a supervised learning setup, without iterating between both. ",
+        "bbox": [
+            174,
+            467,
+            825,
+            536
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            553,
+            362,
+            568
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As stated in Section 5.1, we run training on a subset of 1, 10, 50, 100 and 500 rooms where the surfaces in each room are sampled from the variety of textures available in Vizdoom. The resulting policies are tested on 50 rooms with textures from a hold-out set which are not seen in training. During training we run several agents in parallel to quickly collect observation-action-reward data in multiple environments. Another advantage of this parallelization is the ability to run each agent on a variation of the training environment. Due to hardware limitations, we cannot run one agent for each environment, at least not when we have a large number of training environments, i.e., 100 or 500. Therefore, each agent samples one environment from the training set and runs on it for some $n$ episodes before sampling another one $n = 2 5$ episodes). ",
+        "bbox": [
+            174,
+            579,
+            825,
+            705
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C SUCCESS WEIGHTED SHORTEST PATH LENGTH (SPL) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            578,
+            736
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "SPL was proposed by Anderson et al. (2018) as a way of measuring the navigation agents success rates while taking into account the time it takes agents to succeed 1. ",
+        "bbox": [
+            173,
+            747,
+            825,
+            776
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8c41a7ee586dba413ee4fea8ed773809789805b228385e4c7404fbfb7f1b8cf.jpg",
+        "text": "$$\nS P L = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } S _ { i } \\frac { l _ { i } } { p _ { i } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            782,
+            571,
+            825
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $N$ is the number of runs, $S _ { i }$ is the binary indicator of the success of episode $i , l _ { i }$ is the length of the shortest possible path and $p _ { i }$ is the length of the path taken by the agent. ",
+        "bbox": [
+            173,
+            832,
+            823,
+            861
+        ],
+        "page_idx": 11
+    }
+]

parse/train/Rnn8zoAkrwr/Rnn8zoAkrwr.md

@@ -0,0 +1,406 @@
+# Independent mechanism analysis, a new concept?
+
+Luigi Gresele∗1
+
+Julius von Kügelgen∗1,2
+
+Vincent Stimper 1,2
+
+Bernhard Schölkopf 1 Michel Besserve 1
+
+1 Max Planck Institute for Intelligent Systems, Tübingen, Germany 2 University of Cambridge {luigi.gresele,jvk,vincent.stimper,bs,besserve}@tue.mpg.de
+
+# Abstract
+
+Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.
+
+# 1 Introduction
+
+One of the goals of unsupervised learning is to uncover properties of the data generating process, such as latent structures giving rise to the observed data. Identifiability [55] formalises this desideratum: under suitable assumptions, a model learnt from observations should match the ground truth, up to well-defined ambiguities. Within representation learning, identifiability has been studied mostly in the context of independent component analysis (ICA) [17, 40], which assumes that the observed data $\mathbf { x }$ results from mixing unobserved independent random variables $s _ { i }$ referred to as sources. The aim is to recover the sources based on the observed mixtures alone, also termed blind source separation (BSS). A major obstacle to BSS is that, in the nonlinear case, independent component estimation does not necessarily correspond to recovering the true sources: it is possible to give counterexamples where the observations are transformed into components $y _ { i }$ which are independent, yet still mixed with respect to the true sources $s _ { i }$ [20, 39, 98]. In other words, nonlinear ICA is not identifiable.
+
+In order to achieve identifiability, a growing body of research postulates additional supervision or structure in the data generating process, often in the form of auxiliary variables [28, 30, 37, 38, 41]. In the present work, we investigate a different route to identifiability by drawing inspiration from the field of causal inference [71, 78] which has provided useful insights for a number of machine learning tasks, including semi-supervised [87, 103], transfer [6, 23, 27, 31, 61, 72, 84, 85, 97, 102, 107], reinforcement [7, 14, 22, 26, 53, 59, 60, 106], and unsupervised [9, 10, 54, 70, 88, 91, 104, 105] learning. To this end, we interpret the ICA mixing as a causal process and apply the principle of independent causal mechanisms (ICM) which postulates that the generative process consists of independent modules which do not share information [43, 78, 87]. In this context, “independent” does not refer to statistical independence of random variables, but rather to the notion that the distributions and functions composing the generative process are chosen independently by Nature [43, 48]. While a formalisation of ICM [43, 57] in terms of algorithmic (Kolmogorov) complexity [51] exists, it is not computable, and hence applying ICM in practice requires assessing such non-statistical independence with suitable domain specific criteria [96]. The goal of our work is thus to constrain the nonlinear ICA problem, in particular the mixing function, via suitable ICM measures, thereby ruling out common counterexamples to identifiability which intuitively violate the ICM principle.
+
+![](images/c345db5ab9462a51fb733aea9cf13b579ce81064f822bec38c9bcf0db86d872b.jpg)  
+Figure 1: (Left) For the cocktail party problem, the ICM principle as traditionally understood would say that the content of speech $p _ { \mathbf { s } }$ is independent of the mixing or recording process f (microphone placement, room acoustics). IMA refines, or extends, this idea at the level of the mixing function by postulating that the contributions ${ \partial \mathbf { f } } / { \partial s _ { i } }$ of each source to $\mathbf { f }$ , as captured by the speakers’ positions relative to the recording process, should not be fine-tuned to each other. (Right) We formalise this independence between the ${ \partial \mathbf { f } } / { \partial s _ { i } }$ , which are the columns of the Jacobian $\mathbf { J _ { f } }$ , as an orthogonality condition: the absolute value of the determinant $| \mathbf { J _ { f } } |$ , i.e., the volume of the parallelepiped spanned by ${ \partial \mathbf { f } } / { \partial s _ { i } }$ , should decompose as the product of the norms of the ${ \partial \mathbf { f } } / { \partial s _ { i } }$ .
+
+Traditionally, ICM criteria have been developed for causal discovery, where both cause and effect are observed [18, 45, 46, 110]. They enforce an independence between (i) the cause (source) distribution and (ii) the conditional or mechanism (mixing function) generating the effect (observations), and thus rely on the fact that the observed cause distribution is informative. As we will show, this renders them insufficient for nonlinear ICA, since the constraints they impose are satisfied by common counterexamples to identifiability. With this in mind, we introduce a new way to characterise or refine the ICM principle for unsupervised representation learning tasks such as nonlinear ICA.
+
+Motivating example. To build intuition, we turn to a famous example of ICA and BSS: the cocktail party problem, illustrated in Fig. 1 (Left). Here, a number of conversations are happening in parallel, and the task is to recover the individual voices $s _ { i }$ from the recorded mixtures $x _ { i }$ . The mixing or recording process f is primarily determined by the room acoustics and the locations at which microphones are placed. Moreover, each speaker influences the recording through their positioning in the room, and we may think of this influence as $\partial \mathbf { f } / \partial s _ { i }$ . Our independence postulate then amounts to stating that the speakers’ positions are not fine-tuned to the room acoustics and microphone placement, or to each other, i.e., the contributions $\partial \mathbf { f } / \partial s _ { i }$ should be independent (in a non-statistical sense).1
+
+Our approach. We formalise this notion of independence between the contributions $\partial \mathbf { f } / \partial s _ { i }$ of each source to the mixing process (i.e., the columns of the Jacobian matrix $\mathbf { J _ { f } }$ of partial derivatives) as an orthogonality condition, see Fig. 1 (Right). Specifically, the absolute value of the determinant $| \mathbf { J _ { f } } |$ , which describes the local change in infinitesimal volume induced by mixing the sources, should factorise or decompose as the product of the norms of its columns. This can be seen as a decoupling of the local influence of each partial derivative in the pushforward operation (mixing function) mapping the source distribution to the observed one, and gives rise to a novel framework which we term independent mechanism analysis (IMA). IMA can be understood as a refinement of the ICM principle that applies the idea of independence of mechanisms at the level of the mixing function.
+
+Contributions. The structure and contributions of this paper can be summarised as follows:
+
+• we review well-known obstacles to identifiability of nonlinear ICA $( \ S 2 . 1 )$ , as well as existing ICM criteria $\mathrm { ~ ( ~ \ S ~ } \mathrm { ~ ) ~ }$ , and show that the latter do not sufficiently constrain nonlinear ICA $^ { ( \ S ) }$ ; • we propose a more suitable ICM criterion for unsupervised representation learning which gives rise to a new framework that we term independent mechanism analysis (IMA) $( \ S 4 )$ ; we provide geometric and information-theoretic interpretations of IMA $\mathrm { ~ ( ~ \ S ~ } \mathrm { ~ ) ~ }$ , introduce an IMA contrast function which is invariant to the inherent ambiguities of nonlinear ICA $\mathrm { ~  ~ { ~ \left( \ S ~ \right. ~ } ~ } )$ , and show that it rules out a large class of counterexamples and is consistent with existing identifiability results (§ 4.3); • we experimentally validate our theoretical claims and propose a regularised maximum-likelihood learning approach based on the IMA constrast which outperforms the unregularised baseline $^ { ( \ S ) }$ ; additionally, we introduce a method to learn nonlinear ICA solutions with triangular Jacobian and a metric to assess BSS which can be of independent interest for the nonlinear ICA community.
+
+# 2 Background and preliminaries
+
+Our work builds on and connects related literature from the fields of independent component analysis $\mathrm { ~ ( ~ \ S ~ } \mathrm { ~ ) ~ }$ and causal inference $( \ S 2 . 2 )$ . We review the most important concepts below.
+
+# 2.1 Independent component analysis (ICA)
+
+Assume the following data-generating process for independent component analysis (ICA)
+
+$$
+\begin{array} { r } { { \mathbf { x } } = { \mathbf { f } } \left( { \mathbf { s } } \right) , \qquad p _ { { \mathbf { s } } } ( { \mathbf { s } } ) = \prod _ { i = 1 } ^ { n } p _ { s _ { i } } ( s _ { i } ) , } \end{array}
+$$
+
+where the observed mixtures $\mathbf { x } \in \mathbb { R } ^ { n }$ result from applying a smooth and invertible mixing function $\mathbf { f } : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ to a set of unobserved, independent signals or sources $\mathbf { s } \in \mathbb { R } ^ { n }$ with smooth, factorised density $p _ { \mathbf { s } }$ with connected support (see illustration Fig. 2b). The goal of ICA is to learn an unmixing function $\mathbf { g } : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ such that $\mathbf { y } = \mathbf { g } ( \mathbf { x } )$ has independent components. Blind source separation (BSS), on the other hand, aims to recover the true unmixing f $^ { - 1 }$ and thus the true sources s (up to tolerable ambiguities, see below). Whether performing ICA corresponds to solving BSS is related to the concept of identifiability of the model class. Intuitively, identifiability is the desirable property that all models which give rise to the same mixture distribution should be “equivalent” up to certain ambiguities, formally defined as follows.
+
+Definition 2.1 $\sim$ -identifiability). Let $\mathcal { F }$ be the set of all smooth, invertible functions f $\displaystyle \dot { \mathbf { \beta } } : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ , and $\mathcal { P }$ be the set of all smooth, factorised densities $p _ { \mathbf { s } }$ with connected support on $\mathbb { R } ^ { n }$ . Let $\mathcal { M } \subseteq \mathcal { F } \times \mathcal { P }$ be a subspace of models and let $\sim$ be an equivalence relation on $\mathcal { M }$ . Denote by $\mathbf { f } _ { \ast } p _ { \mathbf { s } }$ the push-forward density of $p _ { \mathbf { s } }$ via f . Then the generative process (1) is said to be $\sim$ -identifiable on $\mathcal { M }$ if
+
+$$
+\forall ( \mathbf { f } , p _ { \mathrm { s } } ) , ( \tilde { \mathbf { f } } , p _ { \mathrm { \tilde { s } } } ) \in \mathcal { M } : \qquad \mathbf { f } _ { \ast } p _ { \mathrm { s } } = \tilde { \mathbf { f } } _ { \ast } p _ { \mathrm { \tilde { s } } } \qquad \implies \qquad ( \mathbf { f } , p _ { \mathrm { s } } ) \sim ( \tilde { \mathbf { f } } , p _ { \mathrm { \tilde { s } } } ) .
+$$
+
+If the true model belongs to the model class $\mathcal { M }$ , then $\sim$ -identifiability ensures that any model in $\mathcal { M }$ learnt from (infinite amounts of) data will be $\sim$ -equivalent to the true one. An example is linear ICA which is identifiable up to permutation and rescaling of the sources on the subspace $\mathcal { M } _ { \mathrm { L I N } }$ of pairs of (i) invertible matrices (constraint on $\mathcal { F }$ ) and (ii) factorizing densities for which at most one $s _ { i }$ is Gaussian (constraint on $\mathcal { P }$ ) [17, 21, 93], see Appendix A for a more detailed account.
+
+In the nonlinear case (i.e., without constraints on $\mathcal { F }$ ), identifiability is much more challenging. If $s _ { i }$ and $s _ { j }$ are independent, then so are $h _ { i } ( s _ { i } )$ and $h _ { j } ( s _ { j } )$ for any functions $h _ { i }$ and $h _ { j }$ . In addition to permutation-ambiguity, such element-wise $\mathbf { h } ( \mathbf { s } ) = { \big ( } h _ { 1 } ( s _ { 1 } ) , . . . , h _ { n } ( s _ { n } ) { \big ) }$ can therefore not be resolved either. We thus define the desired form of identifiability for nonlinear BSS as follows.
+
+Definition 2.2 $( \sim _ { \mathrm { B S S } } )$ . The equivalence relation $\sim _ { \tt B S S }$ on $\mathcal { F } \times \mathcal { P }$ defined as in Defn. 2.1 is given by
+
+$$
+\mathbf { ( f ,  { p _ { \mathrm { s } } } ) \sim _ { \mathrm { { B S S } } } ( \tilde { \mathbf { f } } ,  { p _ { \tilde { \mathbf { s } } } } ) } \iff \exists \mathbf { P } , \mathbf { h } \quad \mathrm { s . t . } \quad ( \mathbf { f } ,  { p _ { \mathbf { s } } } ) = ( \tilde { \mathbf { f } } \circ \mathbf { h } ^ { - 1 } \circ \mathbf { P } ^ { - 1 } , ( \mathbf { P } \circ \mathbf { h } ) _ { \ast } \mathbf { \Lambda } \mathbf { p } _ { \tilde { \mathbf { s } } } )
+$$
+
+here $\mathbf { P }$ is a permutation and $\mathbf { h } ( \mathbf { s } ) = { \big ( } h _ { 1 } ( s _ { 1 } ) , . . . , h _ { n } ( s _ { n } ) { \big ) }$ is an invertible, element-wise function.
+
+A fundamental obstacle—and a crucial difference to the linear problem—is that in the nonlinear case, different mixtures of $s _ { i }$ and $s _ { j }$ can be independent, i.e., solving ICA is not equivalent to solving BSS. A prominent example of this is given by the Darmois construction [20, 39].
+
+Definition 2.3 (Darmois construction). The Darmois construction $\mathbf { g } ^ { \mathrm { D } } : \mathbb { R } ^ { n }  ( 0 , 1 ) ^ { n }$ is obtained by recursively applying the conditional cumulative distribution function (CDF) transform:
+
+$$
+\begin{array} { r } { g _ { i } ^ { \mathrm { D } } ( \mathbf { x } _ { 1 : i } ) : = { \mathbb P } ( X _ { i } \leq x _ { i } | \mathbf { x } _ { 1 : i - 1 } ) = \int _ { - \infty } ^ { x _ { i } } p ( x _ { i } ^ { \prime } | \mathbf { x } _ { 1 : i - 1 } ) d x _ { i } ^ { \prime } ~ ( i = 1 , . . . , n ) . } \end{array}
+$$
+
+The resulting estimated sources ${ \bf y } ^ { \mathrm { D } } = { \bf g } ^ { \mathrm { D } } ( { \bf x } )$ are mutually-independent uniform r.v.s by construction, see Fig. 2a for an illustration. However, they need not be meaningfully related to the true sources s, and will, in general, still be a nonlinear mixing thereof [39].2 Denoting the mixing function corresponding to (4) by $\mathbf { f } ^ { \mathrm { { D } } } = ( \mathbf { g } ^ { \mathrm { { D } } } ) ^ { - 1 }$ and the uniform density on $( 0 , 1 ) ^ { n }$ by $p _ { \mathbf { u } }$ , the Darmois solution $( \mathbf { f } ^ { \mathrm { D } } , p _ { \mathbf { u } } )$ thus allows construction of counterexamples to $\sim _ { \tt B S S }$ -identifiability on $\mathcal { F } \times \mathcal { P }$ . 3
+
+Remark 2.4. $\mathbf { g } ^ { \mathrm { { D } } }$ has lower-triangular Jacobian, i.e., $\partial g _ { i } ^ { \mathrm { D } } / \partial x _ { j } = 0$ for $i < j$ . Since the order of the $x _ { i }$ is arbitrary, applying $\mathbf { g } ^ { \mathrm { { D } } }$ after a permutation yields a different Darmois solution. Moreover, (4) yields independent components $\mathbf { y } ^ { \mathrm { { D } } }$ even if the sources $s _ { i }$ were not independent to begin with.4
+
+![](images/3cdfe122ddcbe9cbd06e82141363d24ddf762882e5522219e6610a9100d5038b.jpg)  
+Figure 2: (a) Any observed density $p _ { x }$ can be mapped to a uniform $p _ { y }$ via the CDF transform $g ( x ) = \mathbb { P } ( X \leq x )$ ; Darmois solutions $( \mathbf { f } ^ { \mathrm { D } } , p _ { \mathbf { u } } )$ constructed from (4) therefore automatically satisfy the independence postulated by IGCI (6). (b) ICA setting with $n = 2$ sources (shaded nodes are observed, white ones are unobserved). (c) Existing ICM criteria typically enforce independence between an observed input or cause distribution $p _ { \mathbf { c } }$ and a mechanism $\scriptstyle p _ { \mathbf { e } | \mathbf { c } }$ (independent objects are highlighted in blue and red). (d) IMA enforces independence between the contributions of different sources $s _ { i }$ to the mixing function f as captured by ${ \partial \mathbf { f } } / { \partial s _ { i } }$ .
+
+Another well-known obstacle to identifiability are measure-preserving automorphisms (MPAs) of the source distribution $p _ { \mathbf { s } }$ : these are functions a which map the source space to itself without affecting its distribution, i.e., $\mathbf { a } _ { * } p _ { \mathbf { s } } = p _ { \mathbf { s } }$ [39]. A particularly instructive class of MPAs is the following [49, 58].
+
+Definition 2.5 (“Rotated-Gaussian” MPA). Let $\mathbf { R } \in O ( n )$ be an orthogonal matrix, and denote by $\mathbf { F _ { s } } ( \mathbf { s } ) = \left( F _ { s _ { 1 } } ( s _ { 1 } ) , . . . , F _ { s _ { n } } ( s _ { n } ) \right)$ and $\Phi ( \mathbf { z } ) = ( \Phi ( z _ { 1 } ) , . . . , \Phi ( z _ { n } ) )$ the element-wise CDFs of a smooth, factorised density $p _ { \mathbf { s } }$ and of a Gaussian, respectively. Then the “rotated-Gaussian” MPA $\mathbf { a ^ { R } } ( p _ { \mathbf { s } } )$ is
+
+$$
+\mathbf { a } ^ { \mathbf { R } } ( p _ { \mathbf { s } } ) = \mathbf { F } _ { \mathbf { s } } ^ { - 1 } \circ \Phi \circ \mathbf { R } \circ \Phi ^ { - 1 } \circ \mathbf { F _ { s } } .
+$$
+
+$\mathbf { a ^ { R } } ( p _ { \mathbf { s } } )$ first maps to the (rotationally invariant) standard isotropic Gaussian (via $\Phi ^ { - 1 } \circ \mathbf { F _ { s } } ,$ ), then applies a rotation, and finally maps back, without affecting the distribution of the estimated sources. Hence, if $( \tilde { \mathbf { f } } , p _ { \tilde { \mathbf { s } } } )$ is a valid solution, then so is $( \tilde { \mathbf { f } } \circ \mathbf { a } ^ { \mathbf { R } } ( p _ { \tilde { \mathbf { s } } } ) , p _ { \tilde { \mathbf { s } } } )$ for any $\mathbf { R } \in O ( n )$ . Unless $\mathbf { R }$ is a permutation, this constitutes another common counterexample to $\sim _ { \tt B S S }$ -identifiability on $\mathcal { F } \times \mathcal { P }$ .
+
+Identifiability results for nonlinear ICA have recently been established for settings where an auxiliary variable $\mathbf { u }$ (e.g., environment index, time stamp, class label) renders the sources conditionally independent [37, 38, 41, 49]. The assumption on $p _ { \mathbf { s } }$ in (1) is replaced with $\begin{array} { r } { p _ { \mathbf { s } | \mathbf { u } } ( \mathbf { s } | \mathbf { u } ) = \prod _ { i = 1 } ^ { n } p _ { s _ { i } | \mathbf { u } } ( s _ { i } | \mathbf { u } ) } \end{array}$ thus restricting $\mathcal { P }$ in Defn. 2.1. In most cases, $\mathbf { u }$ is assumed to be observed, though [30] is a notable exception. Similar results exist given access to a second noisy view $\tilde { \mathbf { x } }$ [28].
+
+# 2.2 Causal inference and the principle of independent causal mechanisms (ICM)
+
+Rather than relying only on additional assumptions on $\mathcal { P }$ (e.g., via auxiliary variables), we seek to further constrain (1) by also placing assumptions on the set $\mathcal { F }$ of mixing functions f . To this end, we draw inspiration from the field of causal inference [71, 78]. Of central importance to our approach is the Principle of Independent Causal Mechanisms (ICM) [43, 56, 87].
+
+Principle 2.6 (ICM principle [78]). The causal generative process of a system’s variables is composed of autonomous modules that do not inform or influence each other.
+
+These “modules” are typically thought of as the conditional distributions of each variable given its direct causes. Intuitively, the principle then states that these causal conditionals correspond to independent mechanisms of nature which do not share information. Crucially, here “independent” does not refer to statistical independence of random variables, but rather to independence of the underlying distributions as algorithmic objects. For a bivariate system comprising a cause c and an effect e, this idea reduces to an independence of cause and mechanism, see Fig. 2c. One way to formalise ICM uses Kolmogorov complexity $K ( \cdot )$ [51] as a measure of algorithmic information [43].
+
+However, since Kolmogorov complexity is is not computable, using ICM in practice requires assessing Principle 2.6 with other suitable proxy criteria [9, 11, 34, 42, 45, 65, 75–78, 90, 110].5 Allowing for deterministic relations between cause (sources) and effect (observations), the criterion which is most closely related to the ICA setting in (1) is information-geometric causal inference (IGCI) [18, 46].6 IGCI assumes a nonlinear relation $\mathbf { e } = \mathbf { f } ( \mathbf { c } )$ and formulates a notion of independence between the cause distribution $p _ { \mathbf { c } }$ and the deterministic mechanism f (which we think of as a degenerate conditional $p _ { \mathbf { e } | \mathbf { c } } $ ) via the following condition (in practice, assumed to hold approximately),
+
+$$
+\begin{array} { r } { C _ { \mathrm { I G C I } } ( \mathbf { f } , p _ { \mathbf { c } } ) : = \int \log \left| \mathbf { J } _ { \mathbf { f } } ( \mathbf { c } ) \right| p _ { \mathbf { c } } ( \mathbf { c } ) d \mathbf { c } - \int \log \left| \mathbf { J } _ { \mathbf { f } } ( \mathbf { c } ) \right| d \mathbf { c } = 0 , } \end{array}
+$$
+
+where $( \mathbf { J _ { f } } ( \mathbf { c } ) ) _ { i j } = \partial f _ { i } / \partial c _ { j } ( \mathbf { c } )$ is the Jacobian matrix and $| \cdot |$ the absolute value of the determinant. $C _ { \mathrm { I G C I } }$ can be understood as the covariance between $p _ { \mathbf { c } }$ and $\log | \mathbf { J _ { f } } |$ (viewed as r.v.s on the unit cube w.r.t. the Lebesgue measure), so that $C _ { \mathrm { { I G C I } } } = 0$ rules out a form of fine-tuning between $p _ { \mathbf { c } }$ and $| \mathbf { J _ { f } } |$ . As its name suggests, IGCI can, from an information-geometric perspective, also be seen as an orthogonality condition between cause and mechanism in the space of probability distributions [46], see Appendix B.2, particularly eq. (19) for further details.
+
+# 3 Existing ICM measures are insufficient for nonlinear ICA
+
+Our aim is to use the ICM Principle 2.6 to further constrain the space of models ${ \mathcal { M } } \subseteq { \mathcal { F } } \times { \mathcal { P } }$ and rule out common counterexamples to identifiability such as those presented in $\ S$ . Intuitively, both the Darmois construction (4) and the rotated Gaussian MPA (5) give rise to “non-generic” solutions which should violate ICM: the former, $( \mathbf { f } ^ { \mathrm { D } } , p _ { \mathbf { u } } )$ , due the triangular Jacobian of $\mathbf { f } ^ { \mathrm { { D } } }$ (see Remark 2.4), meaning that each observation $x _ { i } = f _ { i } ^ { \mathrm { D } } ( \mathbf { y } _ { 1 : i } )$ only depends on a subset of the inferred independent components $\mathbf { y } _ { 1 : i }$ , and the latter, $( \mathbf { f } \circ \mathbf { a } ^ { \mathbf { R } } ( p _ { \mathbf { s } } ) , p _ { \mathbf { s } } )$ , due to the dependence of $\mathbf { f } \circ \mathbf { a } ^ { \mathbf { R } } ( p _ { \mathbf { s } } )$ on $p _ { \mathbf { s } }$ (5).
+
+However, the ICM criteria described in $\ S$ were developed for the task of cause-effect inference where both variables are observed. In contrast, in this work, we consider an unsupervised representation learning task where only the effects (mixtures $\mathbf { x }$ ) are observed, but the causes (sources s) are not. It turns out that this renders existing ICM criteria insufficient for BSS: they can easily be satisfied by spurious solutions which are not equivalent to the true one. We can show this for IGCI. Denote by $\bar { \mathcal { M } } _ { \mathrm { I G C I } } = \{ ( \mathbf { f } , p _ { \mathbf { s } } ) \in \mathcal { F } \times \mathcal { P } : C _ { \mathrm { I G C I } } ( \mathbf { f } , p _ { \mathbf { s } } ) = 0 \} \subset \mathcal { F } \times \mathcal { P }$ the class of nonlinear ICA models satisfying IGCI (6). Then the following negative result holds.
+
+Proposition 3.1 (IGCI is insufficient for $\sim _ { \tt B S S }$ -identifiability). (1) is not $\sim _ { \tt B S S }$ -identifiable on $\mathcal { M } _ { \mathrm { I G C I } }$
+
+Proof. IGCI (6) is satisfied when $p _ { \mathbf { s } }$ is uniform. However, the Darmois construction (4) yields uniform sources, see Fig. 2a. This means that $( \mathbf { f } ^ { \mathrm { D } } \circ \mathbf { a } ^ { \mathbf { R } } ( p _ { \mathbf { u } } ) , p _ { \mathbf { u } } ) \in \mathcal { M } _ { \mathrm { I G C I } }$ , so IGCI can be satisfied by solutions which do not separate the sources in the sense of Defn. 2.2, see footnote 2 and [39].
+
+As illustrated in Fig. 2c, condition (6) and other similar criteria enforce a notion of “genericity” or “decoupling” of the mechanism w.r.t. the observed input distribution.7 They thus rely on the fact that the cause (source) distribution is informative, and are generally not invariant to reparametrisation of the cause variables. In the (nonlinear) ICA setting, on the other hand, the learnt source distribution may be fairly uninformative. This poses a challenge for existing ICM criteria since any mechanism is generic w.r.t. an uninformative (uniform) input distribution.
+
+# 4 Independent mechanism analysis (IMA)
+
+As argued in $\ S$ , enforcing independence between the input distribution and the mechanism (Fig. 2c), as existing ICM criteria do, is insufficient for ruling out spurious solutions to nonlinear ICA. We therefore propose a new ICM-inspired framework which is more suitable for BSS and which we term independent mechanism analysis (IMA).8 All proofs are provided in Appendix C.
+
+# 4.1 Intuition behind IMA
+
+As motivated using the cocktail party example in $\ S$ and Fig. 1 $( L e f t )$ , our main idea is to enforce a notion of independence between the contributions or influences of the different sources $s _ { i }$ on the observations $\mathbf { x } = \mathbf { f } ( \mathbf { s } )$ as illustrated in Fig. 2d—as opposed to between the source distribution and mixing function, cf. Fig. 2c. These contributions or influences are captured by the vectors of partial derivatives ${ \partial \mathbf { f } } / { \partial s _ { i } }$ . IMA can thus be understood as a refinement of ICM at the level of the mixing f : in addition to statistically independent components $s _ { i }$ , we look for a mixing with contributions $\partial \mathbf { f } / \partial s _ { i }$ which are independent, in a non-statistical sense which we formalise as follows.
+
+Principle 4.1 (IMA). The mechanisms by which each source $s _ { i }$ influences the observed distribution, as captured by the partial derivatives ${ \partial \mathbf { f } } / { \partial s _ { i } }$ , are independent of each other in the sense that for all s:
+
+$$
+\underline { { \log | \mathbf { J } _ { \mathbf { f } } ( \mathbf { s } ) | } } = \sum _ { i = 1 } ^ { n } \log \left\| \frac { \partial \mathbf { f } } { \partial s _ { i } } ( \mathbf { s } ) \right\|
+$$
+
+Geometric interpretation. Geometrically, the IMA principle can be understood as an orthogonality condition, as illustrated for $n = 2$ in Fig. 1 $( R i g h t )$ . First, the vectors of partial derivatives ${ \partial \mathbf { f } } / { \partial s _ { i } }$ for which the IMA principle postulates independence, are the columns of $\mathbf { J _ { f } }$ . $| \mathbf { J _ { f } } |$ thus measures the volume of the $n$ −dimensional parallelepiped spanned by these columns, as shown on the right. The product of their norms, on the other hand, corresponds to the volume of an $n$ -dimensional box, or rectangular parallelepiped with side lengths $\| { \partial \mathbf { f } } / { \partial s _ { i } } \|$ , as shown on the left. The two volumes are equal if and only if all columns $\partial \mathbf { f } / \partial s _ { i }$ of $\mathbf { J _ { f } }$ are orthogonal. Note that (7) is trivially satisfied for $n = 1$ , i.e., if there is no mixing, further highlighting its difference from ICM for causal discovery.
+
+Independent influences and orthogonality. In a high dimensional setting (large $n$ ), this orthogonality can be intuitively interpreted from the ICM perspective as Nature choosing the direction of the influence of each source component in the observation space independently and from an isotropic prior. Indeed, it can be shown that the scalar product of two independent isotropic random vectors in $\mathbb { R } ^ { n }$ vanishes as the dimensionality $n$ increases (equivalently: two high-dimensional isotropic vectors are typically orthogonal). This property was previously exploited in other linear ICM-based criteria (see [44, Lemma 5] and [45, Lemma 1 & Thm. 1]).9 The principle in (7) can be seen as a constraint on the function space, enforcing such orthogonality between the columns of the Jacobian of f at all points in the source domain, thus approximating the high-dimensional behavior described above.10
+
+Information-geometric interpretation and comparison to IGCI. The additive contribution of the sources’ influences ${ \partial \mathbf { f } } / { \partial s _ { i } }$ in (7) suggests their local decoupling at the level of the mechanism f . Note that IGCI (6), on the other hand, postulates a different type of decoupling: one between $\log | \mathbf { J _ { f } } |$ and $p _ { \mathbf { s } }$ . There, dependence between cause and mechanism can be conceived as a fine tuning between the derivative of the mechanism and the input density. The IMA principle leads to a complementary, non-statistical measure of independence between the influences ${ \partial \mathbf { f } } / { \partial s _ { i } }$ of the individual sources on the vector of observations. Both the IGCI and IMA postulates have an information-geometric interpretation related to the influence of (“non-statistically”) independent modules on the observations: both lead to an additive decomposition of a KL-divergence between the effect distribution and a reference distribution. For IGCI, independent modules correspond to the cause distribution and the mechanism mapping the cause to the effect (see (19) in Appendix B.2). For IMA, on the other hand, these are the influences of each source component on the observations in an interventional setting (under soft interventions on individual sources), as measured by the KL-divergences between the original and intervened distributions. See Appendix B.3, and especially (22), for a more detailed account.
+
+We finally remark that while recent work based on the ICM principle has mostly used the term “mechanism” to refer to causal Markov kernels $p ( X _ { i } | P A _ { i } )$ or structural equations [78], we employ it in line with the broader use of this concept in the philosophical literature.11 To highlight just two examples, [86] states that “Causal processes, causal interactions, and causal laws provide the mechanisms by which the world works; to understand why certain things happen, we need to see how they are produced by these mechanisms”; and [99] states that “Mechanisms are events that alter relations among some specified set of elements”. Following this perspective, we argue that a causal mechanism can more generally denote any process that describes the way in which causes influence their effects: the partial derivative ${ \partial \mathbf { f } } / { \partial s _ { i } }$ thus reflects a causal mechanism in the sense that it describes the infinitesimal changes in the observations $\mathbf { x }$ , when an infinitesimal perturbation is applied to $s _ { i }$ .
+
+# 4.2 Definition and useful properties of the IMA contrast
+
+We now introduce a contrast function based on the IMA principle (7) and show that it possesses several desirable properties in the context of nonlinear ICA. First, we define a local contrast as the difference between the two integrands of (7) for a particular value of the sources s.
+
+Definition 4.2 (Local IMA contrast). The local IMA contrast $c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } )$ of $\mathbf { f }$ at a point s is given by
+
+$$
+c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } ) = \sum _ { i = 1 } ^ { n } \log \left\| \frac { \partial \mathbf { f } } { \partial s _ { i } } ( \mathbf { s } ) \right\| - \log \left| \mathbf { J } _ { \mathbf { f } } ( \mathbf { s } ) \right| .
+$$
+
+The local IMA contrast $c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } )$ quantifies the extent to which the IMA principle is violated at a given point s. We summarise some of its properties in the following proposition.
+
+Proposition 4.4 (Properties of $C _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } ) )$ . The local IMA contrast $c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } )$ defined in (8) satisfies:
+
+(i) $c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } ) \geq 0$ , with equality if and only if all columns $\partial \mathbf { f } / \partial s _ { i } ( \mathbf { s } )$ of $\mathbf { J } _ { \mathbf { f } } ( \mathbf { s } )$ are orthogonal. (ii) $c _ { \mathrm { I M A } } ( \mathbf { f } , \mathbf { s } )$ is invariant to left multiplication of $\mathbf { J } _ { \mathbf { f } } ( \mathbf { s } )$ by an orthogonal matrix and to right multiplication by permutation and diagonal matrices.
+
+Property $( i )$ formalises the geometric interpretation of IMA as an orthogonality condition on the columns of the Jacobian from $\ S$ , and property $( i i )$ intuitively states that changes of orthonormal basis and permutations or rescalings of the columns of $\mathbf { J _ { f } }$ do not affect their orthogonality. Next, we define a global IMA contrast w.r.t. a source distribution $p _ { \mathbf { s } }$ as the expected local IMA contrast.
+
+Definition 4.5 (Global IMA contrast). The global IMA contrast $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } )$ of f w.r.t. $p _ { \mathbf { s } }$ is given by
+
+$$
+C _ { \mathrm { I M A } } ( \bf f , \it p _ { \mathrm { s } } ) = \mathbb { E } _ { { \bf s } \sim \it { p } _ { \mathrm { s } } } [ c _ { \mathrm { I M A } } ( \bf f , \bf s ) ] = \int c _ { \mathrm { I M A } } ( \bf f , \bf s ) \it { p } _ { \bf s } ( \bf s ) d \bf s .
+$$
+
+The global IMA contrast $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } )$ thus quantifies the extent to which the IMA principle is violated for a particular solution $( \mathbf { f } , p _ { \mathbf { s } } )$ to the nonlinear ICA problem. We summarise its properties as follows.
+
+Proposition 4.6 (Properties of $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } ) )$ . The global IMA contrast $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } )$ from (9) satisfies:
+
+(i) $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } ) ~ \geq ~ 0 ,$ , with equality iff. $\mathbf { J _ { f } } ( \mathbf { s } ) \ = \ \mathbf { O } ( \mathbf { s } ) \mathbf { D } ( \mathbf { s } )$ almost surely w.r.t. $p _ { \mathbf { s } }$ , where $\mathbf { O } ( \mathbf { s } ) , \mathbf { D } ( \mathbf { s } ) \in \mathbb { R } ^ { n \times n }$ are orthogonal and diagonal matrices, respectively;
+
+(ii) $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { S } } ) = C _ { \mathrm { I M A } } ( \tilde { \mathbf { f } } , p _ { \tilde { \mathbf { S } } } )$ for any $\tilde { \mathbf { f } } = \mathbf { f } \circ \mathbf { h } ^ { - 1 } \circ \mathbf { P } ^ { - 1 }$ and $\tilde { \mathbf { s } } = \mathbf { P } \mathbf { h } ( \mathbf { s } )$ , where $\mathbf { P } \in \mathbb { R } ^ { n \times n }$ is $a$ permutation and $\mathbf { h } ( \mathbf { s } ) = ( h _ { 1 } ( s _ { 1 } ) , . . . , h _ { n } ( s _ { n } ) )$ an invertible element-wise function.
+
+Property $( i )$ is the distribution-level analogue to $( i )$ of Prop. 4.4 and only allows for orthogonality violations on sets of measure zero w.r.t. $p _ { \mathbf { s } }$ . This means that $C _ { \mathrm { I M A } }$ can only be zero if f is an orthogonal coordinate transformation almost everywhere [19, 52, 66], see Fig. 3 for an example. We particularly stress property $( i i )$ , as it precisely matches the inherent indeterminacy of nonlinear ICA: $C _ { \mathrm { I M A } }$ is blind to reparametrisation of the sources by permutation and element wise transformation.
+
+![](images/7e5c662ea9f2328e248ab4799df8a6025568b40021ec76b0d73d8f2e27cdc54d.jpg)  
+Figure 3: An example of a (non-conformal) orthogonal coordinate transformation from polar (left) to Cartesian (right) coordinates.
+
+# 4.3 Theoretical analysis and justification of $C _ { \mathbf { I M A } }$
+
+We now show that, under suitable assumptions on the generative model (1), a large class of spurious solutions—such as those based on the Darmois construction (4) or measure preserving automorphisms such as ${ \bf a } ^ { \bf R }$ from (5) as described in $\ S \ : 2 . 1$ —exhibit nonzero IMA contrast. Denote the class of nonlinear ICA models satisfying (7) (IMA) by $\mathcal { M } _ { \mathrm { I M A } } = \{ ( \mathbf { f } , p _ { \mathrm { s } } ) \in \mathcal { F } \times \mathcal { P } : C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathrm { s } } ) = 0 \} \subset \mathcal { F } \times \mathcal { P }$ . Our first main theoretical result is that, under mild assumptions on the observations, Darmois solutions will have strictly positive $C _ { \mathrm { I M A } }$ , making them distinguishable from those in $\mathcal { M } _ { \mathrm { { I M A } } }$ .
+
+Theorem 4.7. Assume the data generating process in (1) and assume that $x _ { i }$ ̸⊥⊥ $x _ { j }$ for some $i \neq j$ Then any Darmois solution $( \mathbf { f } ^ { D } , p _ { \mathbf { u } } )$ based on $\mathbf { g } ^ { D }$ as defined in (4) satisfies $C _ { \mathrm { I M A } } ( \mathbf { f } ^ { D } , p _ { \mathbf { u } } ) > 0$ . Thus $a$ solution satisfying $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } ) = 0$ can be distinguished from $( \mathbf { f } ^ { D } , p _ { \mathbf { u } } )$ based on the contrast $C _ { \mathrm { I M A } }$ .
+
+The proof is based on the fact that the Jacobian of $\mathbf { g } ^ { \mathrm { { D } } }$ is triangular (see Remark 2.4) and on the specific form of (4). A specific example of a mixing process satisfying the IMA assumption is the case where f is a conformal (angle-preserving) map.
+
+Definition 4.8 (Conformal map). A smooth map f : $\mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ is conformal if $\mathbf { J } _ { \mathbf { f } } ( \mathbf { s } ) = \mathbf { O } ( \mathbf { s } ) \lambda ( \mathbf { s } ) \forall \mathbf { s } ,$ where $\lambda : \mathbb { R } ^ { n } \to \mathbb { R }$ is a scalar field, and $\mathbf { O } \in O ( n )$ is an orthogonal matrix.
+
+Corollary 4.9. Under assumptions of Thm. 4.7, if additionally f is a conformal map, then $( \mathbf { f } , p _ { \mathbf { s } } ) \in$ $\mathcal { M } _ { \mathrm { I M A } }$ for any $p _ { \mathbf { s } } ~ \in ~ \mathcal { P }$ due to Prop. $\mathbf { \rho } ( i )$ , see Defn. 4.8. Based on Thm. 4.7, $( \mathbf { f } , p _ { \mathbf { s } } )$ is thus distinguishable from Darmois solutions $( \mathbf { f } ^ { D } , p _ { \mathbf { u } } )$ .
+
+This is consistent with a result that proves identifiability of conformal maps for $n = 2$ and conjectures it in general [39].12 However, conformal maps are only a small subset of all maps for which $C _ { \mathrm { I M A } } = 0$ , as is apparent from the more flexible condition of Prop. $\mathbf { \chi } ( i )$ , compared to the stricter Defn. 4.8.
+
+![](images/afac4d4ed7555f228adb2e08b439d2120f52127d1a2414aac52956bb6944de6c.jpg)  
+Figure 4: Top. Visual comparison of different nonlinear ICA solutions for $n = 2$ : (left to right) true sources; observed mixtures; Darmois solution; true unmixing, composed with the measure preserving automorphism (MPA) from (5) (with rotation by $\pi / 4$ ); Darmois solution composed with the same MPA; maximum likelihood $\lambda = 0$ ); and $C _ { \mathrm { I M A } }$ -regularised approach $( \lambda = 1$ ). Bottom. Quantitative comparison of $C _ { \mathrm { I M A } }$ for different spurious solutions: learnt Darmois solutions for (a) $n = 2$ , and (b) $n \in \{ 2 , 3 , 5 , \hat { 1 } 0 \}$ dimensions; (c) composition of the MPA (5) in $n = 2$ dim. with the true solution (blue) and a Darmois solution (red) for different angles. (d) $C _ { \mathrm { I M A } }$ distribution for true MLP mixing (red) vs. Darmois solution (blue) for $n = 5$ dim., $L \in \{ 2 , 3 , 4 \}$ layers.
+
+Example 4.10 (Polar to Cartesian coordinate transform). Consider the non-conformal transformation from polar to Cartesian coordinates (see Fig. 3), defined as $( x , y ) = \mathbf { f } ( r , \theta ) : = ( r \cos ( \theta ) , r \sin ( \theta ) )$ with independent sources $\mathbf { s } = ( r , \theta )$ , with $r \sim U ( 0 , R )$ and $\theta \sim U ( 0 , 2 \pi )$ .13 Then, $C _ { \mathrm { I M A } } ( \mathbf { f } , p _ { \mathbf { s } } ) = 0$ and $C _ { \mathrm { I M A } } \bar { ( } \mathbf { f } ^ { \mathrm { D } } , p _ { \mathrm { u } } ) > 0$ for any Darmois solution $( \mathbf { f } ^ { \mathrm { D } } , p _ { \mathbf { u } } )$ —see Appendix D for details.
+
+Finally, for the case in which the true mixing is linear, we obtain the following result.
+
+Corollary 4.11. Consider a linear ICA model, $\mathbf { x } = \mathbf { A } \mathbf { s }$ , with $\mathbb { E } [ \mathbf { s } ^ { \top } \mathbf { s } ] = \mathbf { I } ,$ , and $\mathbf { A } \in O ( n )$ an orthogonal, non-trivial mixing matrix, i.e., not the product of a diagonal and a permutation matrix DP. If at most one of the $s _ { i }$ is Gaussian, then $C _ { \mathrm { I M A } } ( \mathbf { A } , p _ { \mathbf { s } } ) = 0$ and $C _ { \mathrm { I M A } } ( \mathbf { f } ^ { D } , p _ { \mathbf { u } } ^ { \top } ) > 0$ .
+
+In a “blind” setting, we may not know a priori whether the true mixing is linear or not, and thus choose to learn a nonlinear unmixing. Corollary 4.11 shows that, in this case, Darmois solutions are still distinguishable from the true mixing via $C _ { \mathrm { I M A } }$ . Note that unlike in Corollary 4.9, the assumption that $x _ { i } \not \perp x _ { j }$ for some $i \neq j$ is not required for Corollary 4.11. In fact, due to Theorem 11 of [17], it follows from the assumed linear ICA model with non-Gaussian sources, and the fact that the mixing matrix is not the product of a diagonal and a permutation matrix (see also Appendix A).
+
+Having shown that the IMA principle allows to distinguish a class of models (including, but not limited to conformal maps) from Darmois solutions, we next turn to a second well-known counterexample to identifiability: the “rotated-Gaussian” MPA $\mathbf { a ^ { R } } ( p _ { \mathbf { s } } )$ (5) from Defn. 2.5. Our second main theoretical result is that, under suitable assumptions, this class of MPAs can also be ruled out for “non-trivial” $\mathbf { R }$ .
+
+Theorem 4.12. Let $( \mathbf { f } , p _ { \mathbf { s } } ) \in \mathcal { M } _ { \mathrm { I M A } }$ and assume that f is a conformal map. Given $\mathbf { R } \in O ( n )$ , assume additionally that $\exists$ at least one non-Gaussian $s _ { i }$ whose associated canonical basis vector $\mathbf { e } _ { i }$ is not transformed by $\mathbf { R } ^ { - 1 } = \mathbf { R } ^ { \top }$ into another canonical basis vector $\mathbf { e } _ { j }$ . Then $C _ { \mathrm { I M A } } ( { \bf f } \circ { \bf a } ^ { \bf R } ( p _ { \bf s } ) , p _ { \bf s } ) > 0$
+
+Thm. 4.12 states that for conformal maps, applying the $\mathbf { a ^ { R } } ( p _ { \mathbf { s } } )$ transformation at the level of the sources leads to an increase in $C _ { \mathrm { I M A } }$ except for very specific rotations $\mathbf { R }$ that are “fine-tuned” to $p _ { \mathbf { s } }$ in the sense that they permute all non-Gaussian sources $s _ { i }$ with another $s _ { j }$ . Interestingly, as for the linear case, non-Gaussianity again plays an important role in the proof of Thm. 4.12.
+
+# 5 Experiments
+
+Our theoretical results from $\ S \ 4$ suggest that $C _ { \mathrm { I M A } }$ is a promising contrast function for nonlinear blind source separation. We test this empirically by evaluating the $C _ { \mathrm { I M A } }$ of spurious nonlinear ICA solutions $\mathrm { ~ ( ~ \ S ~ } \mathrm { ~ ) ~ }$ , and using it as a learning objective to recover the true solution $( \ S 5 . 2 )$ .
+
+We sample the ground truth sources from a uniform distribution in $[ 0 , 1 ] ^ { n }$ ; the reconstructed sources are also mapped to the uniform hypercube as a reference measure via the CDF transform. Unless otherwise specified, the ground truth mixing f is a Möbius transformation [81] (i.e., a conformal map) with randomly sampled parameters, thereby satisfying Principle 4.1. In all of our experiments, we use JAX [12] and Distrax [13]. For additional technical details, equations and plots see Appendix E. The code to reproduce our experiments is available at this link.
+
+# 5.1 Numerical evaluation of the $C _ { \mathbf { I M A } }$ contrast for spurious nonlinear ICA solutions
+
+Learning the Darmois construction. To learn the Darmois construction from data, we use normalising flows, see [35, 69]. Since Darmois solutions have triangular Jacobian (Remark 2.4), we use an architecture based on residual flows [16] which we constrain such that the Jacobian of the full model is triangular. This yields an expressive model which we train effectively via maximum likelihood.
+
+$C _ { \mathbf { I M A } }$ of Darmois solutions. To check whether Darmois solutions (learnt from finite data) can be distinguished from the true one, as predicted by Thm. 4.7, we generate 1000 random mixing functions for $n = 2$ , compute the $C _ { \mathrm { I M A } }$ values of learnt solutions, and find that all values are indeed significantly larger than zero, see Fig. 4 (a). The same holds for higher dimensions, see Fig. 4 (b) for results with 50 random mixings for $n \in \{ 2 , 3 , 5 , 1 0 \}$ : with higher dimensionality, both the mean and variance of the $C _ { \mathrm { I M A } }$ distribution for the learnt Darmois solutions generally attain higher values.14 We confirmed these findings for mappings which are not conformal, while still satisfying (7), in Appendix E.5.
+
+$C _ { \mathbf { I M A } }$ of MPAs. We also investigate the effect on $C _ { \mathrm { I M A } }$ of applying an MPA ${ \bf a } ^ { \bf R } ( \cdot )$ from (5) to the true solution or a learnt Darmois solution. Results for $n = 2$ dim. for different rotation matrices $\mathbf { R }$ (parametrised by the angle $\theta$ ) are shown in Fig. 4 (c). As expected, the behavior is periodic in $\theta$ , and vanishes for the true solution (blue) at multiples of $\pi / 2$ , i.e., when $\mathbf { R }$ is a permutation matrix, as predicted by Thm. 4.12. For the learnt Darmois solution (red, dashed) $C _ { \mathrm { I M A } }$ remains larger than zero.
+
+$C _ { \mathbf { I M A } }$ values for random MLPs. Lastly, we study the behavior of spurious solutions based on the Darmois construction under deviations from our assumption of $C _ { \mathrm { I M A } } = 0$ for the true mixing function. To this end, we use invertible MLPs with orthogonal weight initalisation and leaky_tanh activations [29] as mixing functions; the more layers $L$ are added to the mixing MLP, the larger a deviation from our assumptions is expected. We compare the true mixing and learnt Darmois solutions over 20 realisations for each $\bar { L } \in \{ 2 , 3 , 4 \}$ , $n = 5$ . Results are shown in figure Fig. 4 (d): the $C _ { \mathrm { I M A } }$ of the mixing MLPs grows with $L$ ; still, the one of the Darmois solution is typically higher.
+
+Summary. We verify that spurious solutions can be distinguished from the true one based on $C _ { \mathrm { I M A } }$ .
+
+# 5.2 Learning nonlinear ICA solutions with $C _ { \mathbf { I M A } }$ -regularised maximum likelihood
+
+Experimental setup. To use $C _ { \mathrm { I M A } }$ as a learning signal, we consider a regularised maximum-likelihood approach, with the following objective: $\mathcal { L } ( \mathbf { g } ) \bar { = } \mathbb { E } _ { \mathbf { x } } [ \log p _ { \mathbf { g } } ( \mathbf { x } ) ] - \lambda C _ { \mathrm { I M A } } ^ { \bar { - } } ( \mathbf { g } ^ { - 1 } , p _ { \mathbf { y } } )$ , where $\mathbf { g }$ denotes the learnt unmixing, $\mathbf { y } = \mathbf { g } ( \mathbf { x } )$ the reconstructed sources, and $\lambda \geq 0$ a Lagrange multiplier. For $\lambda = 0$ , this corresponds to standard maximum likelihood estimation, whereas for $\lambda > 0$ , $\mathcal { L }$ lower-bounds the likelihood, and recovers it exactly iff. $( \mathbf { g } ^ { - 1 } , p _ { \mathbf { y } } ) \in \mathcal { M } _ { \mathrm { I M A } }$ . We train a residual flow $\mathbf { g }$ (with full Jacobian) to maximise $\mathcal { L }$ . For evaluation, we compute (i) the KL divergence to the true data likelihood, as a measure of goodness of fit for the learnt flow model; and (ii) the mean correlation coefficient (MCC) between ground truth and reconstructed sources [37, 49]. We also introduce (iii) a nonlinear extension of the Amari distance [5] between the true mixing and the learnt unmixing, which is larger than or equal to zero, with equality iff. the learnt model belongs to the BSS equivalence class (Defn. 2.2) of the true solution, see Appendix E.5 for details.
+
+Results. In Fig. 4 (Top), we show an example of the distortion induced by different spurious solutions for $n = 2$ , and contrast it with a solution learnt using our proposed objective (rightmost plot). Visually, we find that the $C _ { \mathrm { I M A } }$ -regularised solution (with $\lambda = 1$ ) recovers the true sources most faithfully. Quantitative results for 50 learnt models for each $\lambda \in \{ 0 . 0 , 0 . 5 , 1 . 0 \}$ and $n \in \{ 5 , 7 \}$ are summarised in Fig. 5 (see Appendix E for additional plots) . As indicated by the KL divergence values (left), most trained models achieve a good fit to the data across all values of $\lambda$ .15 We observe that using $C _ { \mathrm { I M A } }$ (i.e., $\lambda > 0$ ) is beneficial for BSS, both in terms of our nonlinear Amari distance (center, lower is better) and MCC (right, higher is better), though we do not observe a substantial difference between λ = 0.5 and λ = 1.16
+
+Summary: $C _ { \mathrm { I M A } }$ can be a useful learning signal to recover the true solution.
+
+![](images/9262f36ef530a906eac685329302929beb18aa9c600ac76af282fc4030558598.jpg)  
+Figure 5: BSS via $C _ { \mathrm { I M A } }$ -regularised MLE for, side by side, $n = 5$ (blue) and $n = 7$ (red) dim. with $\lambda \in \{ 0 . 0 , 0 . 5 , 1 . 0 \}$ . (Left) KL-divergence between ground truth likelihood and learnt model; (center) nonlinear Amari distance given true mixing and learnt unmixing; (right) MCC between true and reconstructed sources.
+
+# 6 Discussion
+
+Assumptions on the mixing function. Instead of relying on weak supervision in the form of auxiliary variables [28, 30, 37, 38, 41, 49], our IMA approach places additional constraints on the functional form of the mixing process. In a similar vein, the minimal nonlinear distortion principle [108] proposes to favor solutions that are as close to linear as possible. Another example is the post-nonlinear model [98, 109], which assumes an element-wise nonlinearity applied after a linear mixing. IMA is different in that it still allows for strongly nonlinear mixings (see, e.g., Fig. 3) provided that the columns of their Jacobians are (close to) orthogonal. In the related field of disentanglement [8, 58], a line of work that focuses on image generation with adversarial networks [24] similarly proposes to constrain the “generator” function via regularisation of its Jacobian [82] or Hessian [74], though mostly from an empirically-driven, rather than from an identifiability perspective as in the present work.
+
+Towards identifiability with $C _ { \mathbf { I M A } }$ . The IMA principle rules out a large class of spurious solutions to nonlinear ICA. While we do not present a full identifiability result, our experiments show that $C _ { \mathrm { I M A } }$ can be used to recover the BSS equivalence class, suggesting that identifiability might indeed hold, possibly under additional assumptions—e.g., for conformal maps [39].
+
+IMA and independence of cause and mechanism. While inspired by measures of independence of cause and mechanism as traditionally used for cause-effect inference [18, 45, 46, 110], we view the IMA principle as addressing a different question, in the sense that they evaluate independence between different elements of the causal model. Any nonlinear ICA solution that satisfies the IMA Principle 4.1 can be turned into one with uniform reconstructed sources—thus satisfying IGCI as argued in $\ S \ 3 \cdot$ — through composition with an element-wise transformation which, according to Prop. 4.6 (ii), leaves the $C _ { \mathrm { I M A } }$ value unchanged. Both IGCI (6) and IMA (7) can therefore be fulfilled simultaneosly, while the former on its own is inconsequential for BSS as shown in Prop. 3.1.
+
+BSS through algorithmic information. Algorithmic information theory has previously been proposed as a unifying framework for identifiable approaches to linear BSS [67, 68], in the sense that commonly-used contrast functions could, under suitable assumptions, be interpreted as proxies for the total complexity of the mixing and the reconstructed sources. However, to the best of our knowledge, the problem of specifying suitable proxies for the complexity of nonlinear mixing functions has not yet been addressed. We conjecture that our framework could be linked to this view, based on the additional assumption of algorithmic independence of causal mechanisms [43], thus potentially representing an approach to nonlinear BSS by minimisation of algorithmic complexity.
+
+ICA for causal inference & causality for ICA. Past advances in ICA have inspired novel causal discovery methods [50, 64, 92]. The present work constitutes, to the best of our knowledge, the first effort to use ideas from causality (specifically ICM) for BSS. An application of the IMA principle to causal discovery or causal representation learning [88] is an interesting direction for future work.
+
+Conclusion. We introduce IMA, a path to nonlinear BSS inspired by concepts from causality. We postulate that the influences of different sources on the observed distribution should be approximately independent, and formalise this as an orthogonality condition on the columns of the Jacobian. We prove that this constraint is generally violated by well-known spurious nonlinear ICA solutions, and propose a regularised maximum likelihood approach which we empirically demonstrate to be effective in recovering the true solution. Our IMA principle holds exactly for orthogonal coordinate transformations, and is thus of potential interest for learning spatial representations [33], robot dynamics [63], or physics problems where orthogonal reference frames are common [66].
+
+# Acknowledgements
+
+The authors thank Aapo Hyvärinen, Adrián Javaloy Bornás, Dominik Janzing, Giambattista Parascandolo, Giancarlo Fissore, Nasim Rahaman, Patrick Burauel, Patrik Reizinger, Paul Rubenstein, Shubhangi Ghosh, and the anonymous reviewers for helpful comments and discussions.
+
+# Funding Transparency Statement
+
+This work was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039B; and by the Machine Learning Cluster of Excellence, EXC number 2064/1 - Project number 390727645.
+
+# References
+
+[1] Pierre Ablin, Jean-François Cardoso, and Alexandre Gramfort. Faster independent component analysis by preconditioning with hessian approximations. IEEE Transactions on Signal Processing, 66(15): 4040–4049, 2018.   
+[2] Khaled Alyani, Marco Congedo, and Maher Moakher. Diagonality measures of Hermitian positivedefinite matrices with application to the approximate joint diagonalization problem. Linear Algebra and its Applications, 528:290–320, 2017.   
+[3] Shun-ichi Amari. Information geometry. Japanese Journal of Mathematics, 16(1):1–48, 2021.   
+[4] Shun-ichi Amari and Hiroshi Nagaoka. Methods of information geometry, volume 191. American Mathematical Soc., 2007.   
+[5] Shun-ichi Amari, Andrzej Cichocki, Howard Hua Yang, et al. A new learning algorithm for blind signal separation. In Advances in Neural Information Processing Systems, pages 757–763, 1996.   
+[6] Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019.   
+[7] Elias Bareinboim, Andrew Forney, and Judea Pearl. Bandits with unobserved confounders: A causal approach. Advances in Neural Information Processing Systems, 28:1342–1350, 2015.   
+[8] Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1798–1828, 2013.   
+[9] M Besserve, N Shajarisales, B Schölkopf, and D Janzing. Group invariance principles for causal generative models. In 21st International Conference on Artificial Intelligence and Statistics (AISTATS 2018), pages 557–565. International Machine Learning Society, 2018.   
+[10] M Besserve, A Mehrjou, R Sun, and B Schölkopf. Counterfactuals uncover the modular structure of deep generative models. In Eighth International Conference on Learning Representations (ICLR 2020), 2020.   
+[11] Patrick Blöbaum, Dominik Janzing, Takashi Washio, Shohei Shimizu, and Bernhard Schölkopf. Causeeffect inference by comparing regression errors. In International Conference on Artificial Intelligence and Statistics, pages 900–909. PMLR, 2018.   
+[12] James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, and Skye Wanderman-Milne. JAX: composable transformations of Python+NumPy programs, 2018.   
+[13] Jake Bruce, David Budden, Matteo Hessel, George Papamakarios, and Francisco Ruiz. Distrax: Probability distributions in JAX, 2021.   
+[14] Lars Buesing, Theophane Weber, Yori Zwols, Sebastien Racaniere, Arthur Guez, Jean-Baptiste Lespiau, and Nicolas Heess. Woulda, coulda, shoulda: Counterfactually-guided policy search. arXiv preprint arXiv:1811.06272, 2018.   
+[15] Jean-François Cardoso. The three easy routes to independent component analysis; contrasts and geometry. In Proc. ICA, volume 2001, 2001.   
+[16] Ricky T. Q. Chen, Jens Behrmann, David Duvenaud, and Jörn-Henrik Jacobsen. Residual flows for invertible generative modeling. In Advances in Neural Information Processing Systems, 2019.   
+[17] Pierre Comon. Independent component analysis, a new concept? Signal processing, 36(3):287–314, 1994.   
+[18] Povilas Daniušis, Dominik Janzing, Joris Mooij, Jakob Zscheischler, Bastian Steudel, Kun Zhang, and Bernhard Schölkopf. Inferring deterministic causal relations. In Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence, pages 143–150, 2010.   
+[19] Gaston Darboux. Leçons sur les systemes orthogonaux et les coordonnées curvilignes. Gauthier-Villars, 1910.   
+[20] George Darmois. Analyse des liaisons de probabilité. In Proc. Int. Stat. Conferences 1947, page 231, 1951.   
+[21] George Darmois. Analyse générale des liaisons stochastiques: etude particulière de l’analyse factorielle linéaire. Revue de l’Institut international de statistique, pages 2–8, 1953.   
+[22] Andrew Forney, Judea Pearl, and Elias Bareinboim. Counterfactual data-fusion for online reinforcement learners. In International Conference on Machine Learning, pages 1156–1164. PMLR, 2017.   
+[23] Mingming Gong, Kun Zhang, Tongliang Liu, Dacheng Tao, Clark Glymour, and Bernhard Schölkopf. Domain adaptation with conditional transferable components. In International Conference on Machine Learning, pages 2839–2848. PMLR, 2016.   
+[24] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pages 2672–2680, 2014.   
+[25] Alexander N Gorban and Ivan Yu Tyukin. Blessing of dimensionality: mathematical foundations of the statistical physics of data. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2118):20170237, 2018.   
+[26] Anirudh Goyal, Alex Lamb, Jordan Hoffmann, Shagun Sodhani, Sergey Levine, Yoshua Bengio, and Bernhard Schölkopf. Recurrent independent mechanisms. In 9th International Conference on Learning Representations, 2021.   
+[27] Daniel Greenfeld and Uri Shalit. Robust learning with the hilbert-schmidt independence criterion. In International Conference on Machine Learning, pages 3759–3768. PMLR, 2020.   
+[28] Luigi Gresele, Paul K Rubenstein, Arash Mehrjou, Francesco Locatello, and Bernhard Schölkopf. The Incomplete Rosetta Stone problem: Identifiability results for multi-view nonlinear ICA. In Uncertainty in Artificial Intelligence, pages 217–227. PMLR, 2019.   
+[29] Luigi Gresele, Giancarlo Fissore, Adrián Javaloy, Bernhard Schölkopf, and Aapo Hyvarinen. Relative gradient optimization of the Jacobian term in unsupervised deep learning. Advances in Neural Information Processing Systems, 33, 2020.   
+[30] Hermanni Hälvä and Aapo Hyvärinen. Hidden Markov nonlinear ICA: Unsupervised learning from nonstationary time series. In Conference on Uncertainty in Artificial Intelligence, pages 939–948. PMLR, 2020.   
+[31] Christina Heinze-Deml and Nicolai Meinshausen. Conditional variance penalties and domain shift robustness. Machine Learning, 110(2):303–348, 2021.   
+[32] Tom Hennigan, Trevor Cai, Tamara Norman, and Igor Babuschkin. Haiku: Sonnet for JAX, 2020.   
+[33] Geoffrey E Hinton and Lawrence M Parsons. Frames of reference and mental imagery. Attention and performance IX, pages 261–277, 1981.   
+[34] Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. Advances in Neural Information Processing Systems, 21:689–696, 2008.   
+[35] Chin-Wei Huang, David Krueger, Alexandre Lacoste, and Aaron C. Courville. Neural autoregressive flows. In Jennifer G. Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80, pages 2083–2092. PMLR, 2018.   
+[36] Aapo Hyvärinen. Fast and robust fixed-point algorithms for independent component analysis. IEEE transactions on Neural Networks, 10(3):626–634, 1999.   
+[37] Aapo Hyvärinen and Hiroshi Morioka. Unsupervised feature extraction by time-contrastive learning and nonlinear ICA. In Advances in Neural Information Processing Systems, pages 3765–3773, 2016.   
+[38] Aapo Hyvärinen and Hiroshi Morioka. Nonlinear ICA of temporally dependent stationary sources. In Artificial Intelligence and Statistics, pages 460–469. PMLR, 2017.   
+[39] Aapo Hyvärinen and Petteri Pajunen. Nonlinear independent component analysis: Existence and uniqueness results. Neural Networks, 12(3):429–439, 1999.   
+[40] Aapo Hyvärinen, Juha Karhunen, and Erkki Oja. Independent Component Analysis. John Wiley & Sons, Ltd, 2001.   
+[41] Aapo Hyvärinen, Hiroaki Sasaki, and Richard Turner. Nonlinear ICA using auxiliary variables and generalized contrastive learning. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 859–868. PMLR, 2019.   
+[42] Dominik Janzing. Causal version of principle of insufficient reason and maxent. arXiv preprint arXiv:2102.03906, 2021.   
+[43] Dominik Janzing and Bernhard Schölkopf. Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory, 56(10):5168–5194, 2010.   
+[44] Dominik Janzing and Bernhard Schölkopf. Detecting confounding in multivariate linear models via spectral analysis. Journal of Causal Inference, 6(1), 2018.   
+[45] Dominik Janzing, Patrik O Hoyer, and Bernhard Schölkopf. Telling cause from effect based on highdimensional observations. In International Conference on Machine Learning, 2010.   
+[46] Dominik Janzing, Joris Mooij, Kun Zhang, Jan Lemeire, Jakob Zscheischler, Povilas Daniušis, Bastian Steudel, and Bernhard Schölkopf. Information-geometric approach to inferring causal directions. Artificial Intelligence, 182:1–31, 2012.   
+[47] Dominik Janzing, Bastian Steudel, Naji Shajarisales, and Bernhard Schölkopf. Justifying informationgeometric causal inference. In Measures of complexity, pages 253–265. Springer, 2015.   
+[48] Dominik Janzing, Raphael Chaves, and Bernhard Schölkopf. Algorithmic independence of initial condition and dynamical law in thermodynamics and causal inference. New Journal of Physics, 18(9): 093052, 2016.   
+[49] Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvärinen. Variational autoencoders and nonlinear ICA: A unifying framework. In International Conference on Artificial Intelligence and Statistics, pages 2207–2217. PMLR, 2020.   
+[50] Ilyes Khemakhem, Ricardo Monti, Robert Leech, and Aapo Hyvarinen. Causal autoregressive flows. In International Conference on Artificial Intelligence and Statistics, pages 3520–3528. PMLR, 2021.   
+[51] Andrei N Kolmogorov. On tables of random numbers. Sankhya: The Indian Journal of Statistics, Series ¯ A, pages 369–376, 1963.   
+[52] Gabriel Lamé. Leçons sur les coordonnées curvilignes et leurs diverses applications. Mallet-Bachelier, 1859.   
+[53] Sanghack Lee and Elias Bareinboim. Structural causal bandits: where to intervene? Advances in Neural Information Processing Systems 31, 31, 2018.   
+[54] Felix Leeb, Yashas Annadani, Stefan Bauer, and Bernhard Schölkopf. Structural autoencoders improve representations for generation and transfer. arXiv preprint arXiv:2006.07796, 2020.   
+[55] Erich L Lehmann and George Casella. Theory of point estimation. Springer Science & Business Media, 2006.   
+[56] Jan Lemeire and Erik Dirkx. Causal models as minimal descriptions of multivariate systems, 2006.   
+[57] Jan Lemeire and Dominik Janzing. Replacing causal faithfulness with algorithmic independence of conditionals. Minds and Machines, 23(2):227–249, 2013.   
+[58] Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentangled representations. In International Conference on Machine Learning, pages 4114–4124. PMLR, 2019.   
+[59] Chaochao Lu, Bernhard Schölkopf, and José Miguel Hernández-Lobato. Deconfounding reinforcement learning in observational settings. arXiv preprint arXiv:1812.10576, 2018.   
+[60] Chaochao Lu, Biwei Huang, Ke Wang, José Miguel Hernández-Lobato, Kun Zhang, and Bernhard Schölkopf. Sample-efficient reinforcement learning via counterfactual-based data augmentation. arXiv preprint arXiv:2012.09092, 2020.   
+[61] Sara Magliacane, Thijs van Ommen, Tom Claassen, Stephan Bongers, Philip Versteeg, and Joris M Mooij. Domain adaptation by using causal inference to predict invariant conditional distributions. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 10869–10879, 2018.   
+[62] James Mahoney. Beyond correlational analysis: Recent innovations in theory and method. In Sociological forum, pages 575–593. JSTOR, 2001.   
+[63] Michael Mistry, Jonas Buchli, and Stefan Schaal. Inverse dynamics control of floating base systems using orthogonal decomposition. In 2010 IEEE International Conference on Robotics and Automation, pages 3406–3412. IEEE, 2010.   
+[64] Ricardo Pio Monti, Kun Zhang, and Aapo Hyvärinen. Causal discovery with general non-linear relationships using non-linear ICA. In Uncertainty in Artificial Intelligence, pages 186–195. PMLR, 2020.   
+[65] Joris M Mooij, Jonas Peters, Dominik Janzing, Jakob Zscheischler, and Bernhard Schölkopf. Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research, 17(1):1103–1204, 2016.   
+[66] Parry Moon and Domina Eberle Spencer. Field theory handbook, including coordinate systems, differential equations and their solutions. Springer, 1971.   
+[67] Petteri Pajunen. Blind source separation using algorithmic information theory. Neurocomputing, 22(1-3): 35–48, 1998.   
+[68] Petteri Pajunen. Blind source separation of natural signals based on approximate complexity minimization. In Proc. Int. Workshop on Independent Component Analysis and Signal Separation (ICA’99), page 267. Citeseer, 1999.   
+[69] George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57):1–64, 2021.   
+[70] Giambattista Parascandolo, Niki Kilbertus, Mateo Rojas-Carulla, and Bernhard Schölkopf. Learning independent causal mechanisms. In International Conference on Machine Learning, pages 4036–4044. PMLR, 2018.   
+[71] Judea Pearl. Causality. Cambridge university press, 2009.   
+[72] Judea Pearl and Elias Bareinboim. External validity: From do-calculus to transportability across populations. Statistical Science, 29(4):579–595, 2014.   
+[73] Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikit-learn: Machine learning in python. The Journal of Machine Learning Research, 12:2825–2830, 2011.   
+[74] William S. Peebles, John Peebles, Jun-Yan Zhu, Alexei A. Efros, and Antonio Torralba. The Hessian penalty: A weak prior for unsupervised disentanglement. In ECCV, volume 12351, pages 581–597. Springer, 2020.   
+[75] Jonas Peters and Peter Bühlmann. Identifiability of Gaussian structural equation models with equal error variances. Biometrika, 101(1):219–228, 2014.   
+[76] Jonas Peters, Joris M Mooij, Dominik Janzing, and Bernhard Schölkopf. Causal discovery with continuous additive noise models. Journal of Machine Learning Research, 15:2009–2053, 2014.   
+[77] Jonas Peters, Peter Bühlmann, and Nicolai Meinshausen. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society. Series B (Statistical Methodology), pages 947–1012, 2016.   
+[78] Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.   
+[79] Dinh-Tuan Pham and J-F Cardoso. Blind separation of instantaneous mixtures of nonstationary sources. IEEE Transactions on Signal Processing, 49(9):1837–1848, 2001.   
+[80] Dinh Tuan Pham and Philippe Garat. Blind separation of mixture of independent sources through a quasi-maximum likelihood approach. IEEE Transactions on Signal Processing, 45(7):1712–1725, 1997.   
+[81] Robert Phillips. Liouville’s theorem. Pacific Journal of Mathematics, 28(2):397–405, 1969.   
+[82] Aditya Ramesh, Youngduck Choi, and Yann LeCun. A spectral regularizer for unsupervised disentanglement. arXiv preprint arXiv:1812.01161, 2018.   
+[83] Danilo Jimenez Rezende. Short notes on divergence measures, 2018.   
+[84] Mateo Rojas-Carulla, Bernhard Schölkopf, Richard Turner, and Jonas Peters. Invariant models for causal transfer learning. The Journal of Machine Learning Research, 19(1):1309–1342, 2018.   
+[85] Dominik Rothenhäusler, Nicolai Meinshausen, Peter Bühlmann, and Jonas Peters. Anchor regression: Heterogeneous data meet causality. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 83(2):215–246, 2021.   
+[86] Wesley C Salmon. Scientific explanation and the causal structure of the world. Princeton University Press, 2020.   
+[87] B Schölkopf, D Janzing, J Peters, E Sgouritsa, K Zhang, and J Mooij. On causal and anticausal learning. In 29th International Conference on Machine Learning (ICML 2012), pages 1255–1262. International Machine Learning Society, 2012.   
+[88] Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio. Toward causal representation learning. Proceedings of the IEEE, 2021.   
+[89] Manfred R Schroeder. Listening with two ears. Music perception, 10(3):255–280, 1993.   
+[90] Naji Shajarisales, Dominik Janzing, Bernhard Schoelkopf, and Michel Besserve. Telling cause from effect in deterministic linear dynamical systems. In International Conference on Machine Learning, pages 285–294. PMLR, 2015.   
+[91] Xinwei Shen, Furui Liu, Hanze Dong, Qing Lian, Zhitang Chen, and Tong Zhang. Disentangled generative causal representation learning. arXiv preprint arXiv:2010.02637, 2020.   
+[92] Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(10), 2006.   
+[93] VP Skitovic. On a property of a normal distribution. In ´ Doklady Akad. Nauk, 1953. [94] James R Smart. Modern geometries. Brooks/Cole Pacific Grove, CA, 1998.   
+[95] Mirjam Soeten. Conformal maps and the theorem of Liouville. PhD thesis, Faculty of Science and Engineering, 2011. [96] B. Steudel, D. Janzing, and B. Schölkopf. Causal Markov condition for submodular information measures. In A. Kalai and M. Mohri, editors, Conference on Learning Theory (COLT), pages 464–476, Madison, WI, USA, 2010. OmniPress.   
+[97] Adarsh Subbaswamy, Peter Schulam, and Suchi Saria. Preventing failures due to dataset shift: Learning predictive models that transport. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 3118–3127. PMLR, 2019. [98] Anisse Taleb and Christian Jutten. Source separation in post-nonlinear mixtures. IEEE Transactions on Signal Processing, 47(10):2807–2820, 1999. [99] Charles Tilly. Historical analysis of political processes. In Handbook of sociological theory, pages 567–588. Springer, 2001.   
+[100] Ruy Tojeiro. Liouville’s theorem revisited. Enseignement Mathematique, 53(1/2):67, 2007.   
+[101] Pauli Virtanen, Ralf Gommers, Travis E Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. Scipy 1.0: fundamental algorithms for scientific computing in python. Nature methods, 17(3):261–272, 2020.   
+[102] Julius von Kügelgen, Alexander Mey, and Marco Loog. Semi-generative modelling: Covariate-shift adaptation with cause and effect features. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 1361–1369. PMLR, 2019.   
+[103] Julius von Kügelgen, Alexander Mey, Marco Loog, and Bernhard Schölkopf. Semi-supervised learning, causality, and the conditional cluster assumption. In Conference on Uncertainty in Artificial Intelligence, pages 1–10. PMLR, 2020.   
+[104] Julius von Kügelgen, Ivan Ustyuzhaninov, Peter Gehler, Matthias Bethge, and Bernhard Schölkopf. Towards causal generative scene models via competition of experts. In ICLR 2020 Workshop on "Causal Learning for Decision Making", 2020.   
+[105] Julius von Kügelgen, Yash Sharma, Luigi Gresele, Wieland Brendel, Bernhard Schölkopf, Michel Besserve, and Francesco Locatello. Self-supervised learning with data augmentations provably isolates content from style. In Advances in Neural Information Processing Systems, 2021.   
+[106] Junzhe Zhang and Elias Bareinboim. Near-optimal reinforcement learning in dynamic treatment regimes. Advances in Neural Information Processing Systems, 32:13401–13411, 2019.   
+[107] K. Zhang, B. Schölkopf, K. Muandet, and Z. Wang. Domain adaptation under target and conditional shift. In S. Dasgupta and D. McAllester, editors, Proceedings of the 30th International Conference on Machine Learning, volume 28 of JMLR Workshop and Conference Proceedings, pages 819–827, 2013.   
+[108] Kun Zhang and Laiwan Chan. Minimal nonlinear distortion principle for nonlinear independent component analysis. Journal of Machine Learning Research, 9(Nov):2455–2487, 2008.   
+[109] Kun Zhang and Aapo Hyvärinen. On the identifiability of the post-nonlinear causal model. In Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence, pages 647–655. AUAI Press, 2009.   
+[110] Jakob Zscheischler, Dominik Janzing, and Kun Zhang. Testing whether linear equations are causal: A free probability theory approach. In 27th Conference on Uncertainty in Artificial Intelligence, pages 839–847, 2011.
+
+# Checklist
+
+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] See $\ S$ , where we discuss limitations of our theory (e.g. lines 354-357) and open questions.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work is mainly theoretical, and we believe it does not bear immediate negative societal impacts.   
+(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] We formally define the problem setting in $\ S 2$ and $\ S$ , and transparently state and discuss our assumptions in $\ S 4$ .   
+(b) Did you include complete proofs of all theoretical results? [Yes] Due to space constraints, full proofs and detailed explanations are mainly reported in appendix C and appendix D; the proof of Prop. 3.1 is given in the main text.
+
+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] The code, data, and the configuration files are included in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] All details, including hyperparameters, seed for random number generators, etc. are specified in the configuration files which will be included in the supplemental.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We visualized the distribution of the considered quantities via histograms and violin plots, see e.g. Figure 4 and Figure 5.   
+(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] They are specified appendix E.
+
+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] We use the Python libraries JAX and Distrax and cited the creators in the article.   
+(b) Did you mention the license of the assets? [Yes] Both packages have Apache License 2.0; we report this in the appendices.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Implementations of our proposed methods and metrics will be provided in the supplemental.   
+(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]
+
+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]

parse/train/r7UC-b67YkO/r7UC-b67YkO.md

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+# Rebooting ACGAN: Auxiliary Classifier GANs with Stable Training
+
+Minguk Kang Woohyeon Shim Minsu Cho Jaesik Park
+
+Pohang University of Science and Technology (POSTECH), South Korea {mgkang, wh.shim, mscho, jaesik.park}@postech.ac.kr
+
+# Abstract
+
+Conditional Generative Adversarial Networks (cGAN) generate realistic images by incorporating class information into GAN. While one of the most popular cGANs is an auxiliary classifier GAN with softmax cross-entropy loss (ACGAN), it is widely known that training ACGAN is challenging as the number of classes in the dataset increases. ACGAN also tends to generate easily classifiable samples with a lack of diversity. In this paper, we introduce two cures for ACGAN. First, we identify that gradient exploding in the classifier can cause an undesirable collapse in early training, and projecting input vectors onto a unit hypersphere can resolve the problem. Second, we propose the Data-to-Data Cross-Entropy loss (D2D-CE) to exploit relational information in the class-labeled dataset. On this foundation, we propose the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results show that ReACGAN achieves state-of-the-art generation results on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet datasets. We also verify that ReACGAN benefits from differentiable augmentations and that D2D-CE harmonizes with StyleGAN2 architecture. Model weights and a software package that provides implementations of representative cGANs and all experiments in our paper are available at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN.
+
+# 1 Introduction
+
+Generative Adversarial Networks (GAN) [1] are known for the forefront approach to generating high-fidelity images of diverse categories [2, 3, 4, 5, 6, 7, 8, 9]. Behind the sensational generation ability of GANs, there has been tremendous effort to develop adversarial objectives free from the vanishing gradient problem [10, 11, 12], regularizations for stabilizing adversarial training [11, 13, 14, 3, 15, 16, 17], and conditioning techniques to support the adversarial training using category information of the dataset [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Subsequently, the conditioning techniques have become the de facto standard for high-quality image generation. The models with the conditioning methods are called conditional Generative Adversarial Networks (cGAN), and cGANs can be divided into two groups depending on the discriminator’s conditioning way: classifier-based GANs [18, 20, 23, 25, 27] and projection-based GANs [19, 3, 4, 28].
+
+The classifier-based GANs facilitate an auxiliary classifier to generate class-specific images by penalizing the generator if the synthesized images are not consistent with the conditioned labels. ACGAN [18] has been one of the widely used classifier-based GANs for its simple design and satisfactory generation performance. While ACGAN can exploit class information by pushing and pulling classifier’s weights (proxies) against image embeddings [23], it is well known that ACGAN training is prone to collapsing at the early stage of training as the number of classes increases [19, 23, 25, 27]. In addition, the generator of ACGAN tends to generate easily classifiable images at the cost of reduced diversity [18, 19, 27]. Projection-based GANs, on the other hand, have shown cutting-edge generation results on datasets with a large number of categories. SNGAN [3],
+
+![](images/a77f02b277b4a9b1554bdeb714cab48b71615b6d03f8135fb468d91f2cc6deb5.jpg)  
+Figure 1: Schematics that depict how cGANs perform conditioning. The red color means positive samples/proxy, and the blue color indicates negative samples/proxies. Arrows represent push-pull forces based on the reference sample. The length of an arrow indicates the magnitude of the force.
+
+SAGAN [29], and BigGAN [4] are representatives in this family and can generate realistic images on CIFAR10 [30] and ImageNet [31] datasets. However, projection-based GANs only consider a pairwise relationship between an image and its label proxy (data-to-class relationships). As the result, the projection-based GANs can miss an additional opportunity to consider relation information between data instances (data-to-data relationships) as discovered by [23].
+
+In this paper, we analyze why ACGAN training becomes unstable as the number of classes increases and propose remedies for (1) the instability and (2) the relatively poor generation performance of ACGAN compared with the projection-based models. First, we begin by analytically deriving the gradient of the softmax cross-entropy loss used in ACGAN. By examining the exact values of analytic gradients, we discover that the unboundedness of input feature vectors and poor classification performance in the early training stage can cause an undesirable gradient exploding problem. Second, with alleviating the instability, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) using the Data-to-Data Cross-Entropy loss (D2D-CE). ReACGAN projects image embeddings and proxies onto a unit hypersphere and computes similarities for data-to-data and data-to-class consideration. Additionally, we introduce two margin values for intra-class variations and inter-class separability. In this way, ReACGAN overcomes the training instability and can exploit additional supervisory signals by explicitly considering data-to-class and data-to-data relationships, and also by implicitly looking at class-to-class relationships in the same mini-batch.
+
+To validate our model, we conduct image generation experiments on CIFAR10 [30], TinyImageNet [32], CUB200 [33], and ImageNet [31] datasets. Through extensive experiments, we demonstrate that ReACGAN beats both the classifier-based and projection-based GANs, improving over the state of the art by $2 . 5 \%$ , $1 5 . 8 \%$ , $5 . 1 \%$ , and $1 4 . 5 \%$ in terms of Fréchet Inception Distance (FID) [34] on the four datasets, respectively. We also verify that ReACGAN benefits from consistency regularization [16] and differentiable augmentations [9, 8] for limited data training. Finally, we confirm that D2D-CE harmonizes with the StyleGAN2 architecture [7].
+
+# 2 Background: Generative Adversarial Networks
+
+Generative Adversarial Network (GAN) [1] is an implicit generative model that aims to generate a sample indistinguishable from the real. GAN consists of two networks: a Generator $G : { \mathcal { Z } } \longrightarrow { \mathcal { X } }$ that tries to map a latent variable $\mathbf { z } \sim p ( \mathbf { z } )$ into the real data space $\mathcal { X }$ and a Discriminator $D : \mathcal { X } \longrightarrow$ [0, 1] that strives to discriminate whether a given sample $\mathbf { x }$ is from the real data distribution $p _ { \mathrm { r e a l } } ( \mathbf { x } )$ or from the implicit distribution $p _ { \mathrm { g e n } } ( G ( \mathbf { z } ) )$ derived from the generator $G ( \mathbf { z } )$ . The objective of a vanilla GAN [1] can be expressed as follows:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } [ \log ( D ( \mathbf { x } ) ) ] + \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } [ \log ( 1 - D ( G ( \mathbf { z } ) ) ) ] .
+$$
+
+While GANs have shown impressive results in the image generation task [2, 35, 11, 13], training GANs often ends up encountering a mode-collapse problem [36, 11, 12]. As one of the prescriptions for stabilizing and reinforcing GANs, training GANs with categorical information, named conditional Generative Adversarial Networks (cGAN), is suggested [37, 18, 19]. Depending on the presence of explicit classification losses, cGAN can be divided into two groups: Classifier-based GANs [18, 20, 23, 25, 27] and Projection-based GANs [19, 3, 4, 28]. One of the widely used classifier-based GANs is ACGAN [18], and ACGAN utilizes softmax cross-entropy loss to perform classification task with adversarial training. Although ACGAN has shown satisfactory generation results, training ACGAN becomes unstable as the number of classes in the training dataset increases [19, 23, 25, 27]. Besides, ACGAN tends to generate easily classifiable images at the cost of limited diversity [18, 19, 27]. To alleviate those problems, Zhou et al. [38] have proposed performing adversarial training on the classifier. Gong et al. [20] have introduced an additional classifier to eliminate a conditional entropy minimization process in the adversarial training. However, ACGAN training still suffers from the early-training collapse issue and the reduced diversity problem when trained on datasets with a large number of class categories, such as Tiny-ImageNet [32] and ImageNet [31].
+
+In these circumstances, Miyato et al. [19] have proposed a projection discriminator for cGANs and have shown significant improvement in generating the ImageNet dataset. Motivated by the promising result of the projection discriminator, many projection-based GANs [3, 29, 4, 16, 6, 17] have been proposed and become the standard for conditional image generation. In this paper, we revisit ACGAN and unveil why ACGAN training is so unstable. Coping with the instability, we propose the Rebooted Auxiliary Classifier GANs (ReACGAN) for high-quality and diverse image generation.
+
+# 3 Rebooting Auxiliary Classifier GANs
+
+# 3.1 Feature Normalization
+
+To uncover a nuisance that can cause the instability of ACGAN, we start by analytically deriving the gradients of weight vectors in the softmax classifier. Let the part of the discriminator before the fully connected layer be a Feature extractor $F : \mathcal { X } \longrightarrow \mathcal { F } \in \bar { \mathbb { R } ^ { d } }$ and let Classifier $C : { \mathcal { F } } \longrightarrow \mathbb { R } ^ { c }$ be a single fully connected layer parameterized by $\pmb { W } = [ \pmb { w } _ { 1 } , . . . , \pmb { w } _ { c } ] \in \mathbb { R } ^ { d \times c }$ , where $c$ denotes the number of classess. We sample training images $\pmb { X } = \{ \pmb { x } _ { 1 } , . . . , \pmb { x } _ { N } \}$ and integer labels ${ \textbf { 3 } } =$ $\{ y _ { 1 } , . . . , y _ { N } \}$ from the joint distribution $p ( \mathbf { x } , \mathbf { y } )$ . Using the notations above, we can express the empirical cross-entropy loss used in ACGAN [18] as follows:
+
+$$
+\mathcal { L } _ { \mathrm { C E } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Big ( \frac { \exp \left( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { y _ { i } } \right) } { \sum _ { j = 1 } ^ { c } \exp \left( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { j } \right) } \Big ) .
+$$
+
+Based on Eq. (3), we can derive the derivative of the cross-entropy loss, w.r.t ${ \pmb w } _ { k \in \{ 1 , \dots , c \} }$ as follows:
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { C E } } } { \partial \boldsymbol { w } _ { k } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \bigg \{ F ( \boldsymbol { x } _ { i } ) \bigg ( \mathbf { 1 } _ { y _ { i } = k } - p _ { i , k } \bigg ) \bigg \} ,
+$$
+
+where $\mathbf { 1 } _ { \mathit { y } _ { i } = k }$ is an indicator function that will output 1 if $y _ { i } = k$ is satisfied, and $p _ { i , k }$ is a class probability that represents the probability that $i$ -th sample belongs to class $k$ , mathematically $\frac { \exp { ( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { k } ) } } { \sum _ { j = 1 } ^ { c } \exp { ( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { j } ) } }$ . The equation above implies that the norm of the gradient of the softmax cross-entropy loss is coupled with the norms and directions of each input feature map $F ( { \pmb x } _ { i } )$ and the class probabilities. In the early training stage, the classifier is prone to making incorrect predictions, resulting in low probabilities. This phenomenon occurs more frequently as the number of categories in the dataset increases. As the result, the norm of the gradient $| \frac { \partial \hat { \mathcal { L } } _ { \mathrm { C E } } } { \partial w _ { k } } |$ begins to explode as the vector $F ( { \pmb x } _ { i } )$ stretches out to ${ \pmb w } _ { k }$ direction but being located close to the the other vectors ${ \pmb w } _ { j \in \{ 1 , . . . , c \} \backslash \{ k \} }$ . This often breaks the balance between adversarial learning and classifier training, leading to an early-training collapse. Once the early-training collapse occurs, ACGAN training concentrates on classifying categories of images instead of discriminating the authenticity of given samples. We experimentally demonstrate that the average norm of ACGAN’s input feature maps increases as the training progresses (Fig. 2a). Accordingly, the average norm of the gradients increases sharply at the early training stage and decreases with the high class probabilities of the classifier (Fig. 2b, Fig. A3 in Appendix). While the average norm of gradients decreases at some point, the FID value of ACGAN does not decrease, implying the collapse of ACGAN training (Fig. 2c).
+
+![](images/41f752bd054c08034409d7f5b84c86a0197793ecc9cb9dcf2b22365737c55349.jpg)  
+Figure 2: Merits of integrating feature normalization and data-to-data relationship consideration. All experiments are conducted on Tiny-ImageNet [32] dataset. (a) Average norms of input feature maps $\bar { F } ( \cdot )$ , (b) average norms of gradients of classification losses, and (c) trends of FID scores. Compared with ACGAN [18], the proposed ReACGAN does not experience the training collapse problem caused by excessively large norms of feature maps and gradients at the early training stage. In addition, ReACGAN can converge to better equilibrium by considering data-to-data relationships with easy positive and negative sample suppression.
+
+As one of the remedies for the gradient exploding problem, we find that simply normalizing the feature embeddings onto a unit hypersphere $\frac { F ( { \pmb x } _ { i } ) } { | | F ( { \pmb x } _ { i } ) | | }$ effectively resolves ACGAN’s early-training collapse (see Fig. 2). The motivation is that normalizing the features onto the hypersphere makes the norms of feature maps equal to 1.0. Thus, the discriminator does not experience the gradient exploding problem. From the next section, we will deploy a linear projection layer $P$ on the feature extractor $F$ . And, we will normalize both the embeddings from the projection layer and the weight vectors $( \pmb { w } _ { 1 } , . . . , \pmb { w } _ { c } )$ in the classifier since normalizing both the embeddings and the weight vectors does not degrade image generation performance. We denote the normalized embedding P (F (xi))||P (F (xi))|| as $f _ { i }$ and the normalized weight vector $\frac { { \pmb w } _ { y _ { i } } } { | | { \pmb w } _ { y _ { i } } | | }$ as vyi .
+
+# 3.2 Data-to-Data Cross-Entropy Loss (D2D-CE)
+
+We expand the feature normalized softmax cross-entropy loss described in Sec. 3.1 to the Data-to-Data Cross-Entropy (D2D-CE) loss. The motivations are summarized into two points: (1) replacing datato-class similarities in the denominator of Eq. (3) with data-to-data similarities and (2) introducing two margin values to the modified softmax cross-entropy loss. We expect that point (1) will encourage the feature extractor to consider data-to-class as well as data-to-data relationships, and that point (2) will guarantee inter-class separability and intra-class variations in the feature space while preventing ineffective gradient updates induced by easy negative and positive samples. To develop the feature normalized cross-entropy loss into D2D-CE, we replace the similarities between a sample embedding and all proxies except for the positive one in the denominator, $\begin{array} { r } { \sum _ { j \in \{ 1 , \dots , c \} \backslash \{ y _ { i } \} } \exp \left( \mathbf { { f } } _ { i } ^ { \top } \mathbf { { v } } _ { j } \right) } \end{array}$ with similarities between negative samples in the same mini-batch. The modified cross-entropy loss can be expressed as follows:
+
+$$
+\mathcal { L } _ { \mathrm { C E } } ^ { ' } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Bigg ( \frac { \exp { ( f _ { i } ^ { \top } v _ { y _ { i } } / \tau ) } } { \exp { ( f _ { i } ^ { \top } v _ { y _ { i } } / \tau ) } + \sum _ { j \in N ( i ) } \exp { ( f _ { i } ^ { \top } f _ { j } / \tau ) } } \Bigg ) ,
+$$
+
+where $\tau$ is a temperature, and $\mathcal { N } ( i )$ is the set of indices that point locations of the negative samples whose labels are different from the reference label $\pmb { v } _ { y _ { i } }$ in the mini-batch. The self-similarity matrix of samples in the mini-batch is used to calculate the similarities between negative samples $\pmb { f } _ { i } ^ { \top } \pmb { f } _ { j \in \mathcal { N } ( i ) }$ with a false negative mask (see Fig. 3). Thus, Eq. (5) enables the discriminator to contrastively compare visual differences between multiple images and can supply more informative supervision for image conditioning. Finally, we introduce two margin hyperparameters to $\mathcal { L } _ { \mathrm { C E } } ^ { ' }$ and name it Data-to-Data Cross-Entropy loss (D2D-CE). The proposed D2D-CE can be expressed as follows:
+
+$$
+\mathcal { L } _ { \mathrm { D 2 D - C E } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Bigg ( \frac { \exp \left( [ f _ { i } ^ { \top } v _ { y _ { i } } - m _ { p } ] _ { - } / \tau \right) } { \exp \left( [ f _ { i } ^ { \top } v _ { y _ { i } } - m _ { p } ] _ { - } / \tau \right) + \sum _ { j \in N ( i ) } \exp \left( [ f _ { i } ^ { \top } f _ { j } - m _ { n } ] _ { + } / \tau \right) } \Bigg ) ,
+$$
+
+![](images/ee7bad162e1754a63a5d7f7b5a7a8d9d8e3e367c1fcd2cc101b8a2534e6e145d.jpg)  
+Figure 3: Overview of ReACGAN. ReACGAN performs adversarial training using the loss ${ \mathcal { L } } _ { \mathrm { A d v } }$ . At the same time, ReACGAN tries to minimize data-to-data cross-entropy loss (D2D-CE) on the linear head $( P )$ to exploit relational information in the labeled dataset. $^ { \circ }$ means element-wise product. Note that False negatives mask operates on the similarity matrix between two batches of sample embeddings to compute the similarities between negative samples in the denominator of Eq. (6).
+
+where $m _ { p }$ is a margin for suppressing a high similarity value between a reference sample and its corresponding proxy (easy positive), $m _ { n }$ is a margin for suppressing low similarity values between negatives samples (easy negatives). $[ \cdot ] _ { - }$ and $[ \cdot ] _ { + }$ denote $\operatorname* { m i n } ( \cdot , 0 )$ and $\operatorname* { m a x } ( \cdot , 0 )$ functions, respectively.
+
+# 3.3 Useful Properties of Data-to-Data Cross-Entropy Loss
+
+In this subsection, we explain four useful properties of D2D-CE. Let $s _ { q }$ be a similarity between the normalized embedding $f _ { q }$ and the corresponding normalized proxy ${ \pmb v } _ { y _ { q } }$ , $s _ { q , r }$ be a similarity between $f _ { q }$ and $f _ { r }$ , and $a$ and $\mathit { b }$ be arbitrary indices of negative samples, i.e., $a , b \in \mathcal { N } ( q )$ . Then the properties of D2D-CE can be summarized as follows:
+
+![](images/284973472b7cc1ba4ea65614768295afd00d54dfd0ba532f646e48451266f16e.jpg)  
+Figure 4: Graph showing regions where hard negative mining, easy positive and negative suppression, and no gradient update occur.
+
+Property 1. Hard negative mining. If the value of $s _ { q , a }$ is greater than $s _ { q , b }$ , the derivative of LD2D-CE w.r.t $s _ { q , a }$ is greater than or equal to the derivative $w . r . t \ s _ { q , b }$ ; that is $\begin{array} { r } { \frac { \partial \mathcal { L } _ { D 2 D - C E } } { \partial s _ { q , a } } \geq \frac { \partial \mathcal { L } _ { D 2 D - C E } } { \partial s _ { q , b } } \geq 0 . } \end{array}$
+
+Property 2. Easy positive suppression. If $s _ { q } \mathrm { ~ - ~ }$ $m _ { p } \geq 0$ , the derivative of $\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q }$ is 0.
+
+Property 3. Easy negative suppression. If $s _ { q , r } - m _ { n } \leq 0 ,$ , the derivative of $\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q , r }$ is 0. Property 4. If $s _ { q } - m _ { p } \ge 0$ and $s _ { q , r } - m _ { n } \leq 0$ are satisfied, $\mathcal { L } _ { D 2 D - C E }$ has the global minima of $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \left( 1 + \bar { | \mathcal { N } ( i ) | } \right) } \end{array}$ .
+
+We put proofs of the above properties in Appendix D. Property 1 indicates that D2D-CE implicitly conducts hard negative mining and benefits from comparing samples with each other. Also, Properties 2 and 3 imply that samples will not affect gradient updates if the samples are trained sufficiently. Consequently, the classifier concentrates on pushing and pulling hard negative and hard positive examples without being dominated by easy negative and positive samples.
+
+# 3.4 Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN)
+
+With the proposed D2D-CE objective, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). As ACGAN does, ReACGAN jointly optimizes an adversarial loss and the classification objective (D2D-CE). Specifically, the discriminator, which consists of the feature extractor, adversarial head, and linear head, of ReACGAN tries to discriminate whether a given image is sampled from the real distribution or not. At the same time, the discriminator tries to maximize similarities between the reference samples and corresponding proxies while minimizing similarities between negative samples using real images and D2D-CE loss. After updating the discriminator a predetermined number of times, the generator strives to deceive the discriminator by generating well-conditioned images that will output a low D2D-CE value. By alternatively training the discriminator and generator until convergence, ReACGAN generates high-quality images of diverse categories without early-training collapse. We attach the algorithm table in Appendix A.
+
+Differences between ReACGAN and ContraGAN. The authors of ContraGAN [23] propose a conditional contrastive loss (2C loss) to cover data-to-data relationships when training cGANs. The main differences between 2C loss and D2D-CE objective are summed up into three points: (1) while 2C loss is derived from NT-Xent loss [39], which is popularly used in the field of the self-supervised learning, our D2D-CE is developed to resolve the early-training collapsing problem and the poor generation results of ACGAN, (2) 2C loss holds false-negative samples in the denominator, and they can cause unexpected positive repulsion forces, and (3) 2C loss contains the similarities between all positive samples in the numerator, and they give rise to large gradients on pulling easy positive samples. Consequently, GANs with 2C loss tend to synthesize images of unintended classes as reported in the author’s software document [40]. However, D2D-CE does not contain the false negatives in the denominator and considers only the similarity between a sample and its proxy in the numerator. Therefore, ReACGAN is free from the undesirable repelling forces and does not conduct unnecessary easy positive mining. More detailed explanations are attached in Appendix F.
+
+Consistency Regularization and D2D-CE Loss. Zhang et al. [16] propose a consistency regularization to force the discriminator to make consistent predictions if given two images are visually close to each other. They create a visually similar image pair by augmenting a reference image with pre-defined augmentations. After that, they let the discriminator minimize L2 distance between the logit of the reference image and the logit of the augmented counterpart. While ReACGAN locates an image embedding nearby its corresponding proxy but far apart multiple image embeddings of different classes, consistency regularization only pulls a reference and the augmented image towards each other. Since consistency regularization and D2D-CE can be applied together, we will show that ReACGAN benefits from consistency regularization in the experiments section (Table 1).
+
+# 4 Experiments
+
+# 4.1 Datasets and Evaluation Metrics
+
+To verify the effectiveness of ReACGAN, we conduct conditional image generation experiments using five datasets: CIFAR10 [30], Tiny-ImageNet [32], CUB200 [33], ImageNet [31], and AFHQ [41] datasets and four evaluation metrics: Inception Score (IS) [42], Fréchet Inception Distance (FID) [34], and $\mathrm { F _ { 0 . 1 2 5 } }$ (Precision) and $\mathrm { F } _ { 8 }$ (Recall) [43]. The details on the training datasets are in Appendix C.1.
+
+Inception Score (IS) [42] and Fréchet Inception Distance (FID) [34] are widely used metrics for evaluating generative models. We utilize IS and FID together because some studies [4, 6, 25] have shown that IS has a tendency to measure the fidelity of images better while FID tends to weight capturing the diversity of images.
+
+Precision $\bf ( F _ { 0 . 1 2 5 } )$ and Recall $\bf ( F _ { 8 } )$ [43] are metrics for estimating precision and recall of the approximated distribution $p ( G ( \mathbf { z } ) )$ against the true data distribution $p ( \mathbf { x } )$ . Instead of evaluating generative models using one-dimensional scores, such as IS and FID, Sajjadi et al. [43] have suggested using a two-dimensional score $\mathrm { F _ { 0 . 1 2 5 } }$ and $\mathrm { F } _ { 8 }$ that can quantify how precisely the generated images are and how well the generated images cover the reference distribution.
+
+# 4.2 Experimental Details
+
+The implementation of ReACGAN basically follows details of PyTorch-StudioGAN library [40]1 that supports various experimental setups from ACGAN [18] to StyleGAN2 [7] $^ +$ ADA [8] with different scales of datasets [30, 32, 33, 31, 41] and architectures [2, 13, 4, 7]. For a fair comparison, we use the same backbones over all baselines, except otherwise noted, for both the discriminator and generator. For stable training, we apply spectral normalization (SN) [3] to the generator and discriminator except for experiments using SNGAN (in this case, we apply SN to the discriminator only) and StyleGAN2 [7]. We also use the same conditioning method for generators with conditional batch normalization (cBN) [44, 45, 19]. This allows us to investigate solely the conditioning methods of the discriminator, which are the main interest of our paper. If not specified, we use hinge loss [46] as a default for the adversarial loss.
+
+Table 1: Comparison with classifier-based GANs [18, 23] and projection-based GANs [3, 4, 29] on CIFAR10 [30], Tiny-ImageNet [32], and CUB200 [33] datasets using IS [42], FID [34], $\mathrm { F } _ { 0 . 1 2 5 }$ , and $\mathrm { F } _ { 8 }$ [43] metrics. For baselines, both the numbers from the cited paper (denoted as \* in method) and from our experiments using StudioGAN library [40] are reported. The numbers in bold-faced denote the best performance and in underline indicate the values are in one standard deviation from the best.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">CIFAR10 [30]</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td></tr><tr><td>SNGAN*[3]</td><td>8.22</td><td>21.7</td><td>=</td><td>-</td><td>1</td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>=</td><td>-</td></tr><tr><td>BigGAN*[4]</td><td>9.22</td><td>14.73</td><td>=</td><td>-</td><td>=</td><td>-</td><td></td><td>=</td><td>-</td><td></td><td>-</td><td>1</td></tr><tr><td>ContraGAN* [23]</td><td>1</td><td>10.60</td><td></td><td></td><td></td><td>29.49</td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>ACGAN[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>SNGAN [3]</td><td>8.67</td><td>13.33</td><td>0.985</td><td>0.976</td><td>8.71</td><td>51.15</td><td>0.900</td><td>0.702</td><td>5.41</td><td>47.75</td><td>0.754</td><td>0.912</td></tr><tr><td>SAGAN [29]</td><td>8.66</td><td>14.31</td><td>0.983</td><td>0.973</td><td>8.74</td><td>49.90</td><td>0.872</td><td>0.712</td><td>5.48</td><td>54.29</td><td>0.728</td><td>0.882</td></tr><tr><td>BigGAN[4]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>ContraGAN [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>ReACGAN</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td></td><td>14.06 27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr><tr><td>BigGAN +CR*[16]</td><td>-</td><td>11.48</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BigGAN [4]+ CR[16]</td><td>9.97</td><td>7.18</td><td>0.995</td><td>0.993</td><td></td><td>15.94 19.96</td><td>0.972</td><td>0.950 5.14</td><td></td><td>11.97</td><td>0.978</td><td>0.981</td></tr><tr><td>ContraGAN [23] + CR[16]</td><td>9.59</td><td>8.55</td><td>0.992</td><td>0.972</td><td>15.81</td><td>19.21</td><td>0.983</td><td>0.941</td><td>4.90</td><td>11.08</td><td>0.984</td><td>0.967</td></tr><tr><td>ReACGAN + CR [16]</td><td>10.11</td><td>7.20</td><td>0.996</td><td>0.994 16.56</td><td></td><td>19.69</td><td>0.984</td><td>0.940</td><td>4.87</td><td>10.72</td><td>0.985</td><td>0.971</td></tr><tr><td>BigGAN + DiffAug*[9]</td><td>9.17</td><td>8.49</td><td></td><td></td><td>=</td><td></td><td>=</td><td>=</td><td>=</td><td></td><td></td><td></td></tr><tr><td>BigGAN [4] + DiffAug [9]</td><td>9.94</td><td>7.17</td><td>0.995</td><td>0.992</td><td></td><td>18.0815.70</td><td>0.980</td><td>0.972</td><td>5.53</td><td>12.15</td><td>0.967</td><td>0.981</td></tr><tr><td>ContraGAN [23] + DiffAug [9]</td><td>9.95</td><td>7.27</td><td>0.995</td><td>0.992</td><td>18.20</td><td>15.40</td><td>0.986</td><td>0.963</td><td>5.39</td><td>11.02</td><td>0.978</td><td>0.970</td></tr><tr><td>ReACGAN + DiffAug [9]</td><td>10.22</td><td>6.79</td><td>0.996</td><td>0.993</td><td>20.60</td><td>14.25</td><td>0.988</td><td>0.972</td><td>5.22</td><td>9.27</td><td>0.985</td><td>0.983</td></tr><tr><td>Real Data</td><td colspan="2">11.54</td><td colspan="2">34.11</td><td colspan="2"></td><td colspan="2"></td><td colspan="2">5.49</td><td colspan="2"></td></tr></table>
+
+Before conducting main experiments, we perform hyperparameter search with candidates of a temperature $\tau \in \{ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \}$ and a positive margin $m _ { p } \in \ \{ 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 8 , \bar { 1 . 0 } \}$ . We set a negative margin $m _ { n }$ as $1 \ : - \ : m _ { p }$ to reduce search time. Through extensive experiments with 3 runs per each setting, we select $\tau$ with $\{ 0 . 5 , 0 . 7 5 , 0 . 2 5 , 0 . 5 , 0 . 2 5 \}$ and $m _ { p }$ with $\{ 0 . 9 8 , 1 . 0 , 0 . 9 5 , 0 . 9 \bar { 8 } , 0 . 9 0 \}$ for CIFAR10, Tiny-ImageNet, CUB200, ImageNet 256 B.S., and ImageNet 2048 B.S. experiments, respectively. A low temperature seems to work well on fine-grained image generation tasks, but generally, ReACGAN is robust to the choice of hyperparameters. The results of the hyperparameter search and other hyperparameter setups are provided in Appendix C.2.
+
+We evaluate all methods through the same protocol of [9, 16, 17], which uses the same amounts of generated images from the reference split specialized for each dataset.2 Besides, we run all the experiments three times with random seeds and report the averaged best performances for reliable evaluation with the lone exception of ImageNet and StyleGAN2 related experiments. Please refer to Appendix C.2 for other experimental details. The numbers in bold-faced denote the best performance and in underline indicate that the values are in one standard deviation from the best.
+
+# 4.3 Evaluation Results
+
+Comparison with Other cGANs. We compare ReACGAN with previous state-of-the-art cGANs in Tables 1 and 2. We employ the implementations of GANs in PyTorch-StudioGAN library as it provides improved results on standard benchmark datasets [30, 31]. For a fair comparison, we provide results from each original paper (denoted as \* in method) as well as those from StudioGAN library [40]. We also conduct experiments with popular augmentation-based methods: consistency regularization (CR) [16] and differentiable augmentation (DiffAug) [9].
+
+Table 2: Experiments using ImageNet [31] dataset. B.S. means batch size for training. (Left): Comparisons with previous cGAN approaches. (Right): Learning curves of BigGAN [4], ContraGAN [23], and ReACGAN (ours) which are trained using the batch size of 256.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">ImageNet [31]</td></tr><tr><td>IS↑</td><td>FID↓</td><td></td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>ACGAN*</td><td>[20]</td><td>7.26</td><td>184.41</td><td></td><td></td></tr><tr><td>SNGAN*[3]</td><td></td><td>36.80</td><td>27.62</td><td></td><td></td></tr><tr><td>SAGAN*[29]</td><td></td><td>52.52</td><td>18.28</td><td></td><td></td></tr><tr><td>BigGAN*[20]</td><td></td><td>38.05</td><td>22.77</td><td></td><td></td></tr><tr><td></td><td>TAC-GAN*[20]</td><td>28.86</td><td>23.75</td><td></td><td></td></tr><tr><td>ContraGAN*[23]</td><td></td><td>31.10</td><td>19.69</td><td>0.951</td><td>0.927</td></tr><tr><td></td><td>ACGAN[18]</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>35=58</td><td>SNGAN [3]</td><td>32.25</td><td>26.79</td><td>0.938</td><td>0.913</td></tr><tr><td></td><td>SAGAN [29]</td><td>29.85</td><td>34.73</td><td>0.849</td><td>0.914</td></tr><tr><td></td><td>BigGAN [4]</td><td>43.97</td><td>16.36</td><td>0.964</td><td>0.955</td></tr><tr><td></td><td>ContraGAN [23]</td><td>25.25</td><td>25.16</td><td>0.947</td><td>0.855</td></tr><tr><td>ReACGAN</td><td></td><td>68.27</td><td>13.98</td><td>0.976</td><td>0.977</td></tr><tr><td></td><td>BigGAN [4] + DiffAug [9]</td><td>36.97</td><td>18.57</td><td>0.956</td><td>0.941</td></tr><tr><td></td><td>ReACGAN +DiffAug[9]</td><td>69.74</td><td>11.95</td><td>0.977</td><td>0.975</td></tr><tr><td></td><td>BigGAN*[4]</td><td>99.31</td><td>8.51</td><td>-</td><td>-</td></tr><tr><td>P55=44</td><td>BigGAN*[25]</td><td>104.57</td><td>9.18</td><td></td><td>■</td></tr><tr><td></td><td>BigGAN[4]</td><td>99.71</td><td>7.89</td><td>0.985</td><td>0.989</td></tr><tr><td></td><td>ReACGAN</td><td>92.74</td><td>8.23</td><td>0.991</td><td>0.990</td></tr><tr><td></td><td>Real Data</td><td>173.33</td><td></td><td></td><td></td></tr></table>
+
+![](images/53ae9768944e305563820e0c40eca80c05a3af3fed3352bd392c4954a6225bf3.jpg)
+
+Compared with other cGANs, ReACGAN performs the best on most benchmarks, surpassing the previous methods by $2 . 5 \%$ , $5 . 1 \%$ , $1 5 . 8 \%$ , and $1 4 . 5 \%$ in FID on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet (256 B.S.), respectively. ReACGAN also harmonizes with augmentation-based regularizations, bringing incremental improvements on all the metrics. For the ImageNet experiments using a batch size of 256, ReACGAN reaches higher IS and lower FID with fewer iterations than other models in comparison. Finally, we demonstrate that ReACGAN can learn with a larger batch size on ImageNet. While some recent methods [20, 38, 27] have been built on ACGAN to improve the generation performance of ACGAN, large-scale image generation experiments with the batch size of 2048 have never been reported. Table 2 shows that our ReACGAN reaches FID score of 8.23 on ImageNet, being comparable with the value of 7.89 from BigGAN implementation in PyTorch-StudioGAN library. ReACGAN, however, provides better synthesis results than other implementations of BigGAN [4, 25]. Note that our result on ImageNet is obtained in only two runs while the training setup and architecture of BigGAN have been extensively searched and finely tuned.
+
+Comparison with Other Conditioning Losses. We investigate how the generation qualities vary with different conditioning losses while keeping the other configurations fixed. We compare D2D-CE loss with cross-entropy loss of ACGAN (AC) [18], loss used in the projection discriminator (PD) [19], multi-hinge loss (MH) [26], and conditional contrastive loss (2C) [23]. The results are shown in Table 3. AC- and MH-based models present decent results on CIFAR10, but undergo early-training collapse on Tiny-ImageNet and CUB200 datasets. Replacing them with PD, 2C, and D2D-CE losses produce satisfactory performances across all datasets, where PD loss makes the best $\mathrm { F } _ { 8 }$ (recall) on CUB200 dataset while giving third $\mathrm { F _ { 0 . 1 2 5 } }$ (precision) value. The noticeable point is that the proposed D2D-CE shows consistent results across all datasets, showing the lowest FID and the highest $\mathrm { F _ { 8 } }$ and $\mathrm { F _ { 0 . 1 2 5 } }$ values in most cases. This means ReACGAN can generate high-fidelity images and is relatively free from the precision and recall trade-off [43] than the others.
+
+Consistent Performance of ReACGAN on Adversarial Loss Selection. We validate the consistent performance of ReACGAN on four adversarial losses: non-saturation loss [1], least square loss [10], Wasserstein loss with gradient penalty regularization (W-GP) [13], and hinge loss [46] on CIFAR10 and Tiny-ImageNet datasets in Table 4. The experimental results show that $\mathrm { B i g G A N } + \mathrm { D } 2 \mathrm { D } { \cdot } \mathrm { C E }$ (ReACGAN) consistently outperforms the projection discriminator (PD) and conditional contrastive loss (2C) counterparts over three adversarial losses [1, 13, 46]. However, for the experiments using the least square loss [10], ReACGAN exhibits inferior generation performances to the projection discriminator. We speculate that minimizing the least square distance between an adversarial logit and the target scalar (1 or 0) might affect the norms of feature maps and spoil the classifier training performed by D2D-CE loss.
+
+Table 3: Experiments on the effectiveness of D2D-CE loss compared with other conditioning losses.   
+
+<table><tr><td rowspan="2">Conditioning Method</td><td colspan="4">CIFAR10 [30]</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>BigGAN w/o Condition [4] (Abbreviated to Big)</td><td>9.46</td><td>12.21</td><td>0.987</td><td>0.982</td><td>7.38</td><td>76.15</td><td>0.804</td><td>0.576</td><td>5.16</td><td>35.17</td><td>0.852</td><td>0.936</td></tr><tr><td>Big + AC[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>Big + PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>Big + MH[26]</td><td>10.05</td><td>7.94</td><td>0.994</td><td>0.990</td><td>4.37</td><td>140.74</td><td>0.282</td><td>0.156</td><td>5.18</td><td>245.69</td><td>0.625</td><td>0.832</td></tr><tr><td>Big + 2C[23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>Big + D2D-CE (ReACGAN)</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr></table>
+
+Table 4: Experiments to identify the consistent performance of D2D-CE on adversarial loss selection.   
+
+<table><tr><td rowspan="2">Adversarial Loss</td><td rowspan="2">Conditioning Method</td><td colspan="5">CIFAR10 [30]</td><td colspan="5">Tiny-ImageNet [32]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>Better?</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td><td>Better?</td></tr><tr><td rowspan="3">Non-saturation [1]</td><td>PD[19]</td><td>9.75</td><td>8.29</td><td>0.993</td><td>0.991</td><td>√</td><td>8.27</td><td>58.85</td><td>0.816</td><td>0.713</td><td></td></tr><tr><td>2C [23]</td><td>9.30</td><td>10.47</td><td>0.990</td><td>0.959</td><td></td><td>6.57</td><td>84.27</td><td>0.745</td><td>0.556</td><td></td></tr><tr><td>D2D-CE</td><td>9.79</td><td>8.27</td><td>0.993</td><td>0.991</td><td>√</td><td>11.76</td><td>39.32</td><td>0.942</td><td>0.852</td><td>√</td></tr><tr><td rowspan="3">Least square [10]</td><td>PD[19]</td><td>9.94</td><td>8.26</td><td>0.993</td><td>0.992</td><td>√</td><td>12.74</td><td>37.14</td><td>0.920</td><td>0.900</td><td>√</td></tr><tr><td>2C[23]</td><td>8.66</td><td>12.18</td><td>0.986</td><td>0.941</td><td></td><td>9.58</td><td>53.10</td><td>0.916</td><td>0.706</td><td></td></tr><tr><td>D2D-CE</td><td>9.70</td><td>9.56</td><td>0.991</td><td>0.987</td><td></td><td>9.50</td><td>57.67</td><td>0.848</td><td>0.692</td><td></td></tr><tr><td rowspan="3">W-GP [13]</td><td>PD[19]</td><td>5.71</td><td>64.75</td><td>0.792</td><td>0.652</td><td></td><td>6.67</td><td>84.16</td><td>0.696</td><td>0.498</td><td></td></tr><tr><td>2C [23]</td><td>5.93</td><td>55.99</td><td>0.842</td><td>0.709</td><td></td><td>6.89</td><td>74.45</td><td>0.812</td><td>0.536</td><td></td></tr><tr><td>D2D-CE</td><td>7.30</td><td>35.94</td><td>0.942</td><td>0.847</td><td>√</td><td>8.92</td><td> 52.74</td><td>0.856</td><td>0.689</td><td>√</td></tr><tr><td rowspan="3">Hinge [46]</td><td>PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td></td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td></td></tr><tr><td>2C [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td></td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td></td></tr><tr><td>D2D-CE</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>√</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>√</td></tr></table>
+
+Table 5: FID [34] values of conditional StyleGAN2 (cStyleGAN2) [7] and StyleGAN2 $[ 7 ] + \mathrm { D } 2 \mathrm { D } { \cdot } \mathrm { C E }$ on CIFAR10 [30] and AFHQ [41] datasets. ADA means the adaptive discriminator augmentation [8]. FID is exceptionally computed using the training split for calculating the reference moments since FID value of StyleGAN2 is often calculated using the moments of the training dataset.   
+
+<table><tr><td>Conditioning method</td><td>CIFAR10 [30]</td><td>AFHQ [41]</td></tr><tr><td>cStyleGAN2 [7]</td><td>3.88</td><td>=</td></tr><tr><td>StyleGAN2 [7] + D2D-CE</td><td> 3.34</td><td></td></tr><tr><td>cStyleGAN2 [7] + ADA [8]</td><td>2.43</td><td>4.99</td></tr><tr><td>StyleGAN2 [7] + ADA [8] + D2D-CE</td><td>2.38</td><td>4.95</td></tr><tr><td> StyleGAN2 [7] + DiffAug [9] + D2D-CE + Tuning</td><td>2.26</td><td>-</td></tr></table>
+
+Effect of D2D-CE for Different GAN Architectures. We study the effect of D2D-CE loss with different GAN architectures. In Table 5, we validate that D2D-CE is effective for StyleGAN2 [7] backbone and also fits well with the adaptive discriminator augmentation (ADA) [8]. StyleGAN2 with D2D-CE loss produces $1 3 . 9 \%$ better generation result than the conditional version of StyleGAN2 (cStyleGAN2) on CIFAR10. Moreover, StyleGAN2 with D2D-CE can be reinforced with ADA or DiffAug when train StyleGAN2 $^ +$ D2D-CE under the limited data situation. Among GANs, ${ \mathrm { S t y l e G A N } } 2 + { \mathrm { D i f f A u g } } + { \mathrm { D } } 2 { \mathrm { D } } { \mathrm { - C E } } + { \mathrm { 1 } }$ Tuning achieves the best performance on CIFAR10, even outperforming some diffusion-based methods [47, 48]. Additional results with other architectures, i.e., a deep convolutional network [2] and a resnet style backbone [13], are provided in Appendix E.
+
+# 4.4 Ablation Study
+
+We study how each component of ReACGAN affects ACGAN training. By adding or ablating each part of ReACGAN, as shown in Table 6, we identify four major observations. (1) Feature and weight normalization greatly stabilize ACGAN training and improve generation performances on Tiny-ImageNet and CUB200 datasets. (2) D2D-CE enhances the generation performance by considering data-to-data relationships and by performing easy sample suppression (3th and 4th rows). (3) the suppression technique does not work well on the feature normalized cross-entropy loss (5th row). (4) While ACGAN shows high Inception score on ImageNet experiment, it shows relatively poor FID, $\mathrm { F _ { 0 . 1 2 5 } }$ , and $\mathrm { F _ { 8 } }$ values compared with the model trained with normalization, data-to-data consideration, and the suppression technique. This result is consistent with the qualitative results on ImageNet, where the images from ACGAN are easily classifiable, but the images from ReACGAN are high quality and diverse. We attribute the improvement to the discriminator that successfully leverages informative data-to-data and data-to-class relationships with easy sample suppression.
+
+![](images/a59ce57281fb29c29f8a0af958d8e87aa81a5b050192d0c14ce2973864977a69.jpg)  
+Figure 5: Curated images generated by the proposed ReACGAN. More qualitative results on ReACGAN, BigGAN [4], ContraGAN [23], and ACGAN [18] are attached in Appendix J.
+
+Table 6: Ablation study on feature map normalization, data-to-data consideration, and easy positive and negative sample suppression.   
+
+<table><tr><td rowspan="2">Ablation</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td><td colspan="4">ImageNet [49]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td></tr><tr><td>ACGAN [18]</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>+ Normalization</td><td>13.46</td><td>30.33</td><td>0.955</td><td>0.889</td><td>4.78</td><td>25.54</td><td>0.883</td><td>0.952</td><td>18.16</td><td>36.40</td><td>0.879</td><td>0.787</td></tr><tr><td>+ Data-to-data (Eq. (5))</td><td>12.96</td><td>28.71</td><td>0.967</td><td>0.863</td><td>5.08</td><td>25.12</td><td>0.894</td><td>0.946</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td> + Suppression (Eq. (6))</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td><td>63.16 14.59</td><td></td><td></td><td>0.9740.974</td></tr><tr><td>- Data-to-data</td><td>12.96</td><td>30.79</td><td>0.960</td><td>0.857</td><td>5.39</td><td>30.36</td><td>0.863</td><td>0.947</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we have analyzed why training ACGAN becomes unstable as the number of classes in the dataset increases. By deriving the analytic form of gradient in the classifier and numerically checking the gradient values, we have discovered that the unstable training comes from a gradient exploding problem caused by the unboundedness of input feature vectors and poor classification ability of the classifier in the early training stage. To alleviate the instability and reinforce ACGAN, we have proposed the Data-to-Data Cross-Entropy loss (D2D-CE) and the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results verify the superiority of ReACGAN compared with the existing classifier- and projection-based GANs on five benchmark datasets. Moreover, exhaustive analyses on ReACGAN prove that ReACGAN is robust to hyperparameter selection and harmonizes with various architectures and differentiable augmentations.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the IITP grants (No.2019-0-01906: AI Graduate School Program - POSTECH, No.2021-0-00537: Visual Common Sense, No.2021-0-02068: AI Innovation Hub) funded by Ministry of Science and ICT, Korea.
+
+References   
+[1] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems (NeurIPS), pages 2672–2680, 2014.   
+[2] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv preprint arXiv 1511.06434, 2016.   
+[3] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   
+[4] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large Scale GAN Training for High Fidelity Natural Image Synthesis. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.   
+[5] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 4401–4410, 2019.   
+[6] Yan Wu, Jeff Donahue, David Balduzzi, Karen Simonyan, and Timothy P. Lillicrap. LOGAN: Latent Optimisation for Generative Adversarial Networks. arXiv preprint arXiv 1912.00953, 2019.   
+[7] Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 8110–8119, 2020.   
+[8] Tero Karras, Miika Aittala, Janne Hellsten, Samuli Laine, Jaakko Lehtinen, and Timo Aila. Training Generative Adversarial Networks with Limited Data. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   
+[9] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song Han. Differentiable augmentation for dataefficient gan training. arXiv preprint arXiv 2006.10738, 2020.   
+[10] Xudong Mao, Qing Li, Haoran Xie, Raymond Y. K. Lau, Zhixiang Wang, and Stephen Paul Smolley. Least Squares Generative Adversarial Networks. In Proceedings of the International Conference on Computer Vision (ICCV), pages 2813–2821, 2017.   
+[11] Martín Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein GAN. arXiv preprint arXiv 1701.07875, 2017.   
+[12] Martín Arjovsky and Léon Bottou. Towards Principled Methods for Training Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.   
+[13] Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved Training of Wasserstein GANs. In Advances in Neural Information Processing Systems (NeurIPS), pages 5767–5777, 2017.   
+[14] Naveen Kodali, James Hays, Jacob D. Abernethy, and Zsolt Kira. On Convergence and Stability of GANs. arXiv preprint arXiv 1705.07215, 2018.   
+[15] Zhiming Zhou, Jiadong Liang, Yuxuan Song, Lantao Yu, Hongwei Wang, Weinan Zhang, Yong Yu, and Zhihua Zhang. Lipschitz generative adversarial nets. In Proceedings of the International Conference on Machine Learning (ICML), pages 7584–7593, 2019.   
+[16] Han Zhang, Zizhao Zhang, Augustus Odena, and Honglak Lee. Consistency Regularization for Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2020.   
+[17] Zhengli Zhao, Sameer Singh, Honglak Lee, Zizhao Zhang, Augustus Odena, and Han Zhang. Improved Consistency Regularization for GANs. arXiv preprint arXiv 2002.04724, 2020.   
+[18] Augustus Odena, Christopher Olah, and Jonathon Shlens. Conditional Image Synthesis with Auxiliary Classifier GANs. In Proceedings of the International Conference on Machine Learning (ICML), pages 2642–2651, 2017.   
+[19] Takeru Miyato and Masanori Koyama. cGANs with Projection Discriminator. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   
+[20] Mingming Gong, Yanwu Xu, Chunyuan Li, Kun Zhang, and Kayhan Batmanghelich. Twin Auxilary Classifiers GAN. In Advances in Neural Information Processing Systems (NeurIPS), 2019.   
+[21] Aliaksandr Siarohin, Enver Sangineto, and Nicu Sebe. Whitening and Coloring Batch Transform for GANs. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.   
+[22] Ming-Yu Liu, Xun Huang, Arun Mallya, Tero Karras, Timo Aila, Jaakko Lehtinen, and Jan Kautz. FewShot Unsupervised Image-to-Image Translation. In Proceedings of the International Conference on Computer Vision (ICCV), 2019.   
+[23] Minguk Kang and Jaesik Park. ContraGAN: Contrastive Learning for Conditional Image Generation. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   
+[24] Woohyeon Shim and Minsu Cho. CircleGAN: Generative Adversarial Learning across Spherical Circles. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   
+[25] Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. Omni-GAN: On the Secrets of cGANs and Beyond. arXiv preprint arXiv:2011.13074, 2020.   
+[26] Ilya Kavalerov, Wojciech Czaja, and Rama Chellappa. A multi-class hinge loss for conditional gans. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), 2021.   
+[27] Liang Hou, Qi Cao, Huawei Shen, and Xueqi Cheng. cGANs with Auxiliary Discriminative Classifier. arXiv preprint arXiv:2107.10060, 2021.   
+[28] Han, Ligong and Min, Martin Renqiang and Stathopoulos, Anastasis and Tian, Yu and Gao, Ruijiang and Kadav, Asim and Metaxas, Dimitris N. Dual Projection Generative Adversarial Networks for Conditional Image Generation. In Proceedings of the International Conference on Computer Vision (ICCV), 2021.   
+[29] Han Zhang, Ian Goodfellow, Dimitris Metaxas, and Augustus Odena. Self-Attention Generative Adversarial Networks. In Proceedings of the International Conference on Machine Learning (ICML), pages 7354–7363, 2019.   
+[30] Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images. PhD thesis, University of Toronto, 2012.   
+[31] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Fei-Fei Li. ImageNet: A large-scale hierarchical image database. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 248–255, 2009.   
+[32] Johnson et al. Tiny ImageNet Visual Recognition Challenge. https://tiny-imagenet.herokuapp. com.   
+[33] P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical report, California Institute of Technology, 2010.   
+[34] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In Advances in Neural Information Processing Systems (NeurIPS), pages 6626–6637, 2017.   
+[35] Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. In Advances in Neural Information Processing Systems (NeurIPS), pages 271–279, 2016.   
+[36] Akash Srivastava, Lazar Valkov, Chris Russell, Michael U Gutmann, and Charles A Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In Advances in Neural Information Processing Systems (NeurIPS), 2017.   
+[37] Mehdi Mirza and Simon Osindero. Conditional Generative Adversarial Nets. arXiv preprint arXiv 1411.1784, 2014.   
+[38] Zhiming Zhou, Han Cai, Shu Rong, Yuxuan Song, Kan Ren, Weinan Zhang, Jun Wang, and Yong Yu. Activation Maximization Generative Adversarial Nets. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   
+[39] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey E. Hinton. A Simple Framework for Contrastive Learning of Visual Representations. arXiv preprint arXiv 2002.05709, 2020.   
+[40] Minguk Kang and Jaesik Park. Pytorch-studiogan. https://github.com/POSTECH-CVLab/ PyTorch-StudioGAN, 2020.   
+[41] Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. StarGAN v2: Diverse Image Synthesis for Multiple Domains. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   
+[42] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, and Xi Chen. Improved Techniques for Training GANs. In Advances in Neural Information Processing Systems (NeurIPS), pages 2234–2242, 2016.   
+[43] Mehdi SM Sajjadi, Olivier Bachem, Mario Lucic, Olivier Bousquet, and Sylvain Gelly. Assessing generative models via precision and recall. In Advances in Neural Information Processing Systems (NeurIPS), 2018.   
+[44] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur. A Learned Representation For Artistic Style. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.   
+[45] Harm de Vries, Florian Strub, Jeremie Mary, Hugo Larochelle, Olivier Pietquin, and Aaron C Courville. Modulating early visual processing by language. In Advances in Neural Information Processing Systems (NeurIPS), pages 6594–6604, 2017.   
+[46] Jae Hyun Lim and Jong Chul Ye. Geometric GAN. arXiv preprint arXiv 1705.02894, 2017.   
+[47] Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Proceedings of the International Conference on Machine Learning (ICML), 2021.   
+[48] Dongjun Kim, Seungjae Shin, Kyungwoo Song, Wanmo Kang, and Il-Chul Moon. Score Matching Model for Unbounded Data Score. arXiv preprint arXiv:2106.05527, 2021.   
+[49] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou. Arcface: Additive angular margin loss for deep face recognition. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The main contributions of our paper can be summarized into two points: (1) analyzing why ACGAN training becomes unstable as the number of classes in the training dataset increases, (2) developing a new cGAN model: the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). Both claims are well reflected in the abstract and introduction.   
+(b) Did you describe the limitations of your work? [Yes] The proposed ReACGAN generates high-quality images using informative datato-data and data-to-class relationships in the dataset. However, it requires additional hyperparameter tuning for a temperature and margins values. Also, we analyze the weakness of ReACGAN in Appendix F.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Generative Adversarial Networks can be used to synthesize indistinguishable images from the real. This might cause fake news, sexual harnesses, and hacking machine vision applications. Although the potential negative social impacts are not discussed in our main paper, we deal with this issue in Appendix G.   
+(d) Have you read the ethics review and ensured that your paper conforms to them? [Yes] We carefully read the ethics review guidelines and comply with them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all assumptions that arise from the theoretical results, i.e., Property 1-4 in Sec. 3.3 in the main paper.   
+(b) Did you include complete proofs of all theoretical results? [Yes] We provide the complete proofs in Appendix D.
+
+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 release model weights and the software package used in our paper at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We state details for training models in Sec. 4.2 in the main paper, which include the evaluation procedure, hyperparameter selection, and controlled environment for a fair comparison. In addition, we add how we pre-process training datasets in the Appendix C.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For reliable evaluation, we run all the experiments three times with random seeds. We provide standard deviations of Tables 1, 3, 4, 6 in Appendix I.   
+(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 include the total amount of compute and the type of resources in Appendix H.
+
+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] We cite papers and GitHub repository that we used for our experiments in Sec. 4.2.   
+(b) Did you mention the license of the assets? [No] We utilize five datasets to validate our method: CIFAR10, Tiny-ImageNet, CUB200, ImageNet, and AFHQ. These are widely used public datasets, and we cite them in the paper. However, we could not find any license information for CIFAR10, Tiny-ImageNet, and CUB200. The ImageNet and AFHQ datasets can be used for noncommercial purposes only. In addition, we mention the license of PyTorch-StudioGAN in the main paper, which is used for our experiments.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide the URL of our source code in the abstract.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] The CIFAR10, Tiny-ImageNet, CUB200, ImageNet, and AFHQ datasets are properly curated, and they are popular datasets for research purpose. To our best knowledge, there is no inappropriate contents in the datasets.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] The benchmark datasets we used are widely adopted by others and do not contain any harmful or troublesome contents.
+
+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] Our research has no relevance to the crowdsourcing or human subjects.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] Our research has no relevance to the crowdsourcing or human subjects.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Our research has no relevance to the crowdsourcing or human subjects.

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+        "text": "Conditional Generative Adversarial Networks (cGAN) generate realistic images by incorporating class information into GAN. While one of the most popular cGANs is an auxiliary classifier GAN with softmax cross-entropy loss (ACGAN), it is widely known that training ACGAN is challenging as the number of classes in the dataset increases. ACGAN also tends to generate easily classifiable samples with a lack of diversity. In this paper, we introduce two cures for ACGAN. First, we identify that gradient exploding in the classifier can cause an undesirable collapse in early training, and projecting input vectors onto a unit hypersphere can resolve the problem. Second, we propose the Data-to-Data Cross-Entropy loss (D2D-CE) to exploit relational information in the class-labeled dataset. On this foundation, we propose the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results show that ReACGAN achieves state-of-the-art generation results on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet datasets. We also verify that ReACGAN benefits from differentiable augmentations and that D2D-CE harmonizes with StyleGAN2 architecture. Model weights and a software package that provides implementations of representative cGANs and all experiments in our paper are available at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Generative Adversarial Networks (GAN) [1] are known for the forefront approach to generating high-fidelity images of diverse categories [2, 3, 4, 5, 6, 7, 8, 9]. Behind the sensational generation ability of GANs, there has been tremendous effort to develop adversarial objectives free from the vanishing gradient problem [10, 11, 12], regularizations for stabilizing adversarial training [11, 13, 14, 3, 15, 16, 17], and conditioning techniques to support the adversarial training using category information of the dataset [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Subsequently, the conditioning techniques have become the de facto standard for high-quality image generation. The models with the conditioning methods are called conditional Generative Adversarial Networks (cGAN), and cGANs can be divided into two groups depending on the discriminator’s conditioning way: classifier-based GANs [18, 20, 23, 25, 27] and projection-based GANs [19, 3, 4, 28]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The classifier-based GANs facilitate an auxiliary classifier to generate class-specific images by penalizing the generator if the synthesized images are not consistent with the conditioned labels. ACGAN [18] has been one of the widely used classifier-based GANs for its simple design and satisfactory generation performance. While ACGAN can exploit class information by pushing and pulling classifier’s weights (proxies) against image embeddings [23], it is well known that ACGAN training is prone to collapsing at the early stage of training as the number of classes increases [19, 23, 25, 27]. In addition, the generator of ACGAN tends to generate easily classifiable images at the cost of reduced diversity [18, 19, 27]. Projection-based GANs, on the other hand, have shown cutting-edge generation results on datasets with a large number of categories. SNGAN [3], ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a77f02b277b4a9b1554bdeb714cab48b71615b6d03f8135fb468d91f2cc6deb5.jpg",
+        "image_caption": [
+            "Figure 1: Schematics that depict how cGANs perform conditioning. The red color means positive samples/proxy, and the blue color indicates negative samples/proxies. Arrows represent push-pull forces based on the reference sample. The length of an arrow indicates the magnitude of the force. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "SAGAN [29], and BigGAN [4] are representatives in this family and can generate realistic images on CIFAR10 [30] and ImageNet [31] datasets. However, projection-based GANs only consider a pairwise relationship between an image and its label proxy (data-to-class relationships). As the result, the projection-based GANs can miss an additional opportunity to consider relation information between data instances (data-to-data relationships) as discovered by [23]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we analyze why ACGAN training becomes unstable as the number of classes increases and propose remedies for (1) the instability and (2) the relatively poor generation performance of ACGAN compared with the projection-based models. First, we begin by analytically deriving the gradient of the softmax cross-entropy loss used in ACGAN. By examining the exact values of analytic gradients, we discover that the unboundedness of input feature vectors and poor classification performance in the early training stage can cause an undesirable gradient exploding problem. Second, with alleviating the instability, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) using the Data-to-Data Cross-Entropy loss (D2D-CE). ReACGAN projects image embeddings and proxies onto a unit hypersphere and computes similarities for data-to-data and data-to-class consideration. Additionally, we introduce two margin values for intra-class variations and inter-class separability. In this way, ReACGAN overcomes the training instability and can exploit additional supervisory signals by explicitly considering data-to-class and data-to-data relationships, and also by implicitly looking at class-to-class relationships in the same mini-batch. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To validate our model, we conduct image generation experiments on CIFAR10 [30], TinyImageNet [32], CUB200 [33], and ImageNet [31] datasets. Through extensive experiments, we demonstrate that ReACGAN beats both the classifier-based and projection-based GANs, improving over the state of the art by $2 . 5 \\%$ , $1 5 . 8 \\%$ , $5 . 1 \\%$ , and $1 4 . 5 \\%$ in terms of Fréchet Inception Distance (FID) [34] on the four datasets, respectively. We also verify that ReACGAN benefits from consistency regularization [16] and differentiable augmentations [9, 8] for limited data training. Finally, we confirm that D2D-CE harmonizes with the StyleGAN2 architecture [7]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Background: Generative Adversarial Networks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Network (GAN) [1] is an implicit generative model that aims to generate a sample indistinguishable from the real. GAN consists of two networks: a Generator $G : { \\mathcal { Z } } \\longrightarrow { \\mathcal { X } }$ that tries to map a latent variable $\\mathbf { z } \\sim p ( \\mathbf { z } )$ into the real data space $\\mathcal { X }$ and a Discriminator $D : \\mathcal { X } \\longrightarrow$ [0, 1] that strives to discriminate whether a given sample $\\mathbf { x }$ is from the real data distribution $p _ { \\mathrm { r e a l } } ( \\mathbf { x } )$ or from the implicit distribution $p _ { \\mathrm { g e n } } ( G ( \\mathbf { z } ) )$ derived from the generator $G ( \\mathbf { z } )$ . The objective of a vanilla GAN [1] can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af21913ce94f55d589da37b72bcd0d6981cac1e7074fd7e7c6991e20448adb0b.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\mathbf { x } \\sim p ( \\mathbf { x } ) } [ \\log ( D ( \\mathbf { x } ) ) ] + \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } [ \\log ( 1 - D ( G ( \\mathbf { z } ) ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While GANs have shown impressive results in the image generation task [2, 35, 11, 13], training GANs often ends up encountering a mode-collapse problem [36, 11, 12]. As one of the prescriptions for stabilizing and reinforcing GANs, training GANs with categorical information, named conditional Generative Adversarial Networks (cGAN), is suggested [37, 18, 19]. Depending on the presence of explicit classification losses, cGAN can be divided into two groups: Classifier-based GANs [18, 20, 23, 25, 27] and Projection-based GANs [19, 3, 4, 28]. One of the widely used classifier-based GANs is ACGAN [18], and ACGAN utilizes softmax cross-entropy loss to perform classification task with adversarial training. Although ACGAN has shown satisfactory generation results, training ACGAN becomes unstable as the number of classes in the training dataset increases [19, 23, 25, 27]. Besides, ACGAN tends to generate easily classifiable images at the cost of limited diversity [18, 19, 27]. To alleviate those problems, Zhou et al. [38] have proposed performing adversarial training on the classifier. Gong et al. [20] have introduced an additional classifier to eliminate a conditional entropy minimization process in the adversarial training. However, ACGAN training still suffers from the early-training collapse issue and the reduced diversity problem when trained on datasets with a large number of class categories, such as Tiny-ImageNet [32] and ImageNet [31]. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In these circumstances, Miyato et al. [19] have proposed a projection discriminator for cGANs and have shown significant improvement in generating the ImageNet dataset. Motivated by the promising result of the projection discriminator, many projection-based GANs [3, 29, 4, 16, 6, 17] have been proposed and become the standard for conditional image generation. In this paper, we revisit ACGAN and unveil why ACGAN training is so unstable. Coping with the instability, we propose the Rebooted Auxiliary Classifier GANs (ReACGAN) for high-quality and diverse image generation. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Rebooting Auxiliary Classifier GANs ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Feature Normalization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To uncover a nuisance that can cause the instability of ACGAN, we start by analytically deriving the gradients of weight vectors in the softmax classifier. Let the part of the discriminator before the fully connected layer be a Feature extractor $F : \\mathcal { X } \\longrightarrow \\mathcal { F } \\in \\bar { \\mathbb { R } ^ { d } }$ and let Classifier $C : { \\mathcal { F } } \\longrightarrow \\mathbb { R } ^ { c }$ be a single fully connected layer parameterized by $\\pmb { W } = [ \\pmb { w } _ { 1 } , . . . , \\pmb { w } _ { c } ] \\in \\mathbb { R } ^ { d \\times c }$ , where $c$ denotes the number of classess. We sample training images $\\pmb { X } = \\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { N } \\}$ and integer labels ${ \\textbf { 3 } } =$ $\\{ y _ { 1 } , . . . , y _ { N } \\}$ from the joint distribution $p ( \\mathbf { x } , \\mathbf { y } )$ . Using the notations above, we can express the empirical cross-entropy loss used in ACGAN [18] as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3cde793fe61034f79c6982cfce3d15f0a2d0f599275318cc5332121eac04663.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C E } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Big ( \\frac { \\exp \\left( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { y _ { i } } \\right) } { \\sum _ { j = 1 } ^ { c } \\exp \\left( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { j } \\right) } \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on Eq. (3), we can derive the derivative of the cross-entropy loss, w.r.t ${ \\pmb w } _ { k \\in \\{ 1 , \\dots , c \\} }$ as follows: ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/634f34fe1b9cb8058559e57be9cafdd69cbbcbcc35cc37fe0b2346817ebaa3ee.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { L } _ { \\mathrm { C E } } } { \\partial \\boldsymbol { w } _ { k } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\bigg \\{ F ( \\boldsymbol { x } _ { i } ) \\bigg ( \\mathbf { 1 } _ { y _ { i } = k } - p _ { i , k } \\bigg ) \\bigg \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            630,
+            650,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { 1 } _ { \\mathit { y } _ { i } = k }$ is an indicator function that will output 1 if $y _ { i } = k$ is satisfied, and $p _ { i , k }$ is a class probability that represents the probability that $i$ -th sample belongs to class $k$ , mathematically $\\frac { \\exp { ( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { k } ) } } { \\sum _ { j = 1 } ^ { c } \\exp { ( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { j } ) } }$ . The equation above implies that the norm of the gradient of the softmax cross-entropy loss is coupled with the norms and directions of each input feature map $F ( { \\pmb x } _ { i } )$ and the class probabilities. In the early training stage, the classifier is prone to making incorrect predictions, resulting in low probabilities. This phenomenon occurs more frequently as the number of categories in the dataset increases. As the result, the norm of the gradient $| \\frac { \\partial \\hat { \\mathcal { L } } _ { \\mathrm { C E } } } { \\partial w _ { k } } |$ begins to explode as the vector $F ( { \\pmb x } _ { i } )$ stretches out to ${ \\pmb w } _ { k }$ direction but being located close to the the other vectors ${ \\pmb w } _ { j \\in \\{ 1 , . . . , c \\} \\backslash \\{ k \\} }$ . This often breaks the balance between adversarial learning and classifier training, leading to an early-training collapse. Once the early-training collapse occurs, ACGAN training concentrates on classifying categories of images instead of discriminating the authenticity of given samples. We experimentally demonstrate that the average norm of ACGAN’s input feature maps increases as the training progresses (Fig. 2a). Accordingly, the average norm of the gradients increases sharply at the early training stage and decreases with the high class probabilities of the classifier (Fig. 2b, Fig. A3 in Appendix). While the average norm of gradients decreases at some point, the FID value of ACGAN does not decrease, implying the collapse of ACGAN training (Fig. 2c). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/41f752bd054c08034409d7f5b84c86a0197793ecc9cb9dcf2b22365737c55349.jpg",
+        "image_caption": [
+            "Figure 2: Merits of integrating feature normalization and data-to-data relationship consideration. All experiments are conducted on Tiny-ImageNet [32] dataset. (a) Average norms of input feature maps $\\bar { F } ( \\cdot )$ , (b) average norms of gradients of classification losses, and (c) trends of FID scores. Compared with ACGAN [18], the proposed ReACGAN does not experience the training collapse problem caused by excessively large norms of feature maps and gradients at the early training stage. In addition, ReACGAN can converge to better equilibrium by considering data-to-data relationships with easy positive and negative sample suppression. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As one of the remedies for the gradient exploding problem, we find that simply normalizing the feature embeddings onto a unit hypersphere $\\frac { F ( { \\pmb x } _ { i } ) } { | | F ( { \\pmb x } _ { i } ) | | }$ effectively resolves ACGAN’s early-training collapse (see Fig. 2). The motivation is that normalizing the features onto the hypersphere makes the norms of feature maps equal to 1.0. Thus, the discriminator does not experience the gradient exploding problem. From the next section, we will deploy a linear projection layer $P$ on the feature extractor $F$ . And, we will normalize both the embeddings from the projection layer and the weight vectors $( \\pmb { w } _ { 1 } , . . . , \\pmb { w } _ { c } )$ in the classifier since normalizing both the embeddings and the weight vectors does not degrade image generation performance. We denote the normalized embedding P (F (xi))||P (F (xi))|| as $f _ { i }$ and the normalized weight vector $\\frac { { \\pmb w } _ { y _ { i } } } { | | { \\pmb w } _ { y _ { i } } | | }$ as vyi . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Data-to-Data Cross-Entropy Loss (D2D-CE) ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We expand the feature normalized softmax cross-entropy loss described in Sec. 3.1 to the Data-to-Data Cross-Entropy (D2D-CE) loss. The motivations are summarized into two points: (1) replacing datato-class similarities in the denominator of Eq. (3) with data-to-data similarities and (2) introducing two margin values to the modified softmax cross-entropy loss. We expect that point (1) will encourage the feature extractor to consider data-to-class as well as data-to-data relationships, and that point (2) will guarantee inter-class separability and intra-class variations in the feature space while preventing ineffective gradient updates induced by easy negative and positive samples. To develop the feature normalized cross-entropy loss into D2D-CE, we replace the similarities between a sample embedding and all proxies except for the positive one in the denominator, $\\begin{array} { r } { \\sum _ { j \\in \\{ 1 , \\dots , c \\} \\backslash \\{ y _ { i } \\} } \\exp \\left( \\mathbf { { f } } _ { i } ^ { \\top } \\mathbf { { v } } _ { j } \\right) } \\end{array}$ with similarities between negative samples in the same mini-batch. The modified cross-entropy loss can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2083ef980138ec9937eb51f9829dbe3f337cc0b83827c7b723f912acedff8847.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C E } } ^ { ' } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Bigg ( \\frac { \\exp { ( f _ { i } ^ { \\top } v _ { y _ { i } } / \\tau ) } } { \\exp { ( f _ { i } ^ { \\top } v _ { y _ { i } } / \\tau ) } + \\sum _ { j \\in N ( i ) } \\exp { ( f _ { i } ^ { \\top } f _ { j } / \\tau ) } } \\Bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            702,
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+            746
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a temperature, and $\\mathcal { N } ( i )$ is the set of indices that point locations of the negative samples whose labels are different from the reference label $\\pmb { v } _ { y _ { i } }$ in the mini-batch. The self-similarity matrix of samples in the mini-batch is used to calculate the similarities between negative samples $\\pmb { f } _ { i } ^ { \\top } \\pmb { f } _ { j \\in \\mathcal { N } ( i ) }$ with a false negative mask (see Fig. 3). Thus, Eq. (5) enables the discriminator to contrastively compare visual differences between multiple images and can supply more informative supervision for image conditioning. Finally, we introduce two margin hyperparameters to $\\mathcal { L } _ { \\mathrm { C E } } ^ { ' }$ and name it Data-to-Data Cross-Entropy loss (D2D-CE). The proposed D2D-CE can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f95d96f99ec92483130927216fa5f895d898e210867b75a033db7d5a3213348.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { D 2 D - C E } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Bigg ( \\frac { \\exp \\left( [ f _ { i } ^ { \\top } v _ { y _ { i } } - m _ { p } ] _ { - } / \\tau \\right) } { \\exp \\left( [ f _ { i } ^ { \\top } v _ { y _ { i } } - m _ { p } ] _ { - } / \\tau \\right) + \\sum _ { j \\in N ( i ) } \\exp \\left( [ f _ { i } ^ { \\top } f _ { j } - m _ { n } ] _ { + } / \\tau \\right) } \\Bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/ee7bad162e1754a63a5d7f7b5a7a8d9d8e3e367c1fcd2cc101b8a2534e6e145d.jpg",
+        "image_caption": [
+            "Figure 3: Overview of ReACGAN. ReACGAN performs adversarial training using the loss ${ \\mathcal { L } } _ { \\mathrm { A d v } }$ . At the same time, ReACGAN tries to minimize data-to-data cross-entropy loss (D2D-CE) on the linear head $( P )$ to exploit relational information in the labeled dataset. $^ { \\circ }$ means element-wise product. Note that False negatives mask operates on the similarity matrix between two batches of sample embeddings to compute the similarities between negative samples in the denominator of Eq. (6). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            93,
+            812,
+            332
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $m _ { p }$ is a margin for suppressing a high similarity value between a reference sample and its corresponding proxy (easy positive), $m _ { n }$ is a margin for suppressing low similarity values between negatives samples (easy negatives). $[ \\cdot ] _ { - }$ and $[ \\cdot ] _ { + }$ denote $\\operatorname* { m i n } ( \\cdot , 0 )$ and $\\operatorname* { m a x } ( \\cdot , 0 )$ functions, respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Useful Properties of Data-to-Data Cross-Entropy Loss ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            492,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we explain four useful properties of D2D-CE. Let $s _ { q }$ be a similarity between the normalized embedding $f _ { q }$ and the corresponding normalized proxy ${ \\pmb v } _ { y _ { q } }$ , $s _ { q , r }$ be a similarity between $f _ { q }$ and $f _ { r }$ , and $a$ and $\\mathit { b }$ be arbitrary indices of negative samples, i.e., $a , b \\in \\mathcal { N } ( q )$ . Then the properties of D2D-CE can be summarized as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/284973472b7cc1ba4ea65614768295afd00d54dfd0ba532f646e48451266f16e.jpg",
+        "image_caption": [
+            "Figure 4: Graph showing regions where hard negative mining, easy positive and negative suppression, and no gradient update occur. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Property 1. Hard negative mining. If the value of $s _ { q , a }$ is greater than $s _ { q , b }$ , the derivative of LD2D-CE w.r.t $s _ { q , a }$ is greater than or equal to the derivative $w . r . t \\ s _ { q , b }$ ; that is $\\begin{array} { r } { \\frac { \\partial \\mathcal { L } _ { D 2 D - C E } } { \\partial s _ { q , a } } \\geq \\frac { \\partial \\mathcal { L } _ { D 2 D - C E } } { \\partial s _ { q , b } } \\geq 0 . } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "Property 2. Easy positive suppression. If $s _ { q } \\mathrm { ~ - ~ }$ $m _ { p } \\geq 0$ , the derivative of $\\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q }$ is 0. ",
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+    {
+        "type": "text",
+        "text": "Property 3. Easy negative suppression. If $s _ { q , r } - m _ { n } \\leq 0 ,$ , the derivative of $\\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q , r }$ is 0. Property 4. If $s _ { q } - m _ { p } \\ge 0$ and $s _ { q , r } - m _ { n } \\leq 0$ are satisfied, $\\mathcal { L } _ { D 2 D - C E }$ has the global minima of $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\left( 1 + \\bar { | \\mathcal { N } ( i ) | } \\right) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "We put proofs of the above properties in Appendix D. Property 1 indicates that D2D-CE implicitly conducts hard negative mining and benefits from comparing samples with each other. Also, Properties 2 and 3 imply that samples will not affect gradient updates if the samples are trained sufficiently. Consequently, the classifier concentrates on pushing and pulling hard negative and hard positive examples without being dominated by easy negative and positive samples. ",
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+    {
+        "type": "text",
+        "text": "3.4 Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the proposed D2D-CE objective, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). As ACGAN does, ReACGAN jointly optimizes an adversarial loss and the classification objective (D2D-CE). Specifically, the discriminator, which consists of the feature extractor, adversarial head, and linear head, of ReACGAN tries to discriminate whether a given image is sampled from the real distribution or not. At the same time, the discriminator tries to maximize similarities between the reference samples and corresponding proxies while minimizing similarities between negative samples using real images and D2D-CE loss. After updating the discriminator a predetermined number of times, the generator strives to deceive the discriminator by generating well-conditioned images that will output a low D2D-CE value. By alternatively training the discriminator and generator until convergence, ReACGAN generates high-quality images of diverse categories without early-training collapse. We attach the algorithm table in Appendix A. ",
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+        "type": "text",
+        "text": "Differences between ReACGAN and ContraGAN. The authors of ContraGAN [23] propose a conditional contrastive loss (2C loss) to cover data-to-data relationships when training cGANs. The main differences between 2C loss and D2D-CE objective are summed up into three points: (1) while 2C loss is derived from NT-Xent loss [39], which is popularly used in the field of the self-supervised learning, our D2D-CE is developed to resolve the early-training collapsing problem and the poor generation results of ACGAN, (2) 2C loss holds false-negative samples in the denominator, and they can cause unexpected positive repulsion forces, and (3) 2C loss contains the similarities between all positive samples in the numerator, and they give rise to large gradients on pulling easy positive samples. Consequently, GANs with 2C loss tend to synthesize images of unintended classes as reported in the author’s software document [40]. However, D2D-CE does not contain the false negatives in the denominator and considers only the similarity between a sample and its proxy in the numerator. Therefore, ReACGAN is free from the undesirable repelling forces and does not conduct unnecessary easy positive mining. More detailed explanations are attached in Appendix F. ",
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+        "type": "text",
+        "text": "Consistency Regularization and D2D-CE Loss. Zhang et al. [16] propose a consistency regularization to force the discriminator to make consistent predictions if given two images are visually close to each other. They create a visually similar image pair by augmenting a reference image with pre-defined augmentations. After that, they let the discriminator minimize L2 distance between the logit of the reference image and the logit of the augmented counterpart. While ReACGAN locates an image embedding nearby its corresponding proxy but far apart multiple image embeddings of different classes, consistency regularization only pulls a reference and the augmented image towards each other. Since consistency regularization and D2D-CE can be applied together, we will show that ReACGAN benefits from consistency regularization in the experiments section (Table 1). ",
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+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Datasets and Evaluation Metrics ",
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+        "text": "To verify the effectiveness of ReACGAN, we conduct conditional image generation experiments using five datasets: CIFAR10 [30], Tiny-ImageNet [32], CUB200 [33], ImageNet [31], and AFHQ [41] datasets and four evaluation metrics: Inception Score (IS) [42], Fréchet Inception Distance (FID) [34], and $\\mathrm { F _ { 0 . 1 2 5 } }$ (Precision) and $\\mathrm { F } _ { 8 }$ (Recall) [43]. The details on the training datasets are in Appendix C.1. ",
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+        "type": "text",
+        "text": "Inception Score (IS) [42] and Fréchet Inception Distance (FID) [34] are widely used metrics for evaluating generative models. We utilize IS and FID together because some studies [4, 6, 25] have shown that IS has a tendency to measure the fidelity of images better while FID tends to weight capturing the diversity of images. ",
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+        "type": "text",
+        "text": "Precision $\\bf ( F _ { 0 . 1 2 5 } )$ and Recall $\\bf ( F _ { 8 } )$ [43] are metrics for estimating precision and recall of the approximated distribution $p ( G ( \\mathbf { z } ) )$ against the true data distribution $p ( \\mathbf { x } )$ . Instead of evaluating generative models using one-dimensional scores, such as IS and FID, Sajjadi et al. [43] have suggested using a two-dimensional score $\\mathrm { F _ { 0 . 1 2 5 } }$ and $\\mathrm { F } _ { 8 }$ that can quantify how precisely the generated images are and how well the generated images cover the reference distribution. ",
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+    {
+        "type": "text",
+        "text": "4.2 Experimental Details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The implementation of ReACGAN basically follows details of PyTorch-StudioGAN library [40]1 that supports various experimental setups from ACGAN [18] to StyleGAN2 [7] $^ +$ ADA [8] with different scales of datasets [30, 32, 33, 31, 41] and architectures [2, 13, 4, 7]. For a fair comparison, we use the same backbones over all baselines, except otherwise noted, for both the discriminator and generator. For stable training, we apply spectral normalization (SN) [3] to the generator and discriminator except for experiments using SNGAN (in this case, we apply SN to the discriminator only) and StyleGAN2 [7]. We also use the same conditioning method for generators with conditional batch normalization (cBN) [44, 45, 19]. This allows us to investigate solely the conditioning methods of the discriminator, which are the main interest of our paper. If not specified, we use hinge loss [46] as a default for the adversarial loss. ",
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+    {
+        "type": "table",
+        "img_path": "images/4b04bdeeb55f3430ef72f3f3db9f9e61e72c6ba6199d770fd52b912b38354bce.jpg",
+        "table_caption": [
+            "Table 1: Comparison with classifier-based GANs [18, 23] and projection-based GANs [3, 4, 29] on CIFAR10 [30], Tiny-ImageNet [32], and CUB200 [33] datasets using IS [42], FID [34], $\\mathrm { F } _ { 0 . 1 2 5 }$ , and $\\mathrm { F } _ { 8 }$ [43] metrics. For baselines, both the numbers from the cited paper (denoted as \\* in method) and from our experiments using StudioGAN library [40] are reported. The numbers in bold-faced denote the best performance and in underline indicate the values are in one standard deviation from the best. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">CIFAR10 [30]</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td></tr><tr><td>SNGAN*[3]</td><td>8.22</td><td>21.7</td><td>=</td><td>-</td><td>1</td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>=</td><td>-</td></tr><tr><td>BigGAN*[4]</td><td>9.22</td><td>14.73</td><td>=</td><td>-</td><td>=</td><td>-</td><td></td><td>=</td><td>-</td><td></td><td>-</td><td>1</td></tr><tr><td>ContraGAN* [23]</td><td>1</td><td>10.60</td><td></td><td></td><td></td><td>29.49</td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>ACGAN[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>SNGAN [3]</td><td>8.67</td><td>13.33</td><td>0.985</td><td>0.976</td><td>8.71</td><td>51.15</td><td>0.900</td><td>0.702</td><td>5.41</td><td>47.75</td><td>0.754</td><td>0.912</td></tr><tr><td>SAGAN [29]</td><td>8.66</td><td>14.31</td><td>0.983</td><td>0.973</td><td>8.74</td><td>49.90</td><td>0.872</td><td>0.712</td><td>5.48</td><td>54.29</td><td>0.728</td><td>0.882</td></tr><tr><td>BigGAN[4]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>ContraGAN [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>ReACGAN</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td></td><td>14.06 27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr><tr><td>BigGAN +CR*[16]</td><td>-</td><td>11.48</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BigGAN [4]+ CR[16]</td><td>9.97</td><td>7.18</td><td>0.995</td><td>0.993</td><td></td><td>15.94 19.96</td><td>0.972</td><td>0.950 5.14</td><td></td><td>11.97</td><td>0.978</td><td>0.981</td></tr><tr><td>ContraGAN [23] + CR[16]</td><td>9.59</td><td>8.55</td><td>0.992</td><td>0.972</td><td>15.81</td><td>19.21</td><td>0.983</td><td>0.941</td><td>4.90</td><td>11.08</td><td>0.984</td><td>0.967</td></tr><tr><td>ReACGAN + CR [16]</td><td>10.11</td><td>7.20</td><td>0.996</td><td>0.994 16.56</td><td></td><td>19.69</td><td>0.984</td><td>0.940</td><td>4.87</td><td>10.72</td><td>0.985</td><td>0.971</td></tr><tr><td>BigGAN + DiffAug*[9]</td><td>9.17</td><td>8.49</td><td></td><td></td><td>=</td><td></td><td>=</td><td>=</td><td>=</td><td></td><td></td><td></td></tr><tr><td>BigGAN [4] + DiffAug [9]</td><td>9.94</td><td>7.17</td><td>0.995</td><td>0.992</td><td></td><td>18.0815.70</td><td>0.980</td><td>0.972</td><td>5.53</td><td>12.15</td><td>0.967</td><td>0.981</td></tr><tr><td>ContraGAN [23] + DiffAug [9]</td><td>9.95</td><td>7.27</td><td>0.995</td><td>0.992</td><td>18.20</td><td>15.40</td><td>0.986</td><td>0.963</td><td>5.39</td><td>11.02</td><td>0.978</td><td>0.970</td></tr><tr><td>ReACGAN + DiffAug [9]</td><td>10.22</td><td>6.79</td><td>0.996</td><td>0.993</td><td>20.60</td><td>14.25</td><td>0.988</td><td>0.972</td><td>5.22</td><td>9.27</td><td>0.985</td><td>0.983</td></tr><tr><td>Real Data</td><td colspan=\"2\">11.54</td><td colspan=\"2\">34.11</td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\">5.49</td><td colspan=\"2\"></td></tr></table>",
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+        "type": "text",
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+        "type": "text",
+        "text": "Before conducting main experiments, we perform hyperparameter search with candidates of a temperature $\\tau \\in \\{ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \\}$ and a positive margin $m _ { p } \\in \\ \\{ 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 8 , \\bar { 1 . 0 } \\}$ . We set a negative margin $m _ { n }$ as $1 \\ : - \\ : m _ { p }$ to reduce search time. Through extensive experiments with 3 runs per each setting, we select $\\tau$ with $\\{ 0 . 5 , 0 . 7 5 , 0 . 2 5 , 0 . 5 , 0 . 2 5 \\}$ and $m _ { p }$ with $\\{ 0 . 9 8 , 1 . 0 , 0 . 9 5 , 0 . 9 \\bar { 8 } , 0 . 9 0 \\}$ for CIFAR10, Tiny-ImageNet, CUB200, ImageNet 256 B.S., and ImageNet 2048 B.S. experiments, respectively. A low temperature seems to work well on fine-grained image generation tasks, but generally, ReACGAN is robust to the choice of hyperparameters. The results of the hyperparameter search and other hyperparameter setups are provided in Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": "We evaluate all methods through the same protocol of [9, 16, 17], which uses the same amounts of generated images from the reference split specialized for each dataset.2 Besides, we run all the experiments three times with random seeds and report the averaged best performances for reliable evaluation with the lone exception of ImageNet and StyleGAN2 related experiments. Please refer to Appendix C.2 for other experimental details. The numbers in bold-faced denote the best performance and in underline indicate that the values are in one standard deviation from the best. ",
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+    {
+        "type": "text",
+        "text": "4.3 Evaluation Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Comparison with Other cGANs. We compare ReACGAN with previous state-of-the-art cGANs in Tables 1 and 2. We employ the implementations of GANs in PyTorch-StudioGAN library as it provides improved results on standard benchmark datasets [30, 31]. For a fair comparison, we provide results from each original paper (denoted as \\* in method) as well as those from StudioGAN library [40]. We also conduct experiments with popular augmentation-based methods: consistency regularization (CR) [16] and differentiable augmentation (DiffAug) [9]. ",
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+    {
+        "type": "table",
+        "img_path": "images/d5bbe5ee1dbe2195a327ca572b5f43ba744dd66b42d23636d28e4a2d3dd61fd8.jpg",
+        "table_caption": [
+            "Table 2: Experiments using ImageNet [31] dataset. B.S. means batch size for training. (Left): Comparisons with previous cGAN approaches. (Right): Learning curves of BigGAN [4], ContraGAN [23], and ReACGAN (ours) which are trained using the batch size of 256. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"5\">ImageNet [31]</td></tr><tr><td>IS↑</td><td>FID↓</td><td></td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>ACGAN*</td><td>[20]</td><td>7.26</td><td>184.41</td><td></td><td></td></tr><tr><td>SNGAN*[3]</td><td></td><td>36.80</td><td>27.62</td><td></td><td></td></tr><tr><td>SAGAN*[29]</td><td></td><td>52.52</td><td>18.28</td><td></td><td></td></tr><tr><td>BigGAN*[20]</td><td></td><td>38.05</td><td>22.77</td><td></td><td></td></tr><tr><td></td><td>TAC-GAN*[20]</td><td>28.86</td><td>23.75</td><td></td><td></td></tr><tr><td>ContraGAN*[23]</td><td></td><td>31.10</td><td>19.69</td><td>0.951</td><td>0.927</td></tr><tr><td></td><td>ACGAN[18]</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>35=58</td><td>SNGAN [3]</td><td>32.25</td><td>26.79</td><td>0.938</td><td>0.913</td></tr><tr><td></td><td>SAGAN [29]</td><td>29.85</td><td>34.73</td><td>0.849</td><td>0.914</td></tr><tr><td></td><td>BigGAN [4]</td><td>43.97</td><td>16.36</td><td>0.964</td><td>0.955</td></tr><tr><td></td><td>ContraGAN [23]</td><td>25.25</td><td>25.16</td><td>0.947</td><td>0.855</td></tr><tr><td>ReACGAN</td><td></td><td>68.27</td><td>13.98</td><td>0.976</td><td>0.977</td></tr><tr><td></td><td>BigGAN [4] + DiffAug [9]</td><td>36.97</td><td>18.57</td><td>0.956</td><td>0.941</td></tr><tr><td></td><td>ReACGAN +DiffAug[9]</td><td>69.74</td><td>11.95</td><td>0.977</td><td>0.975</td></tr><tr><td></td><td>BigGAN*[4]</td><td>99.31</td><td>8.51</td><td>-</td><td>-</td></tr><tr><td>P55=44</td><td>BigGAN*[25]</td><td>104.57</td><td>9.18</td><td></td><td>■</td></tr><tr><td></td><td>BigGAN[4]</td><td>99.71</td><td>7.89</td><td>0.985</td><td>0.989</td></tr><tr><td></td><td>ReACGAN</td><td>92.74</td><td>8.23</td><td>0.991</td><td>0.990</td></tr><tr><td></td><td>Real Data</td><td>173.33</td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "image",
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+        "text": "Compared with other cGANs, ReACGAN performs the best on most benchmarks, surpassing the previous methods by $2 . 5 \\%$ , $5 . 1 \\%$ , $1 5 . 8 \\%$ , and $1 4 . 5 \\%$ in FID on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet (256 B.S.), respectively. ReACGAN also harmonizes with augmentation-based regularizations, bringing incremental improvements on all the metrics. For the ImageNet experiments using a batch size of 256, ReACGAN reaches higher IS and lower FID with fewer iterations than other models in comparison. Finally, we demonstrate that ReACGAN can learn with a larger batch size on ImageNet. While some recent methods [20, 38, 27] have been built on ACGAN to improve the generation performance of ACGAN, large-scale image generation experiments with the batch size of 2048 have never been reported. Table 2 shows that our ReACGAN reaches FID score of 8.23 on ImageNet, being comparable with the value of 7.89 from BigGAN implementation in PyTorch-StudioGAN library. ReACGAN, however, provides better synthesis results than other implementations of BigGAN [4, 25]. Note that our result on ImageNet is obtained in only two runs while the training setup and architecture of BigGAN have been extensively searched and finely tuned. ",
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+        "type": "text",
+        "text": "Comparison with Other Conditioning Losses. We investigate how the generation qualities vary with different conditioning losses while keeping the other configurations fixed. We compare D2D-CE loss with cross-entropy loss of ACGAN (AC) [18], loss used in the projection discriminator (PD) [19], multi-hinge loss (MH) [26], and conditional contrastive loss (2C) [23]. The results are shown in Table 3. AC- and MH-based models present decent results on CIFAR10, but undergo early-training collapse on Tiny-ImageNet and CUB200 datasets. Replacing them with PD, 2C, and D2D-CE losses produce satisfactory performances across all datasets, where PD loss makes the best $\\mathrm { F } _ { 8 }$ (recall) on CUB200 dataset while giving third $\\mathrm { F _ { 0 . 1 2 5 } }$ (precision) value. The noticeable point is that the proposed D2D-CE shows consistent results across all datasets, showing the lowest FID and the highest $\\mathrm { F _ { 8 } }$ and $\\mathrm { F _ { 0 . 1 2 5 } }$ values in most cases. This means ReACGAN can generate high-fidelity images and is relatively free from the precision and recall trade-off [43] than the others. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Consistent Performance of ReACGAN on Adversarial Loss Selection. We validate the consistent performance of ReACGAN on four adversarial losses: non-saturation loss [1], least square loss [10], Wasserstein loss with gradient penalty regularization (W-GP) [13], and hinge loss [46] on CIFAR10 and Tiny-ImageNet datasets in Table 4. The experimental results show that $\\mathrm { B i g G A N } + \\mathrm { D } 2 \\mathrm { D } { \\cdot } \\mathrm { C E }$ (ReACGAN) consistently outperforms the projection discriminator (PD) and conditional contrastive loss (2C) counterparts over three adversarial losses [1, 13, 46]. However, for the experiments using the least square loss [10], ReACGAN exhibits inferior generation performances to the projection discriminator. We speculate that minimizing the least square distance between an adversarial logit and the target scalar (1 or 0) might affect the norms of feature maps and spoil the classifier training performed by D2D-CE loss. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5f9662b078f6d463fde84887bf9dcf884f89b25fc053625a9e84dc7b9f48ef56.jpg",
+        "table_caption": [
+            "Table 3: Experiments on the effectiveness of D2D-CE loss compared with other conditioning losses. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Conditioning Method</td><td colspan=\"4\">CIFAR10 [30]</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>BigGAN w/o Condition [4] (Abbreviated to Big)</td><td>9.46</td><td>12.21</td><td>0.987</td><td>0.982</td><td>7.38</td><td>76.15</td><td>0.804</td><td>0.576</td><td>5.16</td><td>35.17</td><td>0.852</td><td>0.936</td></tr><tr><td>Big + AC[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>Big + PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>Big + MH[26]</td><td>10.05</td><td>7.94</td><td>0.994</td><td>0.990</td><td>4.37</td><td>140.74</td><td>0.282</td><td>0.156</td><td>5.18</td><td>245.69</td><td>0.625</td><td>0.832</td></tr><tr><td>Big + 2C[23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>Big + D2D-CE (ReACGAN)</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8543ec2bce142bf9ba668eff880dd37e8f3be1d789b055b0f875b5f0e5b0a8a8.jpg",
+        "table_caption": [
+            "Table 4: Experiments to identify the consistent performance of D2D-CE on adversarial loss selection. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Adversarial Loss</td><td rowspan=\"2\">Conditioning Method</td><td colspan=\"5\">CIFAR10 [30]</td><td colspan=\"5\">Tiny-ImageNet [32]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>Better?</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td><td>Better?</td></tr><tr><td rowspan=\"3\">Non-saturation [1]</td><td>PD[19]</td><td>9.75</td><td>8.29</td><td>0.993</td><td>0.991</td><td>√</td><td>8.27</td><td>58.85</td><td>0.816</td><td>0.713</td><td></td></tr><tr><td>2C [23]</td><td>9.30</td><td>10.47</td><td>0.990</td><td>0.959</td><td></td><td>6.57</td><td>84.27</td><td>0.745</td><td>0.556</td><td></td></tr><tr><td>D2D-CE</td><td>9.79</td><td>8.27</td><td>0.993</td><td>0.991</td><td>√</td><td>11.76</td><td>39.32</td><td>0.942</td><td>0.852</td><td>√</td></tr><tr><td rowspan=\"3\">Least square [10]</td><td>PD[19]</td><td>9.94</td><td>8.26</td><td>0.993</td><td>0.992</td><td>√</td><td>12.74</td><td>37.14</td><td>0.920</td><td>0.900</td><td>√</td></tr><tr><td>2C[23]</td><td>8.66</td><td>12.18</td><td>0.986</td><td>0.941</td><td></td><td>9.58</td><td>53.10</td><td>0.916</td><td>0.706</td><td></td></tr><tr><td>D2D-CE</td><td>9.70</td><td>9.56</td><td>0.991</td><td>0.987</td><td></td><td>9.50</td><td>57.67</td><td>0.848</td><td>0.692</td><td></td></tr><tr><td rowspan=\"3\">W-GP [13]</td><td>PD[19]</td><td>5.71</td><td>64.75</td><td>0.792</td><td>0.652</td><td></td><td>6.67</td><td>84.16</td><td>0.696</td><td>0.498</td><td></td></tr><tr><td>2C [23]</td><td>5.93</td><td>55.99</td><td>0.842</td><td>0.709</td><td></td><td>6.89</td><td>74.45</td><td>0.812</td><td>0.536</td><td></td></tr><tr><td>D2D-CE</td><td>7.30</td><td>35.94</td><td>0.942</td><td>0.847</td><td>√</td><td>8.92</td><td> 52.74</td><td>0.856</td><td>0.689</td><td>√</td></tr><tr><td rowspan=\"3\">Hinge [46]</td><td>PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td></td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td></td></tr><tr><td>2C [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td></td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td></td></tr><tr><td>D2D-CE</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>√</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>√</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/ae88335aa74ef3f4e06044ae71e86fd59505e41e675a879a0dae35e5dc54b2d4.jpg",
+        "table_caption": [
+            "Table 5: FID [34] values of conditional StyleGAN2 (cStyleGAN2) [7] and StyleGAN2 $[ 7 ] + \\mathrm { D } 2 \\mathrm { D } { \\cdot } \\mathrm { C E }$ on CIFAR10 [30] and AFHQ [41] datasets. ADA means the adaptive discriminator augmentation [8]. FID is exceptionally computed using the training split for calculating the reference moments since FID value of StyleGAN2 is often calculated using the moments of the training dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Conditioning method</td><td>CIFAR10 [30]</td><td>AFHQ [41]</td></tr><tr><td>cStyleGAN2 [7]</td><td>3.88</td><td>=</td></tr><tr><td>StyleGAN2 [7] + D2D-CE</td><td> 3.34</td><td></td></tr><tr><td>cStyleGAN2 [7] + ADA [8]</td><td>2.43</td><td>4.99</td></tr><tr><td>StyleGAN2 [7] + ADA [8] + D2D-CE</td><td>2.38</td><td>4.95</td></tr><tr><td> StyleGAN2 [7] + DiffAug [9] + D2D-CE + Tuning</td><td>2.26</td><td>-</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Effect of D2D-CE for Different GAN Architectures. We study the effect of D2D-CE loss with different GAN architectures. In Table 5, we validate that D2D-CE is effective for StyleGAN2 [7] backbone and also fits well with the adaptive discriminator augmentation (ADA) [8]. StyleGAN2 with D2D-CE loss produces $1 3 . 9 \\%$ better generation result than the conditional version of StyleGAN2 (cStyleGAN2) on CIFAR10. Moreover, StyleGAN2 with D2D-CE can be reinforced with ADA or DiffAug when train StyleGAN2 $^ +$ D2D-CE under the limited data situation. Among GANs, ${ \\mathrm { S t y l e G A N } } 2 + { \\mathrm { D i f f A u g } } + { \\mathrm { D } } 2 { \\mathrm { D } } { \\mathrm { - C E } } + { \\mathrm { 1 } }$ Tuning achieves the best performance on CIFAR10, even outperforming some diffusion-based methods [47, 48]. Additional results with other architectures, i.e., a deep convolutional network [2] and a resnet style backbone [13], are provided in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4.4 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We study how each component of ReACGAN affects ACGAN training. By adding or ablating each part of ReACGAN, as shown in Table 6, we identify four major observations. (1) Feature and weight normalization greatly stabilize ACGAN training and improve generation performances on Tiny-ImageNet and CUB200 datasets. (2) D2D-CE enhances the generation performance by considering data-to-data relationships and by performing easy sample suppression (3th and 4th rows). (3) the suppression technique does not work well on the feature normalized cross-entropy loss (5th row). (4) While ACGAN shows high Inception score on ImageNet experiment, it shows relatively poor FID, $\\mathrm { F _ { 0 . 1 2 5 } }$ , and $\\mathrm { F _ { 8 } }$ values compared with the model trained with normalization, data-to-data consideration, and the suppression technique. This result is consistent with the qualitative results on ImageNet, where the images from ACGAN are easily classifiable, but the images from ReACGAN are high quality and diverse. We attribute the improvement to the discriminator that successfully leverages informative data-to-data and data-to-class relationships with easy sample suppression. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/a59ce57281fb29c29f8a0af958d8e87aa81a5b050192d0c14ce2973864977a69.jpg",
+        "image_caption": [
+            "Figure 5: Curated images generated by the proposed ReACGAN. More qualitative results on ReACGAN, BigGAN [4], ContraGAN [23], and ACGAN [18] are attached in Appendix J. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            130,
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d7edc5dfca5366716959d4814eb9bf165b368a9f9e4aa544c93ae94e2b174a7.jpg",
+        "table_caption": [
+            "Table 6: Ablation study on feature map normalization, data-to-data consideration, and easy positive and negative sample suppression. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ablation</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td><td colspan=\"4\">ImageNet [49]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td></tr><tr><td>ACGAN [18]</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>+ Normalization</td><td>13.46</td><td>30.33</td><td>0.955</td><td>0.889</td><td>4.78</td><td>25.54</td><td>0.883</td><td>0.952</td><td>18.16</td><td>36.40</td><td>0.879</td><td>0.787</td></tr><tr><td>+ Data-to-data (Eq. (5))</td><td>12.96</td><td>28.71</td><td>0.967</td><td>0.863</td><td>5.08</td><td>25.12</td><td>0.894</td><td>0.946</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td> + Suppression (Eq. (6))</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td><td>63.16 14.59</td><td></td><td></td><td>0.9740.974</td></tr><tr><td>- Data-to-data</td><td>12.96</td><td>30.79</td><td>0.960</td><td>0.857</td><td>5.39</td><td>30.36</td><td>0.863</td><td>0.947</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we have analyzed why training ACGAN becomes unstable as the number of classes in the dataset increases. By deriving the analytic form of gradient in the classifier and numerically checking the gradient values, we have discovered that the unstable training comes from a gradient exploding problem caused by the unboundedness of input feature vectors and poor classification ability of the classifier in the early training stage. To alleviate the instability and reinforce ACGAN, we have proposed the Data-to-Data Cross-Entropy loss (D2D-CE) and the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results verify the superiority of ReACGAN compared with the existing classifier- and projection-based GANs on five benchmark datasets. Moreover, exhaustive analyses on ReACGAN prove that ReACGAN is robust to hyperparameter selection and harmonizes with various architectures and differentiable augmentations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the IITP grants (No.2019-0-01906: AI Graduate School Program - POSTECH, No.2021-0-00537: Visual Common Sense, No.2021-0-02068: AI Innovation Hub) funded by Ministry of Science and ICT, Korea. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems (NeurIPS), pages 2672–2680, 2014.   \n[2] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv preprint arXiv 1511.06434, 2016.   \n[3] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   \n[4] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large Scale GAN Training for High Fidelity Natural Image Synthesis. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.   \n[5] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 4401–4410, 2019.   \n[6] Yan Wu, Jeff Donahue, David Balduzzi, Karen Simonyan, and Timothy P. Lillicrap. LOGAN: Latent Optimisation for Generative Adversarial Networks. arXiv preprint arXiv 1912.00953, 2019.   \n[7] Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 8110–8119, 2020.   \n[8] Tero Karras, Miika Aittala, Janne Hellsten, Samuli Laine, Jaakko Lehtinen, and Timo Aila. Training Generative Adversarial Networks with Limited Data. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   \n[9] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song Han. Differentiable augmentation for dataefficient gan training. arXiv preprint arXiv 2006.10738, 2020.   \n[10] Xudong Mao, Qing Li, Haoran Xie, Raymond Y. K. Lau, Zhixiang Wang, and Stephen Paul Smolley. Least Squares Generative Adversarial Networks. In Proceedings of the International Conference on Computer Vision (ICCV), pages 2813–2821, 2017.   \n[11] Martín Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein GAN. arXiv preprint arXiv 1701.07875, 2017.   \n[12] Martín Arjovsky and Léon Bottou. Towards Principled Methods for Training Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.   \n[13] Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved Training of Wasserstein GANs. In Advances in Neural Information Processing Systems (NeurIPS), pages 5767–5777, 2017.   \n[14] Naveen Kodali, James Hays, Jacob D. Abernethy, and Zsolt Kira. On Convergence and Stability of GANs. arXiv preprint arXiv 1705.07215, 2018.   \n[15] Zhiming Zhou, Jiadong Liang, Yuxuan Song, Lantao Yu, Hongwei Wang, Weinan Zhang, Yong Yu, and Zhihua Zhang. Lipschitz generative adversarial nets. In Proceedings of the International Conference on Machine Learning (ICML), pages 7584–7593, 2019.   \n[16] Han Zhang, Zizhao Zhang, Augustus Odena, and Honglak Lee. Consistency Regularization for Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2020.   \n[17] Zhengli Zhao, Sameer Singh, Honglak Lee, Zizhao Zhang, Augustus Odena, and Han Zhang. Improved Consistency Regularization for GANs. arXiv preprint arXiv 2002.04724, 2020.   \n[18] Augustus Odena, Christopher Olah, and Jonathon Shlens. Conditional Image Synthesis with Auxiliary Classifier GANs. In Proceedings of the International Conference on Machine Learning (ICML), pages 2642–2651, 2017.   \n[19] Takeru Miyato and Masanori Koyama. cGANs with Projection Discriminator. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   \n[20] Mingming Gong, Yanwu Xu, Chunyuan Li, Kun Zhang, and Kayhan Batmanghelich. Twin Auxilary Classifiers GAN. In Advances in Neural Information Processing Systems (NeurIPS), 2019.   \n[21] Aliaksandr Siarohin, Enver Sangineto, and Nicu Sebe. Whitening and Coloring Batch Transform for GANs. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.   \n[22] Ming-Yu Liu, Xun Huang, Arun Mallya, Tero Karras, Timo Aila, Jaakko Lehtinen, and Jan Kautz. FewShot Unsupervised Image-to-Image Translation. In Proceedings of the International Conference on Computer Vision (ICCV), 2019.   \n[23] Minguk Kang and Jaesik Park. ContraGAN: Contrastive Learning for Conditional Image Generation. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   \n[24] Woohyeon Shim and Minsu Cho. CircleGAN: Generative Adversarial Learning across Spherical Circles. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   \n[25] Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. Omni-GAN: On the Secrets of cGANs and Beyond. arXiv preprint arXiv:2011.13074, 2020.   \n[26] Ilya Kavalerov, Wojciech Czaja, and Rama Chellappa. A multi-class hinge loss for conditional gans. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), 2021.   \n[27] Liang Hou, Qi Cao, Huawei Shen, and Xueqi Cheng. cGANs with Auxiliary Discriminative Classifier. arXiv preprint arXiv:2107.10060, 2021.   \n[28] Han, Ligong and Min, Martin Renqiang and Stathopoulos, Anastasis and Tian, Yu and Gao, Ruijiang and Kadav, Asim and Metaxas, Dimitris N. Dual Projection Generative Adversarial Networks for Conditional Image Generation. In Proceedings of the International Conference on Computer Vision (ICCV), 2021.   \n[29] Han Zhang, Ian Goodfellow, Dimitris Metaxas, and Augustus Odena. Self-Attention Generative Adversarial Networks. In Proceedings of the International Conference on Machine Learning (ICML), pages 7354–7363, 2019.   \n[30] Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images. PhD thesis, University of Toronto, 2012.   \n[31] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Fei-Fei Li. ImageNet: A large-scale hierarchical image database. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pages 248–255, 2009.   \n[32] Johnson et al. Tiny ImageNet Visual Recognition Challenge. https://tiny-imagenet.herokuapp. com.   \n[33] P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical report, California Institute of Technology, 2010.   \n[34] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In Advances in Neural Information Processing Systems (NeurIPS), pages 6626–6637, 2017.   \n[35] Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. In Advances in Neural Information Processing Systems (NeurIPS), pages 271–279, 2016.   \n[36] Akash Srivastava, Lazar Valkov, Chris Russell, Michael U Gutmann, and Charles A Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In Advances in Neural Information Processing Systems (NeurIPS), 2017.   \n[37] Mehdi Mirza and Simon Osindero. Conditional Generative Adversarial Nets. arXiv preprint arXiv 1411.1784, 2014.   \n[38] Zhiming Zhou, Han Cai, Shu Rong, Yuxuan Song, Kan Ren, Weinan Zhang, Jun Wang, and Yong Yu. Activation Maximization Generative Adversarial Nets. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   \n[39] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey E. Hinton. A Simple Framework for Contrastive Learning of Visual Representations. arXiv preprint arXiv 2002.05709, 2020.   \n[40] Minguk Kang and Jaesik Park. Pytorch-studiogan. https://github.com/POSTECH-CVLab/ PyTorch-StudioGAN, 2020.   \n[41] Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. StarGAN v2: Diverse Image Synthesis for Multiple Domains. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   \n[42] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, and Xi Chen. Improved Techniques for Training GANs. In Advances in Neural Information Processing Systems (NeurIPS), pages 2234–2242, 2016.   \n[43] Mehdi SM Sajjadi, Olivier Bachem, Mario Lucic, Olivier Bousquet, and Sylvain Gelly. Assessing generative models via precision and recall. In Advances in Neural Information Processing Systems (NeurIPS), 2018.   \n[44] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur. A Learned Representation For Artistic Style. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.   \n[45] Harm de Vries, Florian Strub, Jeremie Mary, Hugo Larochelle, Olivier Pietquin, and Aaron C Courville. Modulating early visual processing by language. In Advances in Neural Information Processing Systems (NeurIPS), pages 6594–6604, 2017.   \n[46] Jae Hyun Lim and Jong Chul Ye. Geometric GAN. arXiv preprint arXiv 1705.02894, 2017.   \n[47] Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Proceedings of the International Conference on Machine Learning (ICML), 2021.   \n[48] Dongjun Kim, Seungjae Shin, Kyungwoo Song, Wanmo Kang, and Il-Chul Moon. Score Matching Model for Unbounded Data Score. arXiv preprint arXiv:2106.05527, 2021.   \n[49] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou. Arcface: Additive angular margin loss for deep face recognition. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2019. ",
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