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+# SKIP CONNECTIONS MATTER: ON THE TRANSFERABILITY OF ADVERSARIAL EXAMPLES GENERATED WITH RESNETS
+
+Dongxian $\mathbf { W _ { u } } ^ { 1 , 3 }$ Yisen $\mathbf { W a n g } ^ { 2 \dagger }$ Shu-Tao $\mathbf { X _ { i a } ^ { \bullet , 1 , 3 } }$ James Bailey4 Xingjun Ma4
+
+1Tsinghua University   
+2Shanghai Jiao Tong University   
+3PCL Research Center of Networks and Communications, Peng Cheng Laboratory   
+4The University of Melbourne
+
+# ABSTRACT
+
+Skip connections are an essential component of current state-of-the-art deep neural networks (DNNs) such as ResNet, WideResNet, DenseNet, and ResNeXt. Despite their huge success in building deeper and more powerful DNNs, we identify a surprising security weakness of skip connections in this paper. Use of skip connections allows easier generation of highly transferable adversarial examples. Specifically, in ResNet-like (with skip connections) neural networks, gradients can backpropagate through either skip connections or residual modules. We find that using more gradients from the skip connections rather than the residual modules according to a decay factor, allows one to craft adversarial examples with high transferability. Our method is termed Skip Gradient Method (SGM). We conduct comprehensive transfer attacks against state-of-the-art DNNs including ResNets, DenseNets, Inceptions, Inception-ResNet, Squeeze-and-Excitation Network (SENet) and robustly trained DNNs. We show that employing SGM on the gradient flow can greatly improve the transferability of crafted attacks in almost all cases. Furthermore, SGM can be easily combined with existing black-box attack techniques, and obtain high improvements over state-of-the-art transferability methods. Our findings not only motivate new research into the architectural vulnerability of DNNs, but also open up further challenges for the design of secure DNN architectures.
+
+# 1 INTRODUCTION
+
+In deep neural networks (DNNs), a skip connection builds a short-cut from a shallow layer to a deep layer by connecting the input of a convolutional block (also known as the residual module) directly to its output. While different layers of a neural network learn different “levels” of features, skip connections can help preserve low-level features and avoid performance degradation when adding more layers. This has been shown to be crucial for building very deep and powerful DNNs such as ResNet (He et al., 2016a;b), WideResNet (Zagoruyko & Komodakis, 2016), DenseNet (Huang et al., 2017) and ResNeXt (Xie et al., 2017). In the meantime, despite their superior performance, DNNs have been found extremely vulnerable to adversarial examples (or attacks), which are input examples slightly perturbed with an intention to fool the network to make a wrong prediction (Szegedy et al., 2013; Goodfellow et al., 2014; Ma et al., 2018; Bai et al., 2019; Wang et al., 2019; 2020). Adversarial examples often appear imperceptible to human observers, and are transferable across different models (Liu et al., 2017). This has raised security concerns on the deployment of DNNs in security critical scenarios, such as face recognition (Sharif et al., 2016), autonomous driving (Evtimov et al., 2018), video analysis (Jiang et al., 2019) and medical diagnosis (Ma et al., 2019).
+
+Adversarial examples can be crafted following either a white-box setting (the adversary has full access to the target model) or a black-box setting (the adversary has no information of the target model). White-box methods such as Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014), Basic Iterative Method (BIM) (Kurakin et al., 2016), Projected Gradient Decent (PGD) (Madry et al., 2018) and Carlini and Wagner (CW) (Carlini & Wagner, 2017) often suffer from low transferability in a black-box setting, thus posing only limited threats to DNN models which are usually kept secret in practice (Dong et al., 2018; Xie et al., 2019). Several techniques have been proposed to improve the transferability of black-box attacks crafted on a surrogate model, such as momentum boosting (Dong et al., 2018), diverse input (Xie et al., 2019) and translation invariance (Dong et al., 2019). Although these techniques are effective, they (as well as white-box methods) all treat the entire network (either the target model or the surrogate model) as a single component while ignore its inner architectural characteristics. The question of whether or not the DNN architecture itself can expose more transferability of adversarial attacks is an unexplored problem.
+
+![](images/89948fe9f78fd82465425cf0071bf4e77b2e8e2b2ab3eab2c46f940ffe2c3249.jpg)  
+Figure 1: Left: Illustration of the last 3 skip connections (green lines) and residual modules (black boxes) of a ImageNet-trained ResNet-18. Right: The success rate (in the form of “white-box/blackbox”) of adversarial attacks crafted using gradients flowing through either a skip connection (going upwards) or a residual module (going leftwards) at each junction point (circle). Three example backpropagation paths are highlighted in different colors, with the green path skipping over the last two residual modules having the best attack success rate while the red path through all 3 residual modules having the worst attack success rate. The attacks are crafted by BIM on 5000 ImageNet validation images under maximum $L _ { \infty }$ perturbation $\epsilon \ : = \ : 1 6$ (pixel values are in [0, 255]). The black-box success rate is tested against a VGG19 target model.
+
+In this paper, we identify one such weakness about the skip connections used by many state-of-theart DNNs. We first conduct a toy experiment with the BIM attack and ResNet-18 on the ImageNet validation dataset (Deng et al., 2009) to investigate how skip connections affect the adversarial strength of attacks crafted on the network. At each of the last 3 skip connections and residual modules of ResNet-18, we illustrate the success rate of attacks crafted using gradients backpropagate through either the skip connection or the residual module in Figure 1. As can be observed, the success rate drops more drastically whenever using gradients from a residual module instead of the skip connection. This implies that gradients from the skip connections are more vulnerable (high success rate). In addition, we surprisingly find that skip connections expose more transferable information. For example, the black-box success rate was even improved from $5 2 . 5 2 \%$ to $6 2 . 1 0 \%$ when the attack skips the last two residual modules (following the path in green color).
+
+Motivated by the above observations, in this paper, we propose the Skip Gradient Method (SGM) to generate adversarial examples using gradients more from the skip connections rather than the residual modules. In particular, SGM utilizes a decay factor to reduce gradients from the residual modules. We find that this simple adjustment on the gradient flow can generate highly transferable adversarial examples, and the more skip connections in a network, the more transferable are the crafted attacks. This is in sharp contrast to the design principles (e.g., “going deeper” with skip connections) underpinning many modern DNNs. In particular, our main contributions are:
+
+• We identify one surprising property of skip connections in ResNet-like neural networks, i.e., they allow an easy generation of highly transferable adversarial examples.   
+• We propose the Skip Gradient Method (SGM) to craft adversarial examples using gradients more from the skip connections. Using a single decay factor on gradients, SGM is an appealingly simple and generic technique that can be used by any existing gradient-based attack methods.   
+• We provide comprehensive transfer attack experiments, from different source models against 10 state-of-the-art DNNs, showing that SGM can greatly improve the transferability of crafted adversarial examples. When combined with existing transfer techniques, SGM improves the state-ofthe-art transferability benchmarks by a large margin.
+
+# 2 RELATED WORK
+
+Existing adversarial attacks can be categorized into two groups: 1) white-box attacks and 2) blackbox attacks. In the white-box setting, the adversary has full access to the parameters of the target model, while in the black-box setting, the target model is kept secret from the adversary.
+
+# 2.1 WHITE-BOX ATTACKS
+
+Given a clean example $_ { \textbf { \em x } }$ with class label $y$ and a target DNN model $f$ , the goal of an adversary is to find an adversarial example $\mathbf { { x } } _ { a d v }$ that fools the network into making an incorrect prediction (eg. $f ( { \pmb x } _ { a d v } ) \neq y )$ , while still remaining in the $\epsilon$ -ball centered at $_ { \textbf { \em x } }$ (eg. $\| \pmb { x } _ { a d v } - \pmb { x } \| _ { \infty } \leq \bar { \epsilon } )$ .
+
+Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014). FGSM perturbs clean example $_ { \textbf { \em x } }$ for one step by the amount of $\epsilon$ along the gradient direction:
+
+$$
+\begin{array} { r } { \pmb { x } _ { a d v } = \pmb { x } + \epsilon \cdot \mathrm { s i g n } \big ( \nabla _ { \pmb { x } } \ell ( f ( \pmb { x } ) , y ) \big ) . } \end{array}
+$$
+
+The Basic Iterative Method (BIM) (Kurakin et al., 2016) is an iterative version of FGSM that perturbs for $T$ steps with step size $\epsilon / T$ .
+
+Projected Gradient Descent (PGD) (Madry et al., 2018). PGD perturbs normal example $_ { \textbf { \em x } }$ for $T$ steps with smaller step size. After each step of perturbation, PGD projects the adversarial example back onto the $\epsilon$ -ball of $_ { \textbf { \em x } }$ , if it goes beyond the $\epsilon$ -ball:
+
+$$
+\begin{array} { r } { \pmb { x } _ { a d v } ^ { t + 1 } = \Pi _ { \epsilon } \big ( \pmb { x } _ { a d v } ^ { t } + \alpha \cdot \mathrm { s i g n } ( \nabla _ { \pmb { x } } \ell ( f ( \pmb { x } _ { a d v } ^ { t } ) , y ) ) \big ) , } \end{array}
+$$
+
+where $\Pi _ { \epsilon } ( \cdot )$ is the projection operation. Different to BIM, PGD allows step size $\alpha > \epsilon / T$
+
+There are also other types of white-box attacks including sparsity-based methods such as Jacobianbased Saliency Map Attack (JSMA) (Papernot et al., 2016), sparse attack (Modas et al., 2019), onepixel attack (Su et al., 2019), and optimization-based methods such as Carlini and Wagner (CW) (Carlini & Wagner, 2017) and elastic-net (EAD) (Chen et al., 2018).
+
+# 2.2 BLACK-BOX ATTACKS
+
+Black-box attacks can be generated by either attacking a surrogate model or using gradient estimation methods in combination with queries to the target model. Gradient estimation methods estimate the gradients of the target model using black-box optimization methods such as Finite Differences (FD) (Chen et al., 2017; Bhagoji et al., 2018) or Natural Evolution Strategies (NES) (Ilyas et al., 2018; Jiang et al., 2019). These methods all require a large number of queries to the target model, which not only reduces efficiency but also potentially exposes the attack. Alternatively, black-box adversarial examples can be crafted on a surrogate model then applied to attack the target model. Although the white-box methods can be directly applied on the surrogate model, they are far less effective in the black-box setting (Dong et al., 2018; Xie et al., 2019). Several transfer techniques have been proposed to improve the transferability of black-box attacks.
+
+Momentum Iterative boosting (MI) (Dong et al., 2018). MI incorporates a momentum term into the gradient to boost the transferability:
+
+$$
+\begin{array} { r } { \pmb { x } _ { a d v } ^ { t + 1 } = \Pi _ { \epsilon } \big ( \pmb { x } _ { a d v } ^ { t } + \alpha \cdot \mathrm { s i g n } ( \pmb { g } ^ { t + 1 } ) \big ) , \pmb { g } ^ { t + 1 } = \mu \cdot \pmb { g } ^ { t } + \frac { \nabla _ { \mathbf { x } } \ell \left( f \left( \pmb { x } _ { a d v } ^ { t } \right) , \pmb { y } \right) } { \| \nabla _ { \mathbf { x } } \ell \left( f \left( \pmb { x } _ { a d v } ^ { t } \right) , \pmb { y } \right) \| _ { 1 } } , } \end{array}
+$$
+
+where $g ^ { t }$ is the adversarial gradient at the $t$ -th step, $\alpha = \epsilon / T$ is the step size for a total of $T$ steps, $\mu$ is a decay factor, and $\| \cdot \| _ { 1 }$ is the $L _ { 1 }$ norm.
+
+Diverse Input $\mathbf { \Pi } ( \mathbf { D } \mathbf { I } )$ (Xie et al., 2019). DI proposes to craft adversarial exampels using gradient with respect to the randomly-transformed input example:
+
+$$
+\pmb { x } _ { a d v } ^ { t + 1 } = \Pi _ { \epsilon } \big ( \pmb { x } _ { a d v } ^ { t } + \alpha \cdot \mathrm { s i g n } \big ( \nabla _ { \pmb { x } } \ell \big ( f \big ( H ( \pmb { x } _ { a d v } ^ { t } ; p ) \big ) , y \big ) \big ) \big ) ,
+$$
+
+where $H ( \boldsymbol { \mathbf { x } } _ { a d v } ^ { t } ; \boldsymbol { p } )$ is a stochastic transformation function on $\scriptstyle { \pmb { x } } _ { a d v } ^ { t }$ for a given probability $p$
+
+Translation Invariant (TI) (Dong et al., 2019). TI targets to evade robustly trained DNNs by generating adversarial examples that are less sensitive to the discriminative regions of the surrogate model. More specifically, $\mathrm { T I }$ computes the gradients with respect to a set of translated versions of the original input:
+
+$$
+\pmb { x } _ { a d v } ^ { t + 1 } = \Pi _ { \epsilon } \bigl ( \pmb { x } _ { a d v } ^ { t } + \alpha \cdot \mathrm { s i g n } ( W * \nabla _ { \pmb { x } } \ell ( f ( \pmb { x } _ { a d v } ^ { t } ) , y ) ) \bigr ) ,
+$$
+
+where $W$ is a predefined kernel (e.g., uniform, linear, and Gaussian) matrix of size $( 2 k + 1 ) ( 2 k +$ 1) $k$ being the maximal number of pixels to shift). This kernel convolution is equivalent to the weighted sum of gradients over $( 2 k + 1 ) ^ { 2 }$ number of shifted input examples.
+
+Furthermore, there are other studies focusing on intermediate feature representations. For example, Activation Attack (Inkawhich et al., 2019) drives the activation of a specified layer on a given image towards the layer of a target image, to yield a highly transferable targeted example. Intermediate Level Attack (Huang et al., 2019) attempts to fine-tune an existing adversarial example for greater black-box transferability by increasing its perturbation on a pre-specified layer of the source model.
+
+Although the above transfer techniques are effective, they (including white-box attacks) either 1) treat the network (either the surrogate model or the target model) as a single component or 2) only use the intermediate layer output of the network. In other words, they do not directly consider the effects of different DNN architectural characteristics. Li et al. (2018) investigated the use of skip connections and dropout layers for sampling networks, which generates a huge set of ghost networks to perform an ensemble attack. Here, we focus on the architectural property of skip connections from the gradient view without modifying or generating any extra networks.
+
+# 3 PROPOSED SKIP GRADIENT ATTACK
+
+In this section, we first introduce the gradient decomposition of skip connection and residual module. Following that, we propose our Skip Gradient Method (SGM), then demonstrate the adversarial transferability property of skip connection via a case study.
+
+# 3.1 GRADIENT DECOMPOSITION WITH SKIP CONNECTIONS
+
+In ResNet-like neural networks, a skip connection uses identity mapping to bypass residual layers, allowing data flow from a shallow layer directly to subsequent deep layers. Thus, we can decompose the network into a collection of paths of different lengths (Veit et al., 2016). We denote a skip connection together with its associated residual module as a building block (residual block) of a network. Considering three successive building blocks (eg. $z _ { i + 1 } = z _ { i } + f _ { i + 1 } ( z _ { i } ) )$ in a residual network from input $z _ { \mathrm { 0 } }$ to output $z _ { 3 }$ , the output $z _ { 3 }$ can be expanded as:
+
+$$
+\begin{array} { r l } & { z _ { 3 } = z _ { 2 } + f _ { 3 } ( z _ { 2 } ) = [ z _ { 1 } + f _ { 2 } ( z _ { 1 } ) ] + f _ { 3 } ( z _ { 1 } + f _ { 2 } ( z _ { 1 } ) ) } \\ & { \quad = [ z _ { 0 } + f _ { 1 } ( z _ { 0 } ) + f _ { 2 } ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) ] + f _ { 3 } \big ( ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) + f _ { 2 } ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) \big ) . } \end{array}
+$$
+
+According to the chain rule in calculus, the gradient of a loss function $\ell$ with respect to input $z _ { \mathrm { 0 } }$ can then be decomposed as,
+
+$$
+\frac { \partial \ell } { \partial z _ { 0 } } = \frac { \partial \ell } { \partial z _ { 3 } } \frac { \partial z _ { 3 } } { \partial z _ { 2 } } \frac { \partial z _ { 2 } } { \partial z _ { 1 } } \frac { \partial z _ { 1 } } { \partial z _ { 0 } } = \frac { \partial \ell } { \partial z _ { 3 } } ( 1 + \frac { \partial f _ { 3 } } { \partial z _ { 2 } } ) ( 1 + \frac { \partial f _ { 2 } } { \partial z _ { 1 } } ) ( 1 + \frac { \partial f _ { 1 } } { \partial z _ { 0 } } ) .
+$$
+
+Extending this toy example to a network with $L$ residual blocks, the gradient can be decomposed from $L$ -th to the $( l + 1 )$ -th $( 0 \leq l < L )$ ) residual block as,
+
+$$
+\frac { \partial \ell } { \partial \pmb { x } } = \frac { \partial \ell } { \partial z _ { L } } \prod _ { i = l } ^ { L - 1 } \big ( \frac { \partial f _ { i + 1 } } { \partial z _ { i } } + 1 \big ) \frac { \partial z _ { l } } { \partial \pmb { x } } .
+$$
+
+The example illustrated in Figure 1 is a the above decomposition of a ResNet-18 at the last 3 building blocks $( l = L - 3 )$ .
+
+# 3.2 SKIP GRADIENT METHOD (SGM)
+
+In order to use more gradient from the skip connections, here, we introduce a decay parameter into the decomposed gradient to reduce the gradient from the residual modules. Following the decomposition in Equation (8), the “skipped” gradient is,
+
+$$
+\nabla _ { \pmb { x } } \ell = \frac { \partial \ell } { \partial z _ { L } } \prod _ { i = 0 } ^ { L - 1 } \big ( \gamma \frac { \partial f _ { i + 1 } } { \partial z _ { i } } + 1 \big ) \frac { \partial z _ { 0 } } { \partial \pmb { x } } ,
+$$
+
+where ${ \boldsymbol { z } } _ { 0 } = { \boldsymbol { x } }$ is the input of the network, and $\gamma \in ( 0 , 1 ]$ is the decay parameter. Accordingly, given a clean example $_ { \textbf { \em x } }$ and a DNN model $f$ , an adversarial example can be crafted iteratively by,
+
+$$
+\pmb { x } _ { a d v } ^ { t + 1 } = \Pi _ { \epsilon } \Bigl ( \pmb { x } _ { a d v } ^ { t } + \alpha \cdot \mathrm { s i g n } \bigl ( \frac { \partial \ell } { \partial z _ { L } } \prod _ { i = 0 } ^ { L - 1 } ( \gamma \frac { \partial f _ { i + 1 } } { \partial z _ { i } } + 1 ) \frac { \partial z _ { 0 } } { \partial x } \bigr ) \Bigr ) .
+$$
+
+Table 1: The success rates ( $\%$ ±std over 5 random runs) of black-box attacks (untargeted) crafted by PGD and its “skip gradient” (SGM) version, on different source models against a Inception V3 target model. The best results are in bold.   
+
+<table><tr><td></td><td>RN18 RN34</td><td>RN50 RN101 RN152</td><td>DN121 DN169</td><td>DN201</td></tr><tr><td>PGD</td><td>23.23±0.69 24.38±0.41</td><td>22.80±0.55 22.98±0.83 26.56±0.75</td><td></td><td>30.71±0.60 30.90±0.31 36.01±0.59</td></tr><tr><td>SGM</td><td>28.92±0.45 43.43±0.32</td><td>36.71±0.55 38.38±0.53 44.84±0.14</td><td>57.38±0.14 60.45±0.42 65.48±0.23</td><td></td></tr></table>
+
+SGM is a generic technique that can be easily implemented on any neural network that has skip connections. During the backpropagation process, SGM simply multiplies the decay parameter to the gradient whenever it passes a residual module. Therefore, SGM does not require any computation overhead, and works efficiently even on densely connected networks such as DenseNets. The reduction of residual gradients is accumulated along the backpropagation path, that is, the residual gradients at lower layers will be reduced more times than those at higher layers. This is because, compared to high-level features, low-level features have already been well preserved by skip connections (see feature decompositions in Equation (6)).
+
+# 3.3 ADVERSARIAL TRANSFERABILITY WITH SKIP CONNECTIONS: A CASE STUDY
+
+To demonstrate the adversarial transferability of skip connections, we conduct a case study on 10- step PGD, and their corresponding SGM versions, to investigate the success rates of black-box attacks crafted with or without manipulating the skip connections. The black-box attacks are generated on 8 different source (surrogate) models ResNet(RN)-18/34/50/101/152 and DenseNet(DN)- 121/169/201, then applied to attack a Inception V3 target model. All models were trained on ImageNet training set. We randomly select 5000 ImageNet validation images that are correctly classified by all source models, and craft untargeted attacks under maximum $L _ { \infty }$ perturbation $\epsilon = 1 6$ , which is a typical black-box setting (Dong et al., 2018; Xie et al., 2019; Dong et al., 2019). The step size of PGD was set to $\alpha = 2$ , and the decay parameter of SGM was set to $\gamma = 0 . 5$ .
+
+We run the attack for 5 times with different random seeds, and report the success rates (transferability) of different methods in Table 1. As can be seen, when the skip connections are manipulated with our SGM, the transferability of PGD is greatly improved across all source models. On all source models except RN18, the improvements are more than $1 3 \%$ . Without SGM, the best transferability against the Inception-V3 target model is $3 5 . 4 8 \%$ which is achieved by PGD on DN201, however, this is improved further by our proposed SGM to $6 5 . 3 8 \%$ $( > 2 9 \%$ gain). This not only highlights the surprising property of skip connections in terms of the generation of highly transferable attacks, but also indicates the significance of this property, as such a huge boost in transferability only takes a single decay factor.
+
+The 8 source models can be interpreted as from 3 ResNet families: 1) RN18/34 are ResNets with normal residual blocks, 2) RN50/101/152 are ResNets with “bottleneck” residual blocks, and 3) DN121/169/201 are densely connected ResNets. Another important observation is that when there are more skip connections in a network within the same ResNet family (e.g., $\mathrm { R N } 3 4 > \mathrm { R N } 1 8$ , $\mathrm { R N } 1 5 2 > \mathrm { R N } 1 0 1 > \mathrm { R N } 5 0$ , and $\mathrm { D N } 2 0 1 > \mathrm { D N } 1 6 9 > \mathrm { D N } 1 2 1 )$ , or from ResNets to DenseNets (e.g. $\mathrm { D N } 1 2 1 / 1 6 9 / 2 0 1 > \mathrm { R N } 1 8 / 3 4$ and $\mathrm { D N } 1 2 1 / 1 6 9 / 2 0 1 > \mathrm { R N } 5 0 / 1 0 1 / 1 5 2 )$ , the crafted adversarial examples become more transferable, especially when the skip connections are manipulated by our SGM. This raises questions about the design principle behind many state-of-the-art DNNs: “going deeper” with techniques like skip connection and $1 \times 1$ convolution.
+
+# 4 COMPARISON TO EXISTING TRANSFER ATTACKS
+
+In this section, we compare the transferability of adversarial examples crafted by our proposed SGM and existing methods on ImageNet against both unsecured and secured target models.
+
+Baselines. We compare SGM with FGSM, PGD, and 3 state-of-the-art transfer attacks: (1) Momentum Iterative (MI) (Dong et al., 2018), (2) Diverse Input (DI) (Xie et al., 2019), and (3) Transition Invariant (TI) (Dong et al., 2019). Note that the TI attack was originally proposed to attack secured models, although here we include TI to attack both unsecured models and secured models. For TI and our SGM, we test both the one-step and the iterative version, however, the other methods DI and MI only have an iterative version. The iteration step is set to 10 and 20 for unsecured and secured target models respectively. For all iterative methods PGD, TI and our SGM, the step size is set to $\alpha = 2$ . For our proposed SGM, the decay parameter is set to $\gamma = 0 . 2$ (0.5) and $\bar { \gamma } = 0 . 5$ (0.7)
+
+on ResNet and DenseNet source models in PGD (FGSM) respectively. For simplicity, we utilize SGM to indicate $\mathrm { F G S M + S G M }$ in one-step attacks, and $\mathrm { P G D + S G M }$ in multi-step attacks. Other parameters of existing methods are configured as in their original papers.
+
+Threat Model. We adopt a black-box threat model in which adversarial examples are generated by attacking a source model and then applied to attack the target model. The target model is of a different architecture (indicated by the model name) to the source model, expect when the source and target models are of the same architecture, where we directly use the source model as the target model (equivalent to a white-box setting). The attacks are crafted on 5000 randomly selected ImageNet validation images that are classified correctly by all source models, and are repeated for 5 times with different random seeds. For all attack methods, we follow the standard setting (Dong et al., 2018; Xie et al., 2019) to craft untargeted attacks under maximum $L _ { \infty }$ perturbation $\epsilon = 1 6$ with respect to pixel values in [0, 255].
+
+Target Models. We consider two types of target models: 1) unsecured models that are trained on ImageNet training set using traditional training; and 2) secured models trained using adversarial training. For unsecured target model, we choose 7 state-of-the-art DNNs: VGG19 (with batch normalization) (Simonyan & Zisserman, 2015), ResNet-152 (RN152) (He et al., 2016a), DenseNet201 (DN152), 154 layer Squeeze-and-Excitation network (SE154) (Hu et al., 2018), Inception V3 (IncV3) (Szegedy et al., 2016), Inception V4 (IncV3) (Szegedy et al., 2017) and Inception-ResNet V2 (IncResV2) (Szegedy et al., 2017). For secured target models, we consider 3 robustly trained DNNs using ensemble adversarial training (Tramer et al., 2018):\` $\mathrm { I n c V } 3 _ { e n s 3 }$ (ensemble of $3 \ \mathrm { I n c V 3 }$ networks), $\mathrm { I n c V } 3 _ { e n s 4 }$ (ensemble of 4 IncV3 networks) and IncRes $\mathrm { V } 2 _ { e n s 3 }$ (ensemble of 3 IncResV2 networks).
+
+Source Models. We choose 8 different source models from the ResNet family: ResNet(RN)- 18/34/50/101/152 and DenseNet(DN)-121/169/201. Whenever the input size of the source model does not match the target model, we resize the crafted adversarial images to the input size of the target model. For VGG19, ResNet and DenseNet models, images are cropped and resized to $2 2 4 \times 2 2 4$ , while for Inception/Inception-ResNet models, images are cropped and resized to $2 9 9 \times 2 9 9$ .
+
+# 4.1 TRANSFERABILITY AGAINST UNSECURED MODELS
+
+We first investigate the transferability of all attack methods against 7 unsecured models, which is to find the best method that can generate the most transferable attacks on one source model against all target models.
+
+One-step Transferability. The one-step transferability is measured by the success rate of one-step attacks, as reported in Table 2. Here, we only show the results on two source models: 1) RN152 which is the best ResNet source model with the highest success rate on average against all target models, and 2) DN201 which is the best DenseNet source model. Also note that, when the source and target models are the same, the result represents the white-box success rate. Overall, adversarial examples crafted on DN201 have significantly better transferability than those crafted on RN152, especially for our SGM method. This is because there are $\sim 3 0 \times$ more skip connections that can be manipulated by our SGM in DN201 compared to RN152. In comparison to to both FGSM and TI, transferability is improved considerably by SGM in almost all test scenarios, except when transferring from RN152 to VGG19/IncV3/IncV4 where SGM is outperformed by TI. This implies that, when transfereing across different architectures (eg. ResNet VGG/Inception), translation adaptation may help increase the transferability of one-step perturbations. However, this advantage of TI disappears when there are more skip connections, as is the case for the DN201 source model.
+
+Table 2: One-step transferability: the success rates ( $\% \pm$ std over 5 random runs) of black-box attacks crafted by different methods on 2 source models against 7 unsecured target models. The best results are in bold.   
+
+<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncResV2</td></tr><tr><td rowspan="3">RN152</td><td>FGSM</td><td>41.96±0.52</td><td>71.53±0.34</td><td>37.49±0.10</td><td>30.00±0.56</td><td>25.66±0.07</td><td>21.55±0.16</td><td>19.90±0.49</td></tr><tr><td>TI</td><td>49.61±0.11</td><td>49.33±0.35</td><td>36.87±0.42</td><td>29.95±0.32</td><td>33.59±0.73</td><td>29.05±0.34</td><td>20.62±0.09</td></tr><tr><td>SGM</td><td>47.54±0.14</td><td>76.90±0.60</td><td>43.73±0.21</td><td>31.16±0.45</td><td>29.41±0.24</td><td>25.11±0.20</td><td>22.63±0.15</td></tr><tr><td rowspan="3">DN201</td><td>FGSM</td><td>49.87±0.17</td><td>38.89±0.29</td><td>81.51±0.33</td><td>34.94±0.53</td><td>31.21±0.47</td><td>27.08±0.23</td><td>23.87±0.45</td></tr><tr><td>TI</td><td>54.37±0.58</td><td>33.49±0.18</td><td>57.71±0.05</td><td>34.46±0.47</td><td>34.45±0.25</td><td>30.17±0.23</td><td>20.36±0.33</td></tr><tr><td>SGM</td><td>56.97±0.25</td><td>47.54±0.14</td><td>87.73±0.76</td><td>42.31±0.67</td><td>37.91±0.56</td><td>32.83±0.38</td><td>29.64±0.25</td></tr></table>
+
+![](images/ddd5746ae51d7c4389a483eff0be8007bf1101f993b029921ace4f8de190fd80.jpg)  
+Figure 2: The attack success rates of black-box attacks crafted by different attack methods on 8 source models against 3 unsecured target models: VGG19 (left), SE154 (middle) and IncV3 (right).
+
+Table 3: Multi-step transferability: the success rates ( $\% \pm$ std over 5 random runs) of black-box attacks crafted by different methods on 2 source models against 7 unsecured target models. The best results are in bold.   
+
+<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td rowspan="5">RN152</td><td>PGD</td><td>45.03±0.21</td><td>99.91±0.04 51.49±0.51</td><td></td><td>29.35±0.49</td><td>26.56±0.75</td><td>21.03±0.19</td><td>19.10±0.37</td></tr><tr><td>TI</td><td>54.59±0.63</td><td>99.73±0.11</td><td>63.77±0.93</td><td>41.89±0.58 37.97±0.39</td><td></td><td></td><td>36.25±1.13 28.90±0.52</td></tr><tr><td>MI</td><td>65.42±0.60</td><td>99.77±0.02</td><td>75.79±0.69</td><td>53.07±0.44</td><td>50.22±0.07</td><td>43.32±0.27</td><td>41.71±0.36</td></tr><tr><td>DI</td><td>74.01±0.48</td><td>99.90±0.02</td><td>277.81±0.80</td><td>57.49±1.22</td><td>53.95±0.68</td><td>47.16±0.52</td><td>43.47±0.30</td></tr><tr><td>SGM</td><td>79.90±0.69</td><td>99.87±0.03</td><td>81.56±0.35</td><td>61.83±0.17</td><td>57.22±0.51</td><td>48.57±0.09</td><td>45.44±0.36</td></tr><tr><td rowspan="5">DN201</td><td>PGD</td><td>57.61±0.82</td><td>59.84±0.76</td><td>99.89±0.02</td><td>39.78±0.69</td><td>36.01±0.59</td><td>31.76±0.27</td><td>25.92±0.13</td></tr><tr><td>TI</td><td>54.90±0.75</td><td>50.63±1.02</td><td>99.64±0.07</td><td>40.40±0.31</td><td>39.13±0.52</td><td>37.03±1.00</td><td>28.83±0.49</td></tr><tr><td>MI</td><td>75.09±0.80</td><td>76.39±0.61</td><td>99.84±0.05</td><td>64.38±0.69</td><td>59.62±0.36</td><td>54.85±0.56</td><td>50.05±0.25</td></tr><tr><td>DI</td><td>78.11±0.56 78.18±0.91</td><td></td><td>99.81±0.05</td><td>61.75±0.88</td><td>60.04±0.81</td><td>56.15±0.36 49.00±0.83</td><td></td></tr><tr><td>SGM</td><td>82.66±0.29</td><td>86.65±0.50</td><td>99.67±0.08</td><td>72.03±0.53</td><td>65.48±0.23</td><td>58.77±0.78</td><td>54.97±0.25</td></tr></table>
+
+Multi-step Transferability. First we provide a detailed study about the transferability of all attack methods from the 8 source models to the 3 representative unsecured target models. We then compare different attack methods on two best source models against all unsecured target models: the best ResNet source model and the best DenseNet source model. The multi-step (e.g., 10 step) transferability from all source models to three representative target models (VGG19, SE154 and IncV3) is illustrated in Figure 2. In all transfer scenarios, our proposed SGM outperforms existing methods consistently on almost all source models except RN18. Adversarial attacks crafted by SGM become more transferable when there are more skip connections in the source model (e.g., from RN18 to DN201). An interesting observation is that, when the target model is shallow such as VGG19 (left figure in Figure 2), shallow source models transfer better, however, when the target model is deep such as SE154 and IncV3 (middle and right figures in Figure 2), deeper source models tend to have better transferability. We suspect this is due to the architectural similarities shared by the target and source models. Note that against the VGG19 target model, the success rate of baseline methods all drop significantly when the ResNet source models become more complex (from RN18 to RN152). The small variations at RN50 and DN121 source models may be caused by the architectural difference between RN18/34 which consist of normal residual blocks, RN50/101/152 which consist of “bottleneck” residual blocks and DN121/169/201 which has dense skip connections.
+
+Results for the best source models RN152 and DN201 against unsecured target models are reported in Table 3. The proposed SGM attack outperforms existing methods by a large margin consistently against different target models. Particularly, for transfer $\mathrm { D N } 2 0 1  \mathrm { S E } 1 5 4$ (a recent state-of-the-art DNN with only $2 . 2 5 1 \%$ top-5 error on ImageNet), SGM achieves a success rate of $7 2 . 0 3 \%$ , which is $> 7 \%$ and $> 1 0 \%$ higher than MI and DI respectively.
+
+Combining with Existing Methods. We further demonstrate that the adversarial transferability of skip connections can be exploited in combination with existing techniques. The experiments are conducted on DN201 (the best source model in the above multi-step experiments), and TI attack is excluded as it was originally proposed against secured models and demonstrates limited improvement over PGD against unsecured models. The results are reported in Table $4 ^ { * }$ . The transferability of MI and DI is improved remarkably of $1 1 . 9 8 \% \sim 2 1 . 9 8 \%$ by SGM. When combined with both MI and DI, SGM improves the state-of-the-art $\left( \mathrm { M I + D I } \right)$ transferability by a huge margin consistently against all target models. In particular, SGM pushes the new state-of-the-art to at least $8 0 . 5 2 \%$ which previously was only $71 \%$ . This illustrates that skip connections can be easily manipulated to craft highly transferable attacks against many state-of-the-art DNN models.
+
+Table 4: Combined with existing methods: the success rates $( \% )$ of attacks crafted on source model DN201 against 7 unsecured target models. The best results are in bold and $^ +$ indicates improvement.   
+
+<table><tr><td>Attack\Target</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td>MI</td><td>75.09</td><td>76.39</td><td>99.84</td><td>64.38</td><td>59.62</td><td>54.85</td><td>50.05</td></tr><tr><td>MI+SGM DI</td><td>+12.01 78.11</td><td>+13.24 78.18</td><td>99.52 99.81</td><td>+17.16 61.75</td><td>+21.88 60.04</td><td>+15.57 56.15</td><td>+18.35 49.00</td></tr><tr><td>DI+SGM</td><td>+12.28</td><td>+13.76</td><td>99.52</td><td>+20.92</td><td>+17.66</td><td>+15.78</td><td>+20.20</td></tr><tr><td>MI+DI</td><td>87.16</td><td>87.28</td><td>99.76</td><td>79.80</td><td>76.68</td><td>75.20</td><td>71.05</td></tr><tr><td>MI+DI+SGM</td><td>93.00</td><td>93.92</td><td>99.42</td><td>89.86</td><td>85.72</td><td></td><td>80.50</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>81.23</td><td></td></tr></table>
+
+Table 5: Transferability against secured models: the success rates $\% \pm$ std over 5 random runs) of multi-step attacks crafted on RN152 and DN201 source models against 3 secured models. The best results are in bold.   
+
+<table><tr><td>Source</td><td>Attack</td><td>IncV3ens3</td><td>IncVens4</td><td>IncResens3</td></tr><tr><td rowspan="2">RN152</td><td>PGD TI</td><td>12.47±1.27 45.36±0.97</td><td>10.72±1.37 45.81±0.93</td><td>6.97±0.71 38.19±0.81</td></tr><tr><td>MI DI</td><td>24.20±1.15 28.48±1.21</td><td>22.04±0.98 24.19±1.22</td><td>16.10±0.56</td></tr><tr><td rowspan="2"></td><td>SGM</td><td>31.57±0.55</td><td>27.77±0.47</td><td>17.31±0.77 20.02±0.66</td></tr><tr><td>TI+SGM</td><td>52.62±0.40</td><td>52.80±0.79</td><td>43.96±0.62</td></tr><tr><td rowspan="2"></td><td>PGD TI</td><td>18.16±0.56</td><td>15.30±0.62</td><td>10.40±0.49</td></tr><tr><td>MI</td><td>42.76±0.91</td><td>42.01±0.79</td><td>34.28±0.88</td></tr><tr><td rowspan="4">DN201</td><td>DI</td><td>31.79±0.83</td><td>28.21±0.15</td><td>20.60±0.38</td></tr><tr><td></td><td>34.84±1.35</td><td>29.23±0.83</td><td>21.64±0.80</td></tr><tr><td>SGM</td><td>41.45±0.30</td><td>37.85±0.22</td><td>29.41±0.02</td></tr><tr><td>TI+SGM</td><td>46.11±1.23</td><td>47.38±0.89</td><td>39.32±0.80</td></tr></table>
+
+# 4.2 TRANSFERABILITY AGAINST ROBUSTLY TRAINED MODELS
+
+The success rates of our SGM and other baseline methods against the 3 secured target models are reported in Table 5. Overall, with translation adaptation specifically designed for evading adversarially trained models, TI achieves the best standalone transferability, while SGM is the second best with higher success rates than either PGD, MI or DI. When combined with TI, SGM also improves the TI attack by a considerable margin across all transfer scenarios. This indicates that, although manipulating the skip connections alone may not sufficient to attack secured models, it still can make existing attacks more powerful. One interesting observation is that attacks crafted here on RN152 are more transferable than those crafted on DN201, which is quite the opposite to attacking unsecured models.
+
+# 4.3 A CLOSER LOOK AT SGM
+
+In this part, we conduct more experiments to investigate the gradient decay factor of our proposed SGM, and explore the potential use of SGM for ensemble-based attacks and white-box attacks.
+
+Effect of Residual Gradient Decay $\gamma$ . We test the transferability of our proposed SGM with varying decay parameter $\gamma \in \ [ 0 . 1 , 1 . 0 ]$ , where $\gamma = 1 . 0$ means no decay on the residual gradients. The attacks are crafted by 10-step SGM on 5000 random ImageNet validation images. The results against 3 target models (VGG19, SE154 and IncV3) are illustrated in Figure 3. As can be observed, the trends are very consistent against different target models. On DenseNet source models, decreasing decay parameter (increasing decay strength) tends to improve transferability until it exceeds a certain threshold, e.g., $\gamma = 0 . 5$ . This is because the decay encourages the attack to focus on more transferable low-level information, however, it becomes less sufficient if all high-level class-relevant information is ignored. On ResNet source models, decreasing decay parameter can constantly improve transferability for $\gamma \geq 0 . 2$ . Compared to DenseNet source models, ResNets require more decay on the residual gradients. Recalling that skip connections reveal more transferable information of the source model, ResNets require more penalty on the residual gradients to increase the importance of skip gradients that reveal more transferable information of the source model.
+
+![](images/77df147172ccb0b33b337486c0853bc3dac36248c6d98f8856e7b9983faf5869.jpg)  
+Figure 3: Parameter tuning: the success rates of black-box attacks crafted by 10-step SGM with varying decay parameter $\gamma \in [ 0 . 1 , 1 . 0 ]$ . The solid and dash curves represent results on ResNet and DenseNet source models respectively.
+
+Table 6: Multi-step transferability of ensemble-based attack: the success rates ( $\%$ ±std over 5 random runs) of multi-step attacks crafted by different methods on an ensemble of 3 source models (e.g. RN34, RN152 and DN201) against 7 unsecured target models. The best results are in bold.   
+
+<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td rowspan="2">RN34</td><td>PGD</td><td>86.69±0.20</td><td>99.99±0.01</td><td>99.99±0.02</td><td>69.65±0.71</td><td>65.95±0.35</td><td>59.30±0.32</td><td>53.91±0.40</td></tr><tr><td>TI</td><td>84.35±0.21</td><td>99.59±0.11</td><td>99.77±0.06</td><td>571.67±0.19</td><td>67.22±0.82</td><td></td><td>66.02±0.66 56.83±0.89</td></tr><tr><td>+ RN152</td><td>MI</td><td>92.86±0.19</td><td>99.91±0.04</td><td>99.91±0.06</td><td>86.11±0.38</td><td>83.25±0.35</td><td>79.25±1.16</td><td>76.53±0.66</td></tr><tr><td>+</td><td>DI</td><td>96.34±0.23</td><td>99.84±0.20</td><td>99.84±0.20</td><td>89.72±0.52</td><td>87.53±0.29</td><td>85.04±0.75</td><td>81.11±0.44</td></tr><tr><td rowspan="2">DN201</td><td>SGM</td><td>97.36±0.17</td><td>99.87±0.07</td><td>99.86±0.09</td><td>90.40±0.26</td><td>87.86±0.51</td><td>82.97±0.71</td><td>80.93±0.55</td></tr><tr><td>DI+SGM</td><td>98.65±0.08</td><td>99.84±0.04</td><td>99.86±0.04</td><td>94.36±0.19</td><td>93.08±0.41</td><td>89.56±0.07</td><td>88.27±0.43</td></tr></table>
+
+Table 7: Transferability of ensemble-based attack against secured models: the success rates (%±std over 5 random runs) of black-box attacks crafted on an ensemble of 3 source models (e.g. RN34, RN152 and DN201). The best results are in bold.   
+
+<table><tr><td>Source</td><td>Attack</td><td>IncV3ens3</td><td>IncVens4</td><td>IncResens3</td></tr><tr><td rowspan="3">RN34 +</td><td>PGD</td><td>37.63±0.37</td><td>32.69±0.62</td><td>23.49±0.55</td></tr><tr><td>TI</td><td>75.04±0.50</td><td>75.94±0.63</td><td>66.24±0.45</td></tr><tr><td>MI</td><td>54.68±0.27</td><td>50.24±0.48</td><td>39.27±0.33</td></tr><tr><td>RN152 +</td><td>DI</td><td>65.29±0.31</td><td>57.48±0.45</td><td>46.41±0.42</td></tr><tr><td rowspan="2">DN201</td><td>SGM</td><td>66.08±0.42</td><td>62.22±0.73</td><td>51.16±0.12</td></tr><tr><td>TI+SGM</td><td>87.65±1.00</td><td>85.11±0.27</td><td>77.75±0.41</td></tr></table>
+
+As for the selection of $\gamma$ under a scenario without knowing the target model, from Figure 3 and Appendix C, we can see that the influence of $\gamma$ is more related to the source model rather than the target model, that is, given a source model, the best $\gamma$ against different target models are generally the same. This makes the selection of $\gamma$ quite straightforward: choosing the best $\gamma$ on the source model(s). For instance, in Figure 3, suppose the unknown target model is SE154 (middle figure), the adversary could tune $\gamma$ on source model DN201 to attack VGG19 (left figure) and find the best $\gamma = 0 . 5$ . The attacks crafted on DN201 with $\gamma = 0 . 5$ indeed achieved the best success rate against the SE154 target model (and other target models).
+
+Ensemble-based Attacks. It has been shown that attacking multiple source models simultaneously can improve the transferability of the crafted adversarial examples, and is commonly adopted in practice. We follow the ensemble-based strategy (Liu et al., 2017) and craft attacks on an ensemble of RN34, RN152 and DN201. According to the discussion above, we select the best $\gamma$ individually for each source model: choose $\gamma$ for source RN34 and RN152 against target DN201, and $\gamma$ for source DN201 against target RN152. The success rates (transferability) against 7 unsecured models and 3 secured models are reported in Table 6 and Table 7 respectively. Similar to the results of single source model attacks, against unsecured target models, SGM has a similar standalone performance with DI, better than the others (except the two “white-box” scenarios against RN152 and DN201). When combined with other methods, e.g. DI, it improves the success rates again by a large margin.
+
+![](images/fd7af6e1000782748912c2d39b08b3926209699706aa20993c284144c2df03b2.jpg)  
+Figure 4: White-box success rate for FGSM versus SGM. In (b) and (c), each color corresponds to one model, with FGSM is represented by solid curve and SGM is represented by dashed curve.
+
+Against secured models, SGM achieves the second best standalone transferability, with TI is still the best. When combined with TI, SGM improves the success rate by $\sim 1 0 \%$ consistently against all secured target models. In particular, against $\mathrm { I n c V } 3 _ { e n s 3 }$ , $\mathrm { T I } { + } \mathrm { S G M }$ achieves higher success rate $( 8 7 . 6 5 \% )$ than reported in (Dong et al., 2019) $( 8 4 . 8 \% )$ , even if only 3 source models are used here and the source models (e.g. RN34, RN152 and DN201) are all of different architecture to IncV3 target model ((Dong et al., 2019) uses 6 source models including even the IncV3 model). From all aspects analyzed above, the existence of skip connections makes transferable attacks much easier to craft in practice.
+
+Improving Weak White-box Attacks. In addition to the black-box transferability, we next show that SGM can also improve the weak (one-step) white-box attack FGSM. Note that the one-step version of SGM is equivalent to FGSM plus residual gradient decay. Our experiments are conducted on the 8 source models, and the white-box success rates under maximum $L _ { \infty }$ perturbation $\epsilon = 8$ (a typical white-box setting) are shown in Figure 4a. As can be observed, using SGM can help improve the adversarial strength (i.e., higher success rate). We then vary the maximum perturbation $\epsilon \dot { \in } [ 1 , 6 4 ]$ , and show the results on ResNet and DenseNet models separately in Figure 4b and Figure $_ \mathrm { 4 c }$ . Compared to FGSM, SGM can always give better adversarial strength, except when $\epsilon$ is extremely small $( \epsilon \leq 2 )$ ). When the perturbation space becomes infinitely small, the loss landscape within the space becomes flat and the gradient points to the optimal perturbation direction. However, when the perturbation space expands, one-step gradient becomes less accurate due to changes in the loss landscape (success rate decreases as $\epsilon$ increases from 4 to 16), and in this case, the skip gradient which contains more low-level information is more reliable than the residual gradient (the improvement is more significant for $\epsilon \in [ 4 , 1 6 ]$ ). Another interesting observation is that adversarial strength decreases when the model becomes more complex from RN18 to RN152, or DN121 to DN201. This is likely because the loss landscape of complex models is steeper than shallow models, making one-step gradient less reliable.
+
+# 5 CONCLUSION
+
+In this paper, we have identified a surprising property of the skip connections used by many state-ofthe-art ResNet-like neural networks, that is, they can be easily used to generate highly transferable adversarial examples. To demonstrate this architectural “weakness”, we proposed the Skip Gradient Method (SGM) to craft adversarial examples using more gradients from the skip connections rather than the residual ones, via a decay factor on gradients. We conducted a series of transfer attack experiments with 8 source models and 10 target models including 7 unsecured and 3 secured models, and showed that attacks crafted by SGM have significantly better transferability than those crafted by existing methods. When combined with existing techniques, SGM can also boost state-of-the-art transferability by a huge margin. We believe the high adversarial transferability of skip connections is due to the fact that they expose extra low-level information which is more transferable across different DNNs. Our findings in this paper not only remind researchers in adversarial research to pay attention to the architectural vulnerability of DNNs, but also raise new challenges for secure DNN architecture design.
+
+# ACKNOWLEDGEMENT
+
+Shu-Tao Xia is supported in part by National Key Research and Development Program of China under Grant 2018YFB1800204, National Natural Science Foundation of China under Grant 61771273, R&D Program of Shenzhen under Grant JCYJ20180508152204044, and research fund of PCL Future Regional Network Facilities for Large-scale Experiments and Applications (PCL2018KP001).
+
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+
+# A VISUALIZATION OF ADVERSARIAL EXAMPLES CRAFTED BY SGM
+
+In this section, we visualize 6 clean images and their corresponding adversarial examples crafted using our SGM on either a ResNet-152 or a DenseNet201 in Figure 5. These visualization results show that the generated adversarial perturbations are human imperceptible.
+
+![](images/6643c5baa262f9dea9b86ebf084c63e78bd350983d85de61b79faf17880fc5c9.jpg)  
+Figure 5: Visualization of 6 clean images and their corresponding adversarial examples. The clean images are shown in the top row, adversarial images crafted on ResNet-152 are shown in the middle row, while those crafted on DenseNet-201 are shown in the bottom row. All adversarial images are crafted using our proposed SGM (10-step) under maximum perturbation $\epsilon = 1 6$ .
+
+# B COMPARISON WITH PREVIOUSLY PUBLISHED RESULTS
+
+In this section, we compare the experimental settings in previous and our works, and discuss some small discrepancies of the baseline performance reported in ours and previous works.
+
+Table 8 and 9 summarizes these differences for single-source and ensemble-based attack respectively. Out of all these works (Dong et al., 2018; 2019; Xie et al., 2019), results reported in (Xie et al., 2019) are more complete. Our reported success rate of baseline attacks (e.g. MI and DI) matches that reported in (Xie et al., 2019), sometimes even higher. The slight discrepancy is caused by the difference in experimental settings. Table 10 summarizes the different source models used by baseline attacks, and Table 11 summarizes the difference in dataset, number of test images, input image size, maximum $L _ { \infty }$ perturbation $\epsilon$ , number of attack steps $N$ and attack step size $\alpha$ . Compared to $2 9 9 \times 2 9 9$ image size, here we use a more standard image size $2 2 4 \times 2 2 4$ on ImageNet. The use of smaller input size may reduce the effectiveness of existing attacks (Simon-Gabriel et al., 2019).
+
+In another work by Liu et al. (2017), $81 \%$ success rate was reported for optimization-based attack crafted on ResNet-152 against target VGG16, which is higher than our $6 5 . 5 2 \%$ from ResNet-152 to VGG19. This is because they did not restrict the maximum perturbation $\epsilon$ . The root mean square deviation (RMSD) of their attacks is 22.83,which indicates that many pixels are perturbed more than 16 pixel values. In our experiments, the RMSD is 6.29 for PGD, 7.71 for SGM, and 12.55 for MI. This appears to be another reason for the performance discrepancy. Note that the advantage of bounded small perturbation is increasing imperceptibility to human observers (see Figure 5).
+
+Table 8: Previously reported attack success rates $( \% )$ of baseline single-source attacks against 6 target models. “-” means no results were reported.   
+
+<table><tr><td>Reference</td><td>Attack</td><td>IncV3</td><td>IncV4</td><td>IncRes</td><td>IncV3ens3</td><td>IncV3ens4</td><td>IncResens3</td></tr><tr><td rowspan="3">Dong et al. (2018)</td><td>FGSM</td><td>35.0</td><td>28.2</td><td>27.5</td><td>14.6</td><td>13.2</td><td>7.5</td></tr><tr><td>BIM</td><td>26.7</td><td>22.7</td><td>21.2</td><td>9.3</td><td>8.9</td><td>6.2</td></tr><tr><td>MI</td><td>53.6</td><td>48.9</td><td>44.7</td><td>22.1</td><td>21.7</td><td>12.9</td></tr><tr><td rowspan="3">Dong et al. (2019)</td><td>FGSM</td><td>-</td><td>-</td><td>-</td><td>20.2</td><td>17.7</td><td>9.9</td></tr><tr><td>MI</td><td>=</td><td>1</td><td>-</td><td>25.1</td><td>23.7</td><td>13.3</td></tr><tr><td>DI</td><td>-</td><td>-</td><td>-</td><td>40.5</td><td>36.0</td><td>24.1</td></tr><tr><td rowspan="4">Xie et al. (2019)</td><td>FGSM</td><td>34.4</td><td>28.50</td><td>27.1</td><td>12.4</td><td>11.0</td><td>6.0</td></tr><tr><td>PGD</td><td>20.8</td><td>17.2</td><td>14.9</td><td>5.4</td><td>4.6</td><td>2.8</td></tr><tr><td>MI</td><td>50.1</td><td>44.1</td><td>42.2</td><td>18.2</td><td>15.2</td><td>9.0</td></tr><tr><td>DI</td><td>53.8</td><td>49.0</td><td>44.8</td><td>13.0</td><td>11.1</td><td>6.9</td></tr><tr><td rowspan="4">Ours</td><td>FGSM</td><td>26.56</td><td>21.03</td><td>19.10</td><td>1</td><td>-</td><td>1</td></tr><tr><td>PGD</td><td>26.56</td><td>21.03</td><td>19.10</td><td>12.47</td><td>10.72</td><td>6.97</td></tr><tr><td>MI</td><td>50.22</td><td>43.32</td><td>41.71</td><td>24.20</td><td>22.04</td><td>16.10</td></tr><tr><td>DI</td><td>53.95</td><td>47.16</td><td>43.47</td><td>34.84</td><td>29.23</td><td>21.64</td></tr></table>
+
+Table 9: Previously reported attack success rates $( \% )$ of ensemble-based baseline attacks against 6 target models. “-” means no results were reported.   
+
+<table><tr><td>Reference</td><td>Attack</td><td>IncV3</td><td>IncV4</td><td>IncRes</td><td>IncV3ens3</td><td>IncV3ens4</td><td>IncResens3</td></tr><tr><td rowspan="3">Dong et al. (2018)</td><td>FGSM</td><td>45.7</td><td>39.9</td><td>38.8</td><td>15.4</td><td>15.0</td><td>6.4</td></tr><tr><td>BIM</td><td>72.1</td><td>61.0</td><td>54.4</td><td>18.6</td><td>18.7</td><td>9.9</td></tr><tr><td>MI</td><td>87.9</td><td>81.2</td><td>76.5</td><td>37.6</td><td>40.3</td><td>23.3</td></tr><tr><td rowspan="4">Dong et al. (2019)</td><td>FGSM</td><td>-</td><td>-</td><td>-</td><td>27.5</td><td>23.7</td><td>13.4</td></tr><tr><td>MI</td><td></td><td>1</td><td>1</td><td>50.5</td><td>48.3</td><td>32.8</td></tr><tr><td>DI</td><td>-</td><td>-</td><td>-</td><td>66.0</td><td>63.3</td><td>45.9</td></tr><tr><td>TI+DI</td><td>-</td><td>-</td><td>-</td><td>84.8</td><td>82.7</td><td>78.0</td></tr><tr><td rowspan="4">Xie et al. (2019)</td><td>PGD</td><td>43.7</td><td>36.4</td><td>33.3</td><td>12.9</td><td>15.1</td><td>8.8</td></tr><tr><td>MI</td><td>69.9</td><td>67.9</td><td>64.1</td><td>36.3</td><td>35.0</td><td>30.4</td></tr><tr><td>DI</td><td>71.4</td><td>65.9</td><td>64.6</td><td>22.8</td><td>26.1</td><td>15.8</td></tr><tr><td>DI+MI</td><td>80.7</td><td>80.6</td><td>80.7</td><td>44.6</td><td>44.5</td><td>39.4</td></tr><tr><td rowspan="5">Ours</td><td>PGD</td><td>65.95</td><td>59.30</td><td>53.91</td><td>37.63</td><td>32.69</td><td>23.49</td></tr><tr><td>MI</td><td>83.25</td><td>79.25</td><td>76.53</td><td>54.68</td><td>50.24</td><td>39.27</td></tr><tr><td>DI</td><td>87.53</td><td>85.04</td><td>81.11</td><td>65.29</td><td>57.48</td><td>46.41</td></tr><tr><td>DI+SGM</td><td>93.08</td><td>89.56</td><td>88.27</td><td>80.14</td><td>76.52</td><td>66.40</td></tr><tr><td>TI+SGM</td><td>-</td><td>-</td><td>-</td><td>87.65</td><td>85.11</td><td>77.75</td></tr></table>
+
+Table 10: Source models used by existing single-source and ensemble-based black-box attacks. “Hold-out” refers to the hold-out target model from the group, with all remaining models are used as source models. Group 1 consists of ResNet-v2-152, IncV3, IncV4 and IncRes, while group 2 consists of ResNet-v2-152, IncV3, IncV4, IncRes, $\mathrm { I n c V } 3 _ { e n s 3 }$ , $\mathrm { I n c V } 3 _ { e n s 4 }$ and $\mathrm { I n c R e s } _ { e n s 3 }$ .   
+
+<table><tr><td>Reference</td><td>Single-source attack</td><td>Ensemble-basedattack</td></tr><tr><td>Dong et al. (2018)</td><td>ResNet-v2-152</td><td>hold-out from group 1 or group 2</td></tr><tr><td>Dong et al. (2019)</td><td>ResNet-v2-152</td><td>hold-out from group 1</td></tr><tr><td>Xie et al. (2019)</td><td>ResNet-v2-152</td><td>hold-out from group 2</td></tr><tr><td>Ours</td><td>RN152</td><td>RN34+RN152+DN201</td></tr></table>
+
+For proper implementation, we use open-source codes and pretrained models for our experiments, e.g., AdverTorch (Ding et al., 2019) for FGSM, PGD and MI, and source/target models from two GitHub repositories†‡§ for all models. We reproduced DI and TI in PyTorch.
+
+Table 11: Difference in experimental settings of our work compared to previous works.“NeurIPS 2017” indicates the dataset used for NeurIPS 2017 adversarial competition. : maximum per-pixel perturbation; $N$ : number of attack steps; $\alpha$ : attack step size.   
+
+<table><tr><td>Reference</td><td>Dataset</td><td>Number</td><td>Input size</td><td>E</td><td>N</td><td>α</td></tr><tr><td>Dong et al. (2018)</td><td>ImageNet</td><td>1000</td><td>299 × 299</td><td>16</td><td>10</td><td>1.6</td></tr><tr><td>Dong et al. (2019)</td><td>NeurIPS 2017</td><td>1000</td><td>299 × 299</td><td>16</td><td>10</td><td>1.6</td></tr><tr><td>Xie et al. (2019)</td><td>ImageNet</td><td>5000</td><td>299×299</td><td>15</td><td>19</td><td>1.0</td></tr><tr><td>Ours</td><td>ImageNet</td><td>5000</td><td>224×224</td><td>16</td><td>10/20</td><td>2.0</td></tr></table>
+
+# C TRANSFERABILITY OF DECAY PARAMETER $\gamma$
+
+In this section, we study the “transferability” of decay parameter $\gamma$ across different target models. RN152 and DN201 are used as the source model, and the target model are varying to observe trends. As indicated in Figure 6, all black-box target models share the same best selection of $\gamma$ , which makes $\gamma$ selection quite simple. Even if the true target model is unknown, the adversary can tune the gamma for a resnet-like neural network through another model and obtain the best selection as well.
+
+![](images/300f0293aa251c6b51c2049a3d1a099b4b5c4e8f2ff590b6125950515a93bfe7.jpg)  
+Figure 6: “Transferability” of decay parameter: the success rates of black-box attacks crafted by 10-step SGM with varying decay parameter $\gamma \in \ [ 0 . 1 , 1 . 0 ]$ . The curves represent results against different target models respectively.

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+    {
+        "type": "text",
+        "text": "SKIP CONNECTIONS MATTER: ON THE TRANSFERABILITY OF ADVERSARIAL EXAMPLES GENERATED WITH RESNETS ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dongxian $\\mathbf { W _ { u } } ^ { 1 , 3 }$ Yisen $\\mathbf { W a n g } ^ { 2 \\dagger }$ Shu-Tao $\\mathbf { X _ { i a } ^ { \\bullet , 1 , 3 } }$ James Bailey4 Xingjun Ma4 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Tsinghua University   \n2Shanghai Jiao Tong University   \n3PCL Research Center of Networks and Communications, Peng Cheng Laboratory   \n4The University of Melbourne ",
+        "bbox": [
+            184,
+            210,
+            725,
+            266
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            303,
+            544,
+            318
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Skip connections are an essential component of current state-of-the-art deep neural networks (DNNs) such as ResNet, WideResNet, DenseNet, and ResNeXt. Despite their huge success in building deeper and more powerful DNNs, we identify a surprising security weakness of skip connections in this paper. Use of skip connections allows easier generation of highly transferable adversarial examples. Specifically, in ResNet-like (with skip connections) neural networks, gradients can backpropagate through either skip connections or residual modules. We find that using more gradients from the skip connections rather than the residual modules according to a decay factor, allows one to craft adversarial examples with high transferability. Our method is termed Skip Gradient Method (SGM). We conduct comprehensive transfer attacks against state-of-the-art DNNs including ResNets, DenseNets, Inceptions, Inception-ResNet, Squeeze-and-Excitation Network (SENet) and robustly trained DNNs. We show that employing SGM on the gradient flow can greatly improve the transferability of crafted attacks in almost all cases. Furthermore, SGM can be easily combined with existing black-box attack techniques, and obtain high improvements over state-of-the-art transferability methods. Our findings not only motivate new research into the architectural vulnerability of DNNs, but also open up further challenges for the design of secure DNN architectures. ",
+        "bbox": [
+            233,
+            333,
+            764,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            611,
+            336,
+            626
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In deep neural networks (DNNs), a skip connection builds a short-cut from a shallow layer to a deep layer by connecting the input of a convolutional block (also known as the residual module) directly to its output. While different layers of a neural network learn different “levels” of features, skip connections can help preserve low-level features and avoid performance degradation when adding more layers. This has been shown to be crucial for building very deep and powerful DNNs such as ResNet (He et al., 2016a;b), WideResNet (Zagoruyko & Komodakis, 2016), DenseNet (Huang et al., 2017) and ResNeXt (Xie et al., 2017). In the meantime, despite their superior performance, DNNs have been found extremely vulnerable to adversarial examples (or attacks), which are input examples slightly perturbed with an intention to fool the network to make a wrong prediction (Szegedy et al., 2013; Goodfellow et al., 2014; Ma et al., 2018; Bai et al., 2019; Wang et al., 2019; 2020). Adversarial examples often appear imperceptible to human observers, and are transferable across different models (Liu et al., 2017). This has raised security concerns on the deployment of DNNs in security critical scenarios, such as face recognition (Sharif et al., 2016), autonomous driving (Evtimov et al., 2018), video analysis (Jiang et al., 2019) and medical diagnosis (Ma et al., 2019). ",
+        "bbox": [
+            174,
+            636,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adversarial examples can be crafted following either a white-box setting (the adversary has full access to the target model) or a black-box setting (the adversary has no information of the target model). White-box methods such as Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014), Basic Iterative Method (BIM) (Kurakin et al., 2016), Projected Gradient Decent (PGD) (Madry et al., 2018) and Carlini and Wagner (CW) (Carlini & Wagner, 2017) often suffer from low transferability in a black-box setting, thus posing only limited threats to DNN models which are usually kept secret in practice (Dong et al., 2018; Xie et al., 2019). Several techniques have been proposed to improve the transferability of black-box attacks crafted on a surrogate model, such as momentum boosting (Dong et al., 2018), diverse input (Xie et al., 2019) and translation invariance (Dong et al., 2019). Although these techniques are effective, they (as well as white-box methods) all treat the entire network (either the target model or the surrogate model) as a single component while ignore its inner architectural characteristics. The question of whether or not the DNN architecture itself can expose more transferability of adversarial attacks is an unexplored problem. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/89948fe9f78fd82465425cf0071bf4e77b2e8e2b2ab3eab2c46f940ffe2c3249.jpg",
+        "image_caption": [
+            "Figure 1: Left: Illustration of the last 3 skip connections (green lines) and residual modules (black boxes) of a ImageNet-trained ResNet-18. Right: The success rate (in the form of “white-box/blackbox”) of adversarial attacks crafted using gradients flowing through either a skip connection (going upwards) or a residual module (going leftwards) at each junction point (circle). Three example backpropagation paths are highlighted in different colors, with the green path skipping over the last two residual modules having the best attack success rate while the red path through all 3 residual modules having the worst attack success rate. The attacks are crafted by BIM on 5000 ImageNet validation images under maximum $L _ { \\infty }$ perturbation $\\epsilon \\ : = \\ : 1 6$ (pixel values are in [0, 255]). The black-box success rate is tested against a VGG19 target model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            102,
+            805,
+            246
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            410,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we identify one such weakness about the skip connections used by many state-of-theart DNNs. We first conduct a toy experiment with the BIM attack and ResNet-18 on the ImageNet validation dataset (Deng et al., 2009) to investigate how skip connections affect the adversarial strength of attacks crafted on the network. At each of the last 3 skip connections and residual modules of ResNet-18, we illustrate the success rate of attacks crafted using gradients backpropagate through either the skip connection or the residual module in Figure 1. As can be observed, the success rate drops more drastically whenever using gradients from a residual module instead of the skip connection. This implies that gradients from the skip connections are more vulnerable (high success rate). In addition, we surprisingly find that skip connections expose more transferable information. For example, the black-box success rate was even improved from $5 2 . 5 2 \\%$ to $6 2 . 1 0 \\%$ when the attack skips the last two residual modules (following the path in green color). ",
+        "bbox": [
+            174,
+            525,
+            825,
+            674
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the above observations, in this paper, we propose the Skip Gradient Method (SGM) to generate adversarial examples using gradients more from the skip connections rather than the residual modules. In particular, SGM utilizes a decay factor to reduce gradients from the residual modules. We find that this simple adjustment on the gradient flow can generate highly transferable adversarial examples, and the more skip connections in a network, the more transferable are the crafted attacks. This is in sharp contrast to the design principles (e.g., “going deeper” with skip connections) underpinning many modern DNNs. In particular, our main contributions are: ",
+        "bbox": [
+            173,
+            681,
+            825,
+            776
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We identify one surprising property of skip connections in ResNet-like neural networks, i.e., they allow an easy generation of highly transferable adversarial examples.   \n• We propose the Skip Gradient Method (SGM) to craft adversarial examples using gradients more from the skip connections. Using a single decay factor on gradients, SGM is an appealingly simple and generic technique that can be used by any existing gradient-based attack methods.   \n• We provide comprehensive transfer attack experiments, from different source models against 10 state-of-the-art DNNs, showing that SGM can greatly improve the transferability of crafted adversarial examples. When combined with existing transfer techniques, SGM improves the state-ofthe-art transferability benchmarks by a large margin. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Existing adversarial attacks can be categorized into two groups: 1) white-box attacks and 2) blackbox attacks. In the white-box setting, the adversary has full access to the parameters of the target model, while in the black-box setting, the target model is kept secret from the adversary. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 WHITE-BOX ATTACKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            175,
+            369,
+            189
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a clean example $_ { \\textbf { \\em x } }$ with class label $y$ and a target DNN model $f$ , the goal of an adversary is to find an adversarial example $\\mathbf { { x } } _ { a d v }$ that fools the network into making an incorrect prediction (eg. $f ( { \\pmb x } _ { a d v } ) \\neq y )$ , while still remaining in the $\\epsilon$ -ball centered at $_ { \\textbf { \\em x } }$ (eg. $\\| \\pmb { x } _ { a d v } - \\pmb { x } \\| _ { \\infty } \\leq \\bar { \\epsilon } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014). FGSM perturbs clean example $_ { \\textbf { \\em x } }$ for one step by the amount of $\\epsilon$ along the gradient direction: ",
+        "bbox": [
+            171,
+            244,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d64ab0344ffcd26240cb240fda5ca70a20eeb0701be03fe9e3d778006a6e438.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { a d v } = \\pmb { x } + \\epsilon \\cdot \\mathrm { s i g n } \\big ( \\nabla _ { \\pmb { x } } \\ell ( f ( \\pmb { x } ) , y ) \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            279,
+            620,
+            295
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Basic Iterative Method (BIM) (Kurakin et al., 2016) is an iterative version of FGSM that perturbs for $T$ steps with step size $\\epsilon / T$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Projected Gradient Descent (PGD) (Madry et al., 2018). PGD perturbs normal example $_ { \\textbf { \\em x } }$ for $T$ steps with smaller step size. After each step of perturbation, PGD projects the adversarial example back onto the $\\epsilon$ -ball of $_ { \\textbf { \\em x } }$ , if it goes beyond the $\\epsilon$ -ball: ",
+        "bbox": [
+            173,
+            334,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d8287fe52c2380fa68a3a31e0f3c279e44102082c8cd35c56332af58e92a100b.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { a d v } ^ { t + 1 } = \\Pi _ { \\epsilon } \\big ( \\pmb { x } _ { a d v } ^ { t } + \\alpha \\cdot \\mathrm { s i g n } ( \\nabla _ { \\pmb { x } } \\ell ( f ( \\pmb { x } _ { a d v } ^ { t } ) , y ) ) \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            382,
+            658,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\Pi _ { \\epsilon } ( \\cdot )$ is the projection operation. Different to BIM, PGD allows step size $\\alpha > \\epsilon / T$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are also other types of white-box attacks including sparsity-based methods such as Jacobianbased Saliency Map Attack (JSMA) (Papernot et al., 2016), sparse attack (Modas et al., 2019), onepixel attack (Su et al., 2019), and optimization-based methods such as Carlini and Wagner (CW) (Carlini & Wagner, 2017) and elastic-net (EAD) (Chen et al., 2018). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 BLACK-BOX ATTACKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Black-box attacks can be generated by either attacking a surrogate model or using gradient estimation methods in combination with queries to the target model. Gradient estimation methods estimate the gradients of the target model using black-box optimization methods such as Finite Differences (FD) (Chen et al., 2017; Bhagoji et al., 2018) or Natural Evolution Strategies (NES) (Ilyas et al., 2018; Jiang et al., 2019). These methods all require a large number of queries to the target model, which not only reduces efficiency but also potentially exposes the attack. Alternatively, black-box adversarial examples can be crafted on a surrogate model then applied to attack the target model. Although the white-box methods can be directly applied on the surrogate model, they are far less effective in the black-box setting (Dong et al., 2018; Xie et al., 2019). Several transfer techniques have been proposed to improve the transferability of black-box attacks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Momentum Iterative boosting (MI) (Dong et al., 2018). MI incorporates a momentum term into the gradient to boost the transferability: ",
+        "bbox": [
+            173,
+            655,
+            823,
+            683
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6157f4b88c172666fbec43a63df0b2350b915a9ec072aa2539389b1314e7141f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { a d v } ^ { t + 1 } = \\Pi _ { \\epsilon } \\big ( \\pmb { x } _ { a d v } ^ { t } + \\alpha \\cdot \\mathrm { s i g n } ( \\pmb { g } ^ { t + 1 } ) \\big ) , \\pmb { g } ^ { t + 1 } = \\mu \\cdot \\pmb { g } ^ { t } + \\frac { \\nabla _ { \\mathbf { x } } \\ell \\left( f \\left( \\pmb { x } _ { a d v } ^ { t } \\right) , \\pmb { y } \\right) } { \\| \\nabla _ { \\mathbf { x } } \\ell \\left( f \\left( \\pmb { x } _ { a d v } ^ { t } \\right) , \\pmb { y } \\right) \\| _ { 1 } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            689,
+            750,
+            724
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g ^ { t }$ is the adversarial gradient at the $t$ -th step, $\\alpha = \\epsilon / T$ is the step size for a total of $T$ steps, $\\mu$ is a decay factor, and $\\| \\cdot \\| _ { 1 }$ is the $L _ { 1 }$ norm. ",
+        "bbox": [
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+            729,
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+            758
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Diverse Input $\\mathbf { \\Pi } ( \\mathbf { D } \\mathbf { I } )$ (Xie et al., 2019). DI proposes to craft adversarial exampels using gradient with respect to the randomly-transformed input example: ",
+        "bbox": [
+            171,
+            765,
+            823,
+            792
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c06c8e715314122b4576842d9d139a7d15f0bde93c4cc194d80777ba1905afb1.jpg",
+        "text": "$$\n\\pmb { x } _ { a d v } ^ { t + 1 } = \\Pi _ { \\epsilon } \\big ( \\pmb { x } _ { a d v } ^ { t } + \\alpha \\cdot \\mathrm { s i g n } \\big ( \\nabla _ { \\pmb { x } } \\ell \\big ( f \\big ( H ( \\pmb { x } _ { a d v } ^ { t } ; p ) \\big ) , y \\big ) \\big ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            799,
+            681,
+            819
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $H ( \\boldsymbol { \\mathbf { x } } _ { a d v } ^ { t } ; \\boldsymbol { p } )$ is a stochastic transformation function on $\\scriptstyle { \\pmb { x } } _ { a d v } ^ { t }$ for a given probability $p$ ",
+        "bbox": [
+            169,
+            824,
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+            840
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Translation Invariant (TI) (Dong et al., 2019). TI targets to evade robustly trained DNNs by generating adversarial examples that are less sensitive to the discriminative regions of the surrogate model. More specifically, $\\mathrm { T I }$ computes the gradients with respect to a set of translated versions of the original input: ",
+        "bbox": [
+            174,
+            844,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd574f93f89df63ceb544a6e24a122c069a8c4acc24182f82096aebd6e467011.jpg",
+        "text": "$$\n\\pmb { x } _ { a d v } ^ { t + 1 } = \\Pi _ { \\epsilon } \\bigl ( \\pmb { x } _ { a d v } ^ { t } + \\alpha \\cdot \\mathrm { s i g n } ( W * \\nabla _ { \\pmb { x } } \\ell ( f ( \\pmb { x } _ { a d v } ^ { t } ) , y ) ) \\bigr ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            906,
+            676,
+            926
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W$ is a predefined kernel (e.g., uniform, linear, and Gaussian) matrix of size $( 2 k + 1 ) ( 2 k +$ 1) $k$ being the maximal number of pixels to shift). This kernel convolution is equivalent to the weighted sum of gradients over $( 2 k + 1 ) ^ { 2 }$ number of shifted input examples. ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, there are other studies focusing on intermediate feature representations. For example, Activation Attack (Inkawhich et al., 2019) drives the activation of a specified layer on a given image towards the layer of a target image, to yield a highly transferable targeted example. Intermediate Level Attack (Huang et al., 2019) attempts to fine-tune an existing adversarial example for greater black-box transferability by increasing its perturbation on a pre-specified layer of the source model. ",
+        "bbox": [
+            174,
+            151,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Although the above transfer techniques are effective, they (including white-box attacks) either 1) treat the network (either the surrogate model or the target model) as a single component or 2) only use the intermediate layer output of the network. In other words, they do not directly consider the effects of different DNN architectural characteristics. Li et al. (2018) investigated the use of skip connections and dropout layers for sampling networks, which generates a huge set of ghost networks to perform an ensemble attack. Here, we focus on the architectural property of skip connections from the gradient view without modifying or generating any extra networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED SKIP GRADIENT ATTACK ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first introduce the gradient decomposition of skip connection and residual module. Following that, we propose our Skip Gradient Method (SGM), then demonstrate the adversarial transferability property of skip connection via a case study. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 GRADIENT DECOMPOSITION WITH SKIP CONNECTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In ResNet-like neural networks, a skip connection uses identity mapping to bypass residual layers, allowing data flow from a shallow layer directly to subsequent deep layers. Thus, we can decompose the network into a collection of paths of different lengths (Veit et al., 2016). We denote a skip connection together with its associated residual module as a building block (residual block) of a network. Considering three successive building blocks (eg. $z _ { i + 1 } = z _ { i } + f _ { i + 1 } ( z _ { i } ) )$ in a residual network from input $z _ { \\mathrm { 0 } }$ to output $z _ { 3 }$ , the output $z _ { 3 }$ can be expanded as: ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b4afc7842942f6a150be39aebffe07d1f892a8d4032eb132b894c8dcf557ada5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z _ { 3 } = z _ { 2 } + f _ { 3 } ( z _ { 2 } ) = [ z _ { 1 } + f _ { 2 } ( z _ { 1 } ) ] + f _ { 3 } ( z _ { 1 } + f _ { 2 } ( z _ { 1 } ) ) } \\\\ & { \\quad = [ z _ { 0 } + f _ { 1 } ( z _ { 0 } ) + f _ { 2 } ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) ] + f _ { 3 } \\big ( ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) + f _ { 2 } ( z _ { 0 } + f _ { 1 } ( z _ { 0 } ) ) \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            228,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "According to the chain rule in calculus, the gradient of a loss function $\\ell$ with respect to input $z _ { \\mathrm { 0 } }$ can then be decomposed as, ",
+        "bbox": [
+            176,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3746d31c6e7ef3deaccc511c3473155a352180984b4a17b01ac3f962c151529a.jpg",
+        "text": "$$\n\\frac { \\partial \\ell } { \\partial z _ { 0 } } = \\frac { \\partial \\ell } { \\partial z _ { 3 } } \\frac { \\partial z _ { 3 } } { \\partial z _ { 2 } } \\frac { \\partial z _ { 2 } } { \\partial z _ { 1 } } \\frac { \\partial z _ { 1 } } { \\partial z _ { 0 } } = \\frac { \\partial \\ell } { \\partial z _ { 3 } } ( 1 + \\frac { \\partial f _ { 3 } } { \\partial z _ { 2 } } ) ( 1 + \\frac { \\partial f _ { 2 } } { \\partial z _ { 1 } } ) ( 1 + \\frac { \\partial f _ { 1 } } { \\partial z _ { 0 } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Extending this toy example to a network with $L$ residual blocks, the gradient can be decomposed from $L$ -th to the $( l + 1 )$ -th $( 0 \\leq l < L )$ ) residual block as, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e87816afa62c5f9c8f8a46f0eec78c39e411a0a5f0cf0f81f9fe1e0e171b5f44.jpg",
+        "text": "$$\n\\frac { \\partial \\ell } { \\partial \\pmb { x } } = \\frac { \\partial \\ell } { \\partial z _ { L } } \\prod _ { i = l } ^ { L - 1 } \\big ( \\frac { \\partial f _ { i + 1 } } { \\partial z _ { i } } + 1 \\big ) \\frac { \\partial z _ { l } } { \\partial \\pmb { x } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The example illustrated in Figure 1 is a the above decomposition of a ResNet-18 at the last 3 building blocks $( l = L - 3 )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 SKIP GRADIENT METHOD (SGM) ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to use more gradient from the skip connections, here, we introduce a decay parameter into the decomposed gradient to reduce the gradient from the residual modules. Following the decomposition in Equation (8), the “skipped” gradient is, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d77627ff2acda389bf120df9bca12e0af02b1b05beba09101e6ce1eeaba6b4a.jpg",
+        "text": "$$\n\\nabla _ { \\pmb { x } } \\ell = \\frac { \\partial \\ell } { \\partial z _ { L } } \\prod _ { i = 0 } ^ { L - 1 } \\big ( \\gamma \\frac { \\partial f _ { i + 1 } } { \\partial z _ { i } } + 1 \\big ) \\frac { \\partial z _ { 0 } } { \\partial \\pmb { x } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            806,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\boldsymbol { z } } _ { 0 } = { \\boldsymbol { x } }$ is the input of the network, and $\\gamma \\in ( 0 , 1 ]$ is the decay parameter. Accordingly, given a clean example $_ { \\textbf { \\em x } }$ and a DNN model $f$ , an adversarial example can be crafted iteratively by, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc96adf4049f5957637b3b2a37c6c735bf27ce6d5c265b78b1505f1a3e99a8b4.jpg",
+        "text": "$$\n\\pmb { x } _ { a d v } ^ { t + 1 } = \\Pi _ { \\epsilon } \\Bigl ( \\pmb { x } _ { a d v } ^ { t } + \\alpha \\cdot \\mathrm { s i g n } \\bigl ( \\frac { \\partial \\ell } { \\partial z _ { L } } \\prod _ { i = 0 } ^ { L - 1 } ( \\gamma \\frac { \\partial f _ { i + 1 } } { \\partial z _ { i } } + 1 ) \\frac { \\partial z _ { 0 } } { \\partial x } \\bigr ) \\Bigr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/a24f16087c784b162fcc267993f3aad4dd8e332204f3f51ae70a147ba503903a.jpg",
+        "table_caption": [
+            "Table 1: The success rates ( $\\%$ ±std over 5 random runs) of black-box attacks (untargeted) crafted by PGD and its “skip gradient” (SGM) version, on different source models against a Inception V3 target model. The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>RN18 RN34</td><td>RN50 RN101 RN152</td><td>DN121 DN169</td><td>DN201</td></tr><tr><td>PGD</td><td>23.23±0.69 24.38±0.41</td><td>22.80±0.55 22.98±0.83 26.56±0.75</td><td></td><td>30.71±0.60 30.90±0.31 36.01±0.59</td></tr><tr><td>SGM</td><td>28.92±0.45 43.43±0.32</td><td>36.71±0.55 38.38±0.53 44.84±0.14</td><td>57.38±0.14 60.45±0.42 65.48±0.23</td><td></td></tr></table>",
+        "bbox": [
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+            154,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SGM is a generic technique that can be easily implemented on any neural network that has skip connections. During the backpropagation process, SGM simply multiplies the decay parameter to the gradient whenever it passes a residual module. Therefore, SGM does not require any computation overhead, and works efficiently even on densely connected networks such as DenseNets. The reduction of residual gradients is accumulated along the backpropagation path, that is, the residual gradients at lower layers will be reduced more times than those at higher layers. This is because, compared to high-level features, low-level features have already been well preserved by skip connections (see feature decompositions in Equation (6)). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 ADVERSARIAL TRANSFERABILITY WITH SKIP CONNECTIONS: A CASE STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the adversarial transferability of skip connections, we conduct a case study on 10- step PGD, and their corresponding SGM versions, to investigate the success rates of black-box attacks crafted with or without manipulating the skip connections. The black-box attacks are generated on 8 different source (surrogate) models ResNet(RN)-18/34/50/101/152 and DenseNet(DN)- 121/169/201, then applied to attack a Inception V3 target model. All models were trained on ImageNet training set. We randomly select 5000 ImageNet validation images that are correctly classified by all source models, and craft untargeted attacks under maximum $L _ { \\infty }$ perturbation $\\epsilon = 1 6$ , which is a typical black-box setting (Dong et al., 2018; Xie et al., 2019; Dong et al., 2019). The step size of PGD was set to $\\alpha = 2$ , and the decay parameter of SGM was set to $\\gamma = 0 . 5$ . ",
+        "bbox": [
+            174,
+            366,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We run the attack for 5 times with different random seeds, and report the success rates (transferability) of different methods in Table 1. As can be seen, when the skip connections are manipulated with our SGM, the transferability of PGD is greatly improved across all source models. On all source models except RN18, the improvements are more than $1 3 \\%$ . Without SGM, the best transferability against the Inception-V3 target model is $3 5 . 4 8 \\%$ which is achieved by PGD on DN201, however, this is improved further by our proposed SGM to $6 5 . 3 8 \\%$ $( > 2 9 \\%$ gain). This not only highlights the surprising property of skip connections in terms of the generation of highly transferable attacks, but also indicates the significance of this property, as such a huge boost in transferability only takes a single decay factor. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The 8 source models can be interpreted as from 3 ResNet families: 1) RN18/34 are ResNets with normal residual blocks, 2) RN50/101/152 are ResNets with “bottleneck” residual blocks, and 3) DN121/169/201 are densely connected ResNets. Another important observation is that when there are more skip connections in a network within the same ResNet family (e.g., $\\mathrm { R N } 3 4 > \\mathrm { R N } 1 8$ , $\\mathrm { R N } 1 5 2 > \\mathrm { R N } 1 0 1 > \\mathrm { R N } 5 0$ , and $\\mathrm { D N } 2 0 1 > \\mathrm { D N } 1 6 9 > \\mathrm { D N } 1 2 1 )$ , or from ResNets to DenseNets (e.g. $\\mathrm { D N } 1 2 1 / 1 6 9 / 2 0 1 > \\mathrm { R N } 1 8 / 3 4$ and $\\mathrm { D N } 1 2 1 / 1 6 9 / 2 0 1 > \\mathrm { R N } 5 0 / 1 0 1 / 1 5 2 )$ , the crafted adversarial examples become more transferable, especially when the skip connections are manipulated by our SGM. This raises questions about the design principle behind many state-of-the-art DNNs: “going deeper” with techniques like skip connection and $1 \\times 1$ convolution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 COMPARISON TO EXISTING TRANSFER ATTACKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare the transferability of adversarial examples crafted by our proposed SGM and existing methods on ImageNet against both unsecured and secured target models. ",
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+    {
+        "type": "text",
+        "text": "Baselines. We compare SGM with FGSM, PGD, and 3 state-of-the-art transfer attacks: (1) Momentum Iterative (MI) (Dong et al., 2018), (2) Diverse Input (DI) (Xie et al., 2019), and (3) Transition Invariant (TI) (Dong et al., 2019). Note that the TI attack was originally proposed to attack secured models, although here we include TI to attack both unsecured models and secured models. For TI and our SGM, we test both the one-step and the iterative version, however, the other methods DI and MI only have an iterative version. The iteration step is set to 10 and 20 for unsecured and secured target models respectively. For all iterative methods PGD, TI and our SGM, the step size is set to $\\alpha = 2$ . For our proposed SGM, the decay parameter is set to $\\gamma = 0 . 2$ (0.5) and $\\bar { \\gamma } = 0 . 5$ (0.7) ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "on ResNet and DenseNet source models in PGD (FGSM) respectively. For simplicity, we utilize SGM to indicate $\\mathrm { F G S M + S G M }$ in one-step attacks, and $\\mathrm { P G D + S G M }$ in multi-step attacks. Other parameters of existing methods are configured as in their original papers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Threat Model. We adopt a black-box threat model in which adversarial examples are generated by attacking a source model and then applied to attack the target model. The target model is of a different architecture (indicated by the model name) to the source model, expect when the source and target models are of the same architecture, where we directly use the source model as the target model (equivalent to a white-box setting). The attacks are crafted on 5000 randomly selected ImageNet validation images that are classified correctly by all source models, and are repeated for 5 times with different random seeds. For all attack methods, we follow the standard setting (Dong et al., 2018; Xie et al., 2019) to craft untargeted attacks under maximum $L _ { \\infty }$ perturbation $\\epsilon = 1 6$ with respect to pixel values in [0, 255]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Target Models. We consider two types of target models: 1) unsecured models that are trained on ImageNet training set using traditional training; and 2) secured models trained using adversarial training. For unsecured target model, we choose 7 state-of-the-art DNNs: VGG19 (with batch normalization) (Simonyan & Zisserman, 2015), ResNet-152 (RN152) (He et al., 2016a), DenseNet201 (DN152), 154 layer Squeeze-and-Excitation network (SE154) (Hu et al., 2018), Inception V3 (IncV3) (Szegedy et al., 2016), Inception V4 (IncV3) (Szegedy et al., 2017) and Inception-ResNet V2 (IncResV2) (Szegedy et al., 2017). For secured target models, we consider 3 robustly trained DNNs using ensemble adversarial training (Tramer et al., 2018):\\` $\\mathrm { I n c V } 3 _ { e n s 3 }$ (ensemble of $3 \\ \\mathrm { I n c V 3 }$ networks), $\\mathrm { I n c V } 3 _ { e n s 4 }$ (ensemble of 4 IncV3 networks) and IncRes $\\mathrm { V } 2 _ { e n s 3 }$ (ensemble of 3 IncResV2 networks). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Source Models. We choose 8 different source models from the ResNet family: ResNet(RN)- 18/34/50/101/152 and DenseNet(DN)-121/169/201. Whenever the input size of the source model does not match the target model, we resize the crafted adversarial images to the input size of the target model. For VGG19, ResNet and DenseNet models, images are cropped and resized to $2 2 4 \\times 2 2 4$ , while for Inception/Inception-ResNet models, images are cropped and resized to $2 9 9 \\times 2 9 9$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 TRANSFERABILITY AGAINST UNSECURED MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first investigate the transferability of all attack methods against 7 unsecured models, which is to find the best method that can generate the most transferable attacks on one source model against all target models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One-step Transferability. The one-step transferability is measured by the success rate of one-step attacks, as reported in Table 2. Here, we only show the results on two source models: 1) RN152 which is the best ResNet source model with the highest success rate on average against all target models, and 2) DN201 which is the best DenseNet source model. Also note that, when the source and target models are the same, the result represents the white-box success rate. Overall, adversarial examples crafted on DN201 have significantly better transferability than those crafted on RN152, especially for our SGM method. This is because there are $\\sim 3 0 \\times$ more skip connections that can be manipulated by our SGM in DN201 compared to RN152. In comparison to to both FGSM and TI, transferability is improved considerably by SGM in almost all test scenarios, except when transferring from RN152 to VGG19/IncV3/IncV4 where SGM is outperformed by TI. This implies that, when transfereing across different architectures (eg. ResNet VGG/Inception), translation adaptation may help increase the transferability of one-step perturbations. However, this advantage of TI disappears when there are more skip connections, as is the case for the DN201 source model. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c2ffcf5d9007489fed95a5a624aa0fb7b07925966f8211b1fffc537ebfabfbe0.jpg",
+        "table_caption": [
+            "Table 2: One-step transferability: the success rates ( $\\% \\pm$ std over 5 random runs) of black-box attacks crafted by different methods on 2 source models against 7 unsecured target models. The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncResV2</td></tr><tr><td rowspan=\"3\">RN152</td><td>FGSM</td><td>41.96±0.52</td><td>71.53±0.34</td><td>37.49±0.10</td><td>30.00±0.56</td><td>25.66±0.07</td><td>21.55±0.16</td><td>19.90±0.49</td></tr><tr><td>TI</td><td>49.61±0.11</td><td>49.33±0.35</td><td>36.87±0.42</td><td>29.95±0.32</td><td>33.59±0.73</td><td>29.05±0.34</td><td>20.62±0.09</td></tr><tr><td>SGM</td><td>47.54±0.14</td><td>76.90±0.60</td><td>43.73±0.21</td><td>31.16±0.45</td><td>29.41±0.24</td><td>25.11±0.20</td><td>22.63±0.15</td></tr><tr><td rowspan=\"3\">DN201</td><td>FGSM</td><td>49.87±0.17</td><td>38.89±0.29</td><td>81.51±0.33</td><td>34.94±0.53</td><td>31.21±0.47</td><td>27.08±0.23</td><td>23.87±0.45</td></tr><tr><td>TI</td><td>54.37±0.58</td><td>33.49±0.18</td><td>57.71±0.05</td><td>34.46±0.47</td><td>34.45±0.25</td><td>30.17±0.23</td><td>20.36±0.33</td></tr><tr><td>SGM</td><td>56.97±0.25</td><td>47.54±0.14</td><td>87.73±0.76</td><td>42.31±0.67</td><td>37.91±0.56</td><td>32.83±0.38</td><td>29.64±0.25</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ddd5746ae51d7c4389a483eff0be8007bf1101f993b029921ace4f8de190fd80.jpg",
+        "image_caption": [
+            "Figure 2: The attack success rates of black-box attacks crafted by different attack methods on 8 source models against 3 unsecured target models: VGG19 (left), SE154 (middle) and IncV3 (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/69c26b17fcba1a9f1aa5fca9f4f4c8e1096eb67fefc3420d0b0d7a44ca003dd1.jpg",
+        "table_caption": [
+            "Table 3: Multi-step transferability: the success rates ( $\\% \\pm$ std over 5 random runs) of black-box attacks crafted by different methods on 2 source models against 7 unsecured target models. The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td rowspan=\"5\">RN152</td><td>PGD</td><td>45.03±0.21</td><td>99.91±0.04 51.49±0.51</td><td></td><td>29.35±0.49</td><td>26.56±0.75</td><td>21.03±0.19</td><td>19.10±0.37</td></tr><tr><td>TI</td><td>54.59±0.63</td><td>99.73±0.11</td><td>63.77±0.93</td><td>41.89±0.58 37.97±0.39</td><td></td><td></td><td>36.25±1.13 28.90±0.52</td></tr><tr><td>MI</td><td>65.42±0.60</td><td>99.77±0.02</td><td>75.79±0.69</td><td>53.07±0.44</td><td>50.22±0.07</td><td>43.32±0.27</td><td>41.71±0.36</td></tr><tr><td>DI</td><td>74.01±0.48</td><td>99.90±0.02</td><td>277.81±0.80</td><td>57.49±1.22</td><td>53.95±0.68</td><td>47.16±0.52</td><td>43.47±0.30</td></tr><tr><td>SGM</td><td>79.90±0.69</td><td>99.87±0.03</td><td>81.56±0.35</td><td>61.83±0.17</td><td>57.22±0.51</td><td>48.57±0.09</td><td>45.44±0.36</td></tr><tr><td rowspan=\"5\">DN201</td><td>PGD</td><td>57.61±0.82</td><td>59.84±0.76</td><td>99.89±0.02</td><td>39.78±0.69</td><td>36.01±0.59</td><td>31.76±0.27</td><td>25.92±0.13</td></tr><tr><td>TI</td><td>54.90±0.75</td><td>50.63±1.02</td><td>99.64±0.07</td><td>40.40±0.31</td><td>39.13±0.52</td><td>37.03±1.00</td><td>28.83±0.49</td></tr><tr><td>MI</td><td>75.09±0.80</td><td>76.39±0.61</td><td>99.84±0.05</td><td>64.38±0.69</td><td>59.62±0.36</td><td>54.85±0.56</td><td>50.05±0.25</td></tr><tr><td>DI</td><td>78.11±0.56 78.18±0.91</td><td></td><td>99.81±0.05</td><td>61.75±0.88</td><td>60.04±0.81</td><td>56.15±0.36 49.00±0.83</td><td></td></tr><tr><td>SGM</td><td>82.66±0.29</td><td>86.65±0.50</td><td>99.67±0.08</td><td>72.03±0.53</td><td>65.48±0.23</td><td>58.77±0.78</td><td>54.97±0.25</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Multi-step Transferability. First we provide a detailed study about the transferability of all attack methods from the 8 source models to the 3 representative unsecured target models. We then compare different attack methods on two best source models against all unsecured target models: the best ResNet source model and the best DenseNet source model. The multi-step (e.g., 10 step) transferability from all source models to three representative target models (VGG19, SE154 and IncV3) is illustrated in Figure 2. In all transfer scenarios, our proposed SGM outperforms existing methods consistently on almost all source models except RN18. Adversarial attacks crafted by SGM become more transferable when there are more skip connections in the source model (e.g., from RN18 to DN201). An interesting observation is that, when the target model is shallow such as VGG19 (left figure in Figure 2), shallow source models transfer better, however, when the target model is deep such as SE154 and IncV3 (middle and right figures in Figure 2), deeper source models tend to have better transferability. We suspect this is due to the architectural similarities shared by the target and source models. Note that against the VGG19 target model, the success rate of baseline methods all drop significantly when the ResNet source models become more complex (from RN18 to RN152). The small variations at RN50 and DN121 source models may be caused by the architectural difference between RN18/34 which consist of normal residual blocks, RN50/101/152 which consist of “bottleneck” residual blocks and DN121/169/201 which has dense skip connections. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results for the best source models RN152 and DN201 against unsecured target models are reported in Table 3. The proposed SGM attack outperforms existing methods by a large margin consistently against different target models. Particularly, for transfer $\\mathrm { D N } 2 0 1  \\mathrm { S E } 1 5 4$ (a recent state-of-the-art DNN with only $2 . 2 5 1 \\%$ top-5 error on ImageNet), SGM achieves a success rate of $7 2 . 0 3 \\%$ , which is $> 7 \\%$ and $> 1 0 \\%$ higher than MI and DI respectively. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Combining with Existing Methods. We further demonstrate that the adversarial transferability of skip connections can be exploited in combination with existing techniques. The experiments are conducted on DN201 (the best source model in the above multi-step experiments), and TI attack is excluded as it was originally proposed against secured models and demonstrates limited improvement over PGD against unsecured models. The results are reported in Table $4 ^ { * }$ . The transferability of MI and DI is improved remarkably of $1 1 . 9 8 \\% \\sim 2 1 . 9 8 \\%$ by SGM. When combined with both MI and DI, SGM improves the state-of-the-art $\\left( \\mathrm { M I + D I } \\right)$ transferability by a huge margin consistently against all target models. In particular, SGM pushes the new state-of-the-art to at least $8 0 . 5 2 \\%$ which previously was only $71 \\%$ . This illustrates that skip connections can be easily manipulated to craft highly transferable attacks against many state-of-the-art DNN models. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/821c387e6e42df108d6a92eafb5d0af66ed5071df28136a9ed10a4b715a8502e.jpg",
+        "table_caption": [
+            "Table 4: Combined with existing methods: the success rates $( \\% )$ of attacks crafted on source model DN201 against 7 unsecured target models. The best results are in bold and $^ +$ indicates improvement. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack\\Target</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td>MI</td><td>75.09</td><td>76.39</td><td>99.84</td><td>64.38</td><td>59.62</td><td>54.85</td><td>50.05</td></tr><tr><td>MI+SGM DI</td><td>+12.01 78.11</td><td>+13.24 78.18</td><td>99.52 99.81</td><td>+17.16 61.75</td><td>+21.88 60.04</td><td>+15.57 56.15</td><td>+18.35 49.00</td></tr><tr><td>DI+SGM</td><td>+12.28</td><td>+13.76</td><td>99.52</td><td>+20.92</td><td>+17.66</td><td>+15.78</td><td>+20.20</td></tr><tr><td>MI+DI</td><td>87.16</td><td>87.28</td><td>99.76</td><td>79.80</td><td>76.68</td><td>75.20</td><td>71.05</td></tr><tr><td>MI+DI+SGM</td><td>93.00</td><td>93.92</td><td>99.42</td><td>89.86</td><td>85.72</td><td></td><td>80.50</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>81.23</td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/70b27106a8feddc593e669cab46a342f866414497794a8d92b594e8ba481dbfe.jpg",
+        "table_caption": [
+            "Table 5: Transferability against secured models: the success rates $\\% \\pm$ std over 5 random runs) of multi-step attacks crafted on RN152 and DN201 source models against 3 secured models. The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>Attack</td><td>IncV3ens3</td><td>IncVens4</td><td>IncResens3</td></tr><tr><td rowspan=\"2\">RN152</td><td>PGD TI</td><td>12.47±1.27 45.36±0.97</td><td>10.72±1.37 45.81±0.93</td><td>6.97±0.71 38.19±0.81</td></tr><tr><td>MI DI</td><td>24.20±1.15 28.48±1.21</td><td>22.04±0.98 24.19±1.22</td><td>16.10±0.56</td></tr><tr><td rowspan=\"2\"></td><td>SGM</td><td>31.57±0.55</td><td>27.77±0.47</td><td>17.31±0.77 20.02±0.66</td></tr><tr><td>TI+SGM</td><td>52.62±0.40</td><td>52.80±0.79</td><td>43.96±0.62</td></tr><tr><td rowspan=\"2\"></td><td>PGD TI</td><td>18.16±0.56</td><td>15.30±0.62</td><td>10.40±0.49</td></tr><tr><td>MI</td><td>42.76±0.91</td><td>42.01±0.79</td><td>34.28±0.88</td></tr><tr><td rowspan=\"4\">DN201</td><td>DI</td><td>31.79±0.83</td><td>28.21±0.15</td><td>20.60±0.38</td></tr><tr><td></td><td>34.84±1.35</td><td>29.23±0.83</td><td>21.64±0.80</td></tr><tr><td>SGM</td><td>41.45±0.30</td><td>37.85±0.22</td><td>29.41±0.02</td></tr><tr><td>TI+SGM</td><td>46.11±1.23</td><td>47.38±0.89</td><td>39.32±0.80</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 TRANSFERABILITY AGAINST ROBUSTLY TRAINED MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The success rates of our SGM and other baseline methods against the 3 secured target models are reported in Table 5. Overall, with translation adaptation specifically designed for evading adversarially trained models, TI achieves the best standalone transferability, while SGM is the second best with higher success rates than either PGD, MI or DI. When combined with TI, SGM also improves the TI attack by a considerable margin across all transfer scenarios. This indicates that, although manipulating the skip connections alone may not sufficient to attack secured models, it still can make existing attacks more powerful. One interesting observation is that attacks crafted here on RN152 are more transferable than those crafted on DN201, which is quite the opposite to attacking unsecured models. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 A CLOSER LOOK AT SGM ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this part, we conduct more experiments to investigate the gradient decay factor of our proposed SGM, and explore the potential use of SGM for ensemble-based attacks and white-box attacks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of Residual Gradient Decay $\\gamma$ . We test the transferability of our proposed SGM with varying decay parameter $\\gamma \\in \\ [ 0 . 1 , 1 . 0 ]$ , where $\\gamma = 1 . 0$ means no decay on the residual gradients. The attacks are crafted by 10-step SGM on 5000 random ImageNet validation images. The results against 3 target models (VGG19, SE154 and IncV3) are illustrated in Figure 3. As can be observed, the trends are very consistent against different target models. On DenseNet source models, decreasing decay parameter (increasing decay strength) tends to improve transferability until it exceeds a certain threshold, e.g., $\\gamma = 0 . 5$ . This is because the decay encourages the attack to focus on more transferable low-level information, however, it becomes less sufficient if all high-level class-relevant information is ignored. On ResNet source models, decreasing decay parameter can constantly improve transferability for $\\gamma \\geq 0 . 2$ . Compared to DenseNet source models, ResNets require more decay on the residual gradients. Recalling that skip connections reveal more transferable information of the source model, ResNets require more penalty on the residual gradients to increase the importance of skip gradients that reveal more transferable information of the source model. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/77df147172ccb0b33b337486c0853bc3dac36248c6d98f8856e7b9983faf5869.jpg",
+        "image_caption": [
+            "Figure 3: Parameter tuning: the success rates of black-box attacks crafted by 10-step SGM with varying decay parameter $\\gamma \\in [ 0 . 1 , 1 . 0 ]$ . The solid and dash curves represent results on ResNet and DenseNet source models respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/d36bd178b49149c10430fadf9ae1c68a9032c061bbb4bec53b3f863a91e245f2.jpg",
+        "table_caption": [
+            "Table 6: Multi-step transferability of ensemble-based attack: the success rates ( $\\%$ ±std over 5 random runs) of multi-step attacks crafted by different methods on an ensemble of 3 source models (e.g. RN34, RN152 and DN201) against 7 unsecured target models. The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>Attack</td><td>VGG19</td><td>RN152</td><td>DN201</td><td>SE154</td><td>IncV3</td><td>IncV4</td><td>IncRes</td></tr><tr><td rowspan=\"2\">RN34</td><td>PGD</td><td>86.69±0.20</td><td>99.99±0.01</td><td>99.99±0.02</td><td>69.65±0.71</td><td>65.95±0.35</td><td>59.30±0.32</td><td>53.91±0.40</td></tr><tr><td>TI</td><td>84.35±0.21</td><td>99.59±0.11</td><td>99.77±0.06</td><td>571.67±0.19</td><td>67.22±0.82</td><td></td><td>66.02±0.66 56.83±0.89</td></tr><tr><td>+ RN152</td><td>MI</td><td>92.86±0.19</td><td>99.91±0.04</td><td>99.91±0.06</td><td>86.11±0.38</td><td>83.25±0.35</td><td>79.25±1.16</td><td>76.53±0.66</td></tr><tr><td>+</td><td>DI</td><td>96.34±0.23</td><td>99.84±0.20</td><td>99.84±0.20</td><td>89.72±0.52</td><td>87.53±0.29</td><td>85.04±0.75</td><td>81.11±0.44</td></tr><tr><td rowspan=\"2\">DN201</td><td>SGM</td><td>97.36±0.17</td><td>99.87±0.07</td><td>99.86±0.09</td><td>90.40±0.26</td><td>87.86±0.51</td><td>82.97±0.71</td><td>80.93±0.55</td></tr><tr><td>DI+SGM</td><td>98.65±0.08</td><td>99.84±0.04</td><td>99.86±0.04</td><td>94.36±0.19</td><td>93.08±0.41</td><td>89.56±0.07</td><td>88.27±0.43</td></tr></table>",
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+        "img_path": "images/180ed7f6ae99cffeddb5918970b5bef0e4367234eae6f55e3ef3883633289abf.jpg",
+        "table_caption": [
+            "Table 7: Transferability of ensemble-based attack against secured models: the success rates (%±std over 5 random runs) of black-box attacks crafted on an ensemble of 3 source models (e.g. RN34, RN152 and DN201). The best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>Attack</td><td>IncV3ens3</td><td>IncVens4</td><td>IncResens3</td></tr><tr><td rowspan=\"3\">RN34 +</td><td>PGD</td><td>37.63±0.37</td><td>32.69±0.62</td><td>23.49±0.55</td></tr><tr><td>TI</td><td>75.04±0.50</td><td>75.94±0.63</td><td>66.24±0.45</td></tr><tr><td>MI</td><td>54.68±0.27</td><td>50.24±0.48</td><td>39.27±0.33</td></tr><tr><td>RN152 +</td><td>DI</td><td>65.29±0.31</td><td>57.48±0.45</td><td>46.41±0.42</td></tr><tr><td rowspan=\"2\">DN201</td><td>SGM</td><td>66.08±0.42</td><td>62.22±0.73</td><td>51.16±0.12</td></tr><tr><td>TI+SGM</td><td>87.65±1.00</td><td>85.11±0.27</td><td>77.75±0.41</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "As for the selection of $\\gamma$ under a scenario without knowing the target model, from Figure 3 and Appendix C, we can see that the influence of $\\gamma$ is more related to the source model rather than the target model, that is, given a source model, the best $\\gamma$ against different target models are generally the same. This makes the selection of $\\gamma$ quite straightforward: choosing the best $\\gamma$ on the source model(s). For instance, in Figure 3, suppose the unknown target model is SE154 (middle figure), the adversary could tune $\\gamma$ on source model DN201 to attack VGG19 (left figure) and find the best $\\gamma = 0 . 5$ . The attacks crafted on DN201 with $\\gamma = 0 . 5$ indeed achieved the best success rate against the SE154 target model (and other target models). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ensemble-based Attacks. It has been shown that attacking multiple source models simultaneously can improve the transferability of the crafted adversarial examples, and is commonly adopted in practice. We follow the ensemble-based strategy (Liu et al., 2017) and craft attacks on an ensemble of RN34, RN152 and DN201. According to the discussion above, we select the best $\\gamma$ individually for each source model: choose $\\gamma$ for source RN34 and RN152 against target DN201, and $\\gamma$ for source DN201 against target RN152. The success rates (transferability) against 7 unsecured models and 3 secured models are reported in Table 6 and Table 7 respectively. Similar to the results of single source model attacks, against unsecured target models, SGM has a similar standalone performance with DI, better than the others (except the two “white-box” scenarios against RN152 and DN201). When combined with other methods, e.g. DI, it improves the success rates again by a large margin. ",
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+        "type": "image",
+        "img_path": "images/fd7af6e1000782748912c2d39b08b3926209699706aa20993c284144c2df03b2.jpg",
+        "image_caption": [
+            "Figure 4: White-box success rate for FGSM versus SGM. In (b) and (c), each color corresponds to one model, with FGSM is represented by solid curve and SGM is represented by dashed curve. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Against secured models, SGM achieves the second best standalone transferability, with TI is still the best. When combined with TI, SGM improves the success rate by $\\sim 1 0 \\%$ consistently against all secured target models. In particular, against $\\mathrm { I n c V } 3 _ { e n s 3 }$ , $\\mathrm { T I } { + } \\mathrm { S G M }$ achieves higher success rate $( 8 7 . 6 5 \\% )$ than reported in (Dong et al., 2019) $( 8 4 . 8 \\% )$ , even if only 3 source models are used here and the source models (e.g. RN34, RN152 and DN201) are all of different architecture to IncV3 target model ((Dong et al., 2019) uses 6 source models including even the IncV3 model). From all aspects analyzed above, the existence of skip connections makes transferable attacks much easier to craft in practice. ",
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+    {
+        "type": "text",
+        "text": "Improving Weak White-box Attacks. In addition to the black-box transferability, we next show that SGM can also improve the weak (one-step) white-box attack FGSM. Note that the one-step version of SGM is equivalent to FGSM plus residual gradient decay. Our experiments are conducted on the 8 source models, and the white-box success rates under maximum $L _ { \\infty }$ perturbation $\\epsilon = 8$ (a typical white-box setting) are shown in Figure 4a. As can be observed, using SGM can help improve the adversarial strength (i.e., higher success rate). We then vary the maximum perturbation $\\epsilon \\dot { \\in } [ 1 , 6 4 ]$ , and show the results on ResNet and DenseNet models separately in Figure 4b and Figure $_ \\mathrm { 4 c }$ . Compared to FGSM, SGM can always give better adversarial strength, except when $\\epsilon$ is extremely small $( \\epsilon \\leq 2 )$ ). When the perturbation space becomes infinitely small, the loss landscape within the space becomes flat and the gradient points to the optimal perturbation direction. However, when the perturbation space expands, one-step gradient becomes less accurate due to changes in the loss landscape (success rate decreases as $\\epsilon$ increases from 4 to 16), and in this case, the skip gradient which contains more low-level information is more reliable than the residual gradient (the improvement is more significant for $\\epsilon \\in [ 4 , 1 6 ]$ ). Another interesting observation is that adversarial strength decreases when the model becomes more complex from RN18 to RN152, or DN121 to DN201. This is likely because the loss landscape of complex models is steeper than shallow models, making one-step gradient less reliable. ",
+        "bbox": [
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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 identified a surprising property of the skip connections used by many state-ofthe-art ResNet-like neural networks, that is, they can be easily used to generate highly transferable adversarial examples. To demonstrate this architectural “weakness”, we proposed the Skip Gradient Method (SGM) to craft adversarial examples using more gradients from the skip connections rather than the residual ones, via a decay factor on gradients. We conducted a series of transfer attack experiments with 8 source models and 10 target models including 7 unsecured and 3 secured models, and showed that attacks crafted by SGM have significantly better transferability than those crafted by existing methods. When combined with existing techniques, SGM can also boost state-of-the-art transferability by a huge margin. We believe the high adversarial transferability of skip connections is due to the fact that they expose extra low-level information which is more transferable across different DNNs. Our findings in this paper not only remind researchers in adversarial research to pay attention to the architectural vulnerability of DNNs, but also raise new challenges for secure DNN architecture design. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Shu-Tao Xia is supported in part by National Key Research and Development Program of China under Grant 2018YFB1800204, National Natural Science Foundation of China under Grant 61771273, R&D Program of Shenzhen under Grant JCYJ20180508152204044, and research fund of PCL Future Regional Network Facilities for Large-scale Experiments and Applications (PCL2018KP001). ",
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+    },
+    {
+        "type": "text",
+        "text": "Yisen Wang, Difan Zou, Jinfeng Yi, James Bailey, Xingjun Ma, and Quanquan Gu. Improving adversarial robustness requires revisiting misclassified examples. In ICLR, 2020. ",
+        "bbox": [
+            174,
+            736,
+            821,
+            765
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Cihang Xie, Zhishuai Zhang, Yuyin Zhou, Song Bai, Jianyu Wang, Zhou Ren, and Alan L Yuille. Improving transferability of adversarial examples with input diversity. In CVPR, 2019. ",
+        "bbox": [
+            174,
+            773,
+            823,
+            803
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Saining Xie, Ross Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual trans-´ formations for deep neural networks. In CVPR, 2017. ",
+        "bbox": [
+            173,
+            810,
+            823,
+            839
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016. ",
+        "bbox": [
+            174,
+            848,
+            825,
+            876
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A VISUALIZATION OF ADVERSARIAL EXAMPLES CRAFTED BY SGM ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            761,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section, we visualize 6 clean images and their corresponding adversarial examples crafted using our SGM on either a ResNet-152 or a DenseNet201 in Figure 5. These visualization results show that the generated adversarial perturbations are human imperceptible. ",
+        "bbox": [
+            174,
+            127,
+            826,
+            170
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/6643c5baa262f9dea9b86ebf084c63e78bd350983d85de61b79faf17880fc5c9.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of 6 clean images and their corresponding adversarial examples. The clean images are shown in the top row, adversarial images crafted on ResNet-152 are shown in the middle row, while those crafted on DenseNet-201 are shown in the bottom row. All adversarial images are crafted using our proposed SGM (10-step) under maximum perturbation $\\epsilon = 1 6$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            224,
+            825,
+            496
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B COMPARISON WITH PREVIOUSLY PUBLISHED RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            626,
+            666,
+            642
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare the experimental settings in previous and our works, and discuss some small discrepancies of the baseline performance reported in ours and previous works. ",
+        "bbox": [
+            171,
+            651,
+            821,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 8 and 9 summarizes these differences for single-source and ensemble-based attack respectively. Out of all these works (Dong et al., 2018; 2019; Xie et al., 2019), results reported in (Xie et al., 2019) are more complete. Our reported success rate of baseline attacks (e.g. MI and DI) matches that reported in (Xie et al., 2019), sometimes even higher. The slight discrepancy is caused by the difference in experimental settings. Table 10 summarizes the different source models used by baseline attacks, and Table 11 summarizes the difference in dataset, number of test images, input image size, maximum $L _ { \\infty }$ perturbation $\\epsilon$ , number of attack steps $N$ and attack step size $\\alpha$ . Compared to $2 9 9 \\times 2 9 9$ image size, here we use a more standard image size $2 2 4 \\times 2 2 4$ on ImageNet. The use of smaller input size may reduce the effectiveness of existing attacks (Simon-Gabriel et al., 2019). ",
+        "bbox": [
+            173,
+            685,
+            825,
+            821
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In another work by Liu et al. (2017), $81 \\%$ success rate was reported for optimization-based attack crafted on ResNet-152 against target VGG16, which is higher than our $6 5 . 5 2 \\%$ from ResNet-152 to VGG19. This is because they did not restrict the maximum perturbation $\\epsilon$ . The root mean square deviation (RMSD) of their attacks is 22.83,which indicates that many pixels are perturbed more than 16 pixel values. In our experiments, the RMSD is 6.29 for PGD, 7.71 for SGM, and 12.55 for MI. This appears to be another reason for the performance discrepancy. Note that the advantage of bounded small perturbation is increasing imperceptibility to human observers (see Figure 5). ",
+        "bbox": [
+            173,
+            828,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/329e8d96e4e1374a5e642e48cb7e93465b9f561584e4f3c7a7aee073aecd6e2f.jpg",
+        "table_caption": [
+            "Table 8: Previously reported attack success rates $( \\% )$ of baseline single-source attacks against 6 target models. “-” means no results were reported. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reference</td><td>Attack</td><td>IncV3</td><td>IncV4</td><td>IncRes</td><td>IncV3ens3</td><td>IncV3ens4</td><td>IncResens3</td></tr><tr><td rowspan=\"3\">Dong et al. (2018)</td><td>FGSM</td><td>35.0</td><td>28.2</td><td>27.5</td><td>14.6</td><td>13.2</td><td>7.5</td></tr><tr><td>BIM</td><td>26.7</td><td>22.7</td><td>21.2</td><td>9.3</td><td>8.9</td><td>6.2</td></tr><tr><td>MI</td><td>53.6</td><td>48.9</td><td>44.7</td><td>22.1</td><td>21.7</td><td>12.9</td></tr><tr><td rowspan=\"3\">Dong et al. (2019)</td><td>FGSM</td><td>-</td><td>-</td><td>-</td><td>20.2</td><td>17.7</td><td>9.9</td></tr><tr><td>MI</td><td>=</td><td>1</td><td>-</td><td>25.1</td><td>23.7</td><td>13.3</td></tr><tr><td>DI</td><td>-</td><td>-</td><td>-</td><td>40.5</td><td>36.0</td><td>24.1</td></tr><tr><td rowspan=\"4\">Xie et al. (2019)</td><td>FGSM</td><td>34.4</td><td>28.50</td><td>27.1</td><td>12.4</td><td>11.0</td><td>6.0</td></tr><tr><td>PGD</td><td>20.8</td><td>17.2</td><td>14.9</td><td>5.4</td><td>4.6</td><td>2.8</td></tr><tr><td>MI</td><td>50.1</td><td>44.1</td><td>42.2</td><td>18.2</td><td>15.2</td><td>9.0</td></tr><tr><td>DI</td><td>53.8</td><td>49.0</td><td>44.8</td><td>13.0</td><td>11.1</td><td>6.9</td></tr><tr><td rowspan=\"4\">Ours</td><td>FGSM</td><td>26.56</td><td>21.03</td><td>19.10</td><td>1</td><td>-</td><td>1</td></tr><tr><td>PGD</td><td>26.56</td><td>21.03</td><td>19.10</td><td>12.47</td><td>10.72</td><td>6.97</td></tr><tr><td>MI</td><td>50.22</td><td>43.32</td><td>41.71</td><td>24.20</td><td>22.04</td><td>16.10</td></tr><tr><td>DI</td><td>53.95</td><td>47.16</td><td>43.47</td><td>34.84</td><td>29.23</td><td>21.64</td></tr></table>",
+        "bbox": [
+            189,
+            140,
+            805,
+            348
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f8ca3f2d5ad24f2895d93c96920aca4c84541b5e1421666d3fde702a5d6c670f.jpg",
+        "table_caption": [
+            "Table 9: Previously reported attack success rates $( \\% )$ of ensemble-based baseline attacks against 6 target models. “-” means no results were reported. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reference</td><td>Attack</td><td>IncV3</td><td>IncV4</td><td>IncRes</td><td>IncV3ens3</td><td>IncV3ens4</td><td>IncResens3</td></tr><tr><td rowspan=\"3\">Dong et al. (2018)</td><td>FGSM</td><td>45.7</td><td>39.9</td><td>38.8</td><td>15.4</td><td>15.0</td><td>6.4</td></tr><tr><td>BIM</td><td>72.1</td><td>61.0</td><td>54.4</td><td>18.6</td><td>18.7</td><td>9.9</td></tr><tr><td>MI</td><td>87.9</td><td>81.2</td><td>76.5</td><td>37.6</td><td>40.3</td><td>23.3</td></tr><tr><td rowspan=\"4\">Dong et al. (2019)</td><td>FGSM</td><td>-</td><td>-</td><td>-</td><td>27.5</td><td>23.7</td><td>13.4</td></tr><tr><td>MI</td><td></td><td>1</td><td>1</td><td>50.5</td><td>48.3</td><td>32.8</td></tr><tr><td>DI</td><td>-</td><td>-</td><td>-</td><td>66.0</td><td>63.3</td><td>45.9</td></tr><tr><td>TI+DI</td><td>-</td><td>-</td><td>-</td><td>84.8</td><td>82.7</td><td>78.0</td></tr><tr><td rowspan=\"4\">Xie et al. (2019)</td><td>PGD</td><td>43.7</td><td>36.4</td><td>33.3</td><td>12.9</td><td>15.1</td><td>8.8</td></tr><tr><td>MI</td><td>69.9</td><td>67.9</td><td>64.1</td><td>36.3</td><td>35.0</td><td>30.4</td></tr><tr><td>DI</td><td>71.4</td><td>65.9</td><td>64.6</td><td>22.8</td><td>26.1</td><td>15.8</td></tr><tr><td>DI+MI</td><td>80.7</td><td>80.6</td><td>80.7</td><td>44.6</td><td>44.5</td><td>39.4</td></tr><tr><td rowspan=\"5\">Ours</td><td>PGD</td><td>65.95</td><td>59.30</td><td>53.91</td><td>37.63</td><td>32.69</td><td>23.49</td></tr><tr><td>MI</td><td>83.25</td><td>79.25</td><td>76.53</td><td>54.68</td><td>50.24</td><td>39.27</td></tr><tr><td>DI</td><td>87.53</td><td>85.04</td><td>81.11</td><td>65.29</td><td>57.48</td><td>46.41</td></tr><tr><td>DI+SGM</td><td>93.08</td><td>89.56</td><td>88.27</td><td>80.14</td><td>76.52</td><td>66.40</td></tr><tr><td>TI+SGM</td><td>-</td><td>-</td><td>-</td><td>87.65</td><td>85.11</td><td>77.75</td></tr></table>",
+        "bbox": [
+            181,
+            401,
+            812,
+            636
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/9aa418b3ed798a926221bfde59d88db2ca7a4f4fff08a1bd168c736abcd7d965.jpg",
+        "table_caption": [
+            "Table 10: Source models used by existing single-source and ensemble-based black-box attacks. “Hold-out” refers to the hold-out target model from the group, with all remaining models are used as source models. Group 1 consists of ResNet-v2-152, IncV3, IncV4 and IncRes, while group 2 consists of ResNet-v2-152, IncV3, IncV4, IncRes, $\\mathrm { I n c V } 3 _ { e n s 3 }$ , $\\mathrm { I n c V } 3 _ { e n s 4 }$ and $\\mathrm { I n c R e s } _ { e n s 3 }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reference</td><td>Single-source attack</td><td>Ensemble-basedattack</td></tr><tr><td>Dong et al. (2018)</td><td>ResNet-v2-152</td><td>hold-out from group 1 or group 2</td></tr><tr><td>Dong et al. (2019)</td><td>ResNet-v2-152</td><td>hold-out from group 1</td></tr><tr><td>Xie et al. (2019)</td><td>ResNet-v2-152</td><td>hold-out from group 2</td></tr><tr><td>Ours</td><td>RN152</td><td>RN34+RN152+DN201</td></tr></table>",
+        "bbox": [
+            253,
+            717,
+            740,
+            790
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For proper implementation, we use open-source codes and pretrained models for our experiments, e.g., AdverTorch (Ding et al., 2019) for FGSM, PGD and MI, and source/target models from two GitHub repositories†‡§ for all models. We reproduced DI and TI in PyTorch. ",
+        "bbox": [
+            174,
+            816,
+            825,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/15ce5c402d154ebf3f4a08213563eea1040b6857e47b41c7ba7a9f816e10b98a.jpg",
+        "table_caption": [
+            "Table 11: Difference in experimental settings of our work compared to previous works.“NeurIPS 2017” indicates the dataset used for NeurIPS 2017 adversarial competition. \u000f: maximum per-pixel perturbation; $N$ : number of attack steps; $\\alpha$ : attack step size. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reference</td><td>Dataset</td><td>Number</td><td>Input size</td><td>E</td><td>N</td><td>α</td></tr><tr><td>Dong et al. (2018)</td><td>ImageNet</td><td>1000</td><td>299 × 299</td><td>16</td><td>10</td><td>1.6</td></tr><tr><td>Dong et al. (2019)</td><td>NeurIPS 2017</td><td>1000</td><td>299 × 299</td><td>16</td><td>10</td><td>1.6</td></tr><tr><td>Xie et al. (2019)</td><td>ImageNet</td><td>5000</td><td>299×299</td><td>15</td><td>19</td><td>1.0</td></tr><tr><td>Ours</td><td>ImageNet</td><td>5000</td><td>224×224</td><td>16</td><td>10/20</td><td>2.0</td></tr></table>",
+        "bbox": [
+            236,
+            154,
+            756,
+            227
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C TRANSFERABILITY OF DECAY PARAMETER $\\gamma$ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            251,
+            581,
+            267
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study the “transferability” of decay parameter $\\gamma$ across different target models. RN152 and DN201 are used as the source model, and the target model are varying to observe trends. As indicated in Figure 6, all black-box target models share the same best selection of $\\gamma$ , which makes $\\gamma$ selection quite simple. Even if the true target model is unknown, the adversary can tune the gamma for a resnet-like neural network through another model and obtain the best selection as well. ",
+        "bbox": [
+            173,
+            275,
+            826,
+            344
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/300f0293aa251c6b51c2049a3d1a099b4b5c4e8f2ff590b6125950515a93bfe7.jpg",
+        "image_caption": [
+            "Figure 6: “Transferability” of decay parameter: the success rates of black-box attacks crafted by 10-step SGM with varying decay parameter $\\gamma \\in \\ [ 0 . 1 , 1 . 0 ]$ . The curves represent results against different target models respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            358,
+            789,
+            506
+        ],
+        "page_idx": 14
+    }
+]

parse/train/HJeKCi0qYX/HJeKCi0qYX.md

@@ -0,0 +1,348 @@
+# MILE: A Multi-Level Framework for Scalable Graph Embedding
+
+Anonymous authors Paper under double-blind review
+
+# Abstract
+
+Recently there has been a surge of interest in designing graph embedding methods. Few, if any, can scale to a large-sized graph with millions of nodes due to both computational complexity and memory requirements. In this paper, we relax this limitation by introducing the MultI-Level Embedding (MILE) framework – a generic methodology allowing contemporary graph embedding methods to scale to large graphs. MILE repeatedly coarsens the graph into smaller ones using a hybrid matching technique to maintain the backbone structure of the graph. It then applies existing embedding methods on the coarsest graph and refines the embeddings to the original graph through a graph convolution neural network that it learns. The proposed MILE framework is agnostic to the underlying graph embedding techniques and can be applied to many existing graph embedding methods without modifying them. We employ our framework on several popular graph embedding techniques and conduct embedding for real-world graphs. Experimental results on five large-scale datasets demonstrate that MILE significantly boosts the speed (order of magnitude) of graph embedding while generating embeddings of better quality, for the task of node classification. MILE can comfortably scale to a graph with 9 million nodes and 40 million edges, on which existing methods run out of memory or take too long to compute on a modern workstation.
+
+# 1 Introduction
+
+In recent years, graph embedding has attracted much interest due to its broad applicability for various tasks (Perozzi et al., 2014; Wang et al., 2016; Henderson et al., 2012). However, such methods rarely scale to large datasets (e.g., graphs with over 1 million nodes) since they are computationally expensive and often memory intensive. For example, random-walkbased embedding techniques require a large amount of CPU time to generate a sufficient number of walks and train the embedding model. As another example, embedding methods based on matrix factorization, including GraRep (Cao et al., 2015) and NetMF (Qiu et al., 2018), requires constructing an enormous objective matrix (usually much denser than adjacency matrix), on which matrix factorization is performed. Even a medium-size graph with 100K nodes can easily require hundreds of GB of memory using those methods. On the other hand, many graph datasets in the real world tend to be large-scale with millions or even billions of nodes. To the best of our knowledge, none of the existing efforts examines how to scale up graph embedding in a generic way. We make the first attempt to close this gap. We are also interested in the related question of whether the quality of such embeddings can be improved along the way. Specifically, we ask:
+
+1) Can we scale up the existing embedding techniques in an agnostic manner so that they can be directly applied to larger datasets?
+
+2) Can the quality of such embedding methods be strengthened by incorporating the holistic view of the graph?
+
+To tackle these problems, we propose a MultI-Level Embedding (MILE) framework for graph embedding. Our approach relies on a three-step process: first, we repeatedly coarsen the original graph into smaller ones by employing a hybrid matching strategy; second, we compute the embeddings on the coarsest graph using an existing embedding techniques - and third, we propose a novel refinement model based on learning a graph convolution network to refine the embeddings from the coarsest graph to the original graph – learning a graph convolution network allows us to compute a refinement procedure that levers the dependencies inherent to the graph structure and the embedding method of choice. To summarize, we find that:
+
+• MILE is generalizable : Our MILE framework is agnostic to the underlying graph embedding techniques and treats them as black boxes. • MILE is scalable : MILE can significantly improve the scalability of the embedding methods (up to 30-fold), by reducing the running time and memory consumption. • MILE generates high-quality embeddings : In many cases, we find that the quality of embeddings improves by levering MILE (in some cases is in excess of $1 0 \%$ ).
+
+# 2 Related Work
+
+Many techniques for graph or network embedding have been proposed in recent years. DeepWalk and Node2Vec generate truncated random walks on graphs and apply the Skip Gram by treating the walks as sentences (Perozzi et al., 2014; Grover & Leskovec, 2016). LINE learns the node embeddings by preserving the first-order and second-order proximities (Tang et al., 2015). Following LINE, SDNE leverages deep neural networks to capture the highly non-linear structure (Wang et al., 2016). Other methods construct a particular objective matrix and use matrix factorization techniques to generate embeddings, e.g., GraRep (Cao et al., 2015) and NetMF (Qiu et al., 2018). This also led to the proliferation of network embedding methods for information-rich graphs, including heterogeneous information networks (Chang et al., 2015; Dong et al., 2017) and attributed graphs (Pan et al., 2016; Liang et al., 2018; Yang et al., 2015; Kipf & Welling, 2017).
+
+On the other hand, there are very few efforts, focusing on the scalability of network embedding (Yang et al., 2017; Huang et al., 2017). First, such efforts are specific to a particular embedding strategy and do not generalize. Second, the scalability of such efforts is limited to moderately sized datasets. Finally, and notably, these efforts at scalability are actually orthogonal to our strategy and can potentially be employed along with our efforts to afford even greater speedup.
+
+The closest work to this paper is the very recently proposed HARP (Chen et al., 2018), which proposes a hierarchical paradigm for graph embedding based on iterative learning methods (e.g., DeepWalk and Node2Vec). However, HARP focuses on improving the quality of embeddings by using the learned embeddings from the previous level as the initialized embeddings for the next level, which introduces a huge computational overhead. Moreover, it is not immediately obvious how a HARP like methodology would be extended to other graph embedding techniques (e.g., GraRep and NetMF) in an agnostic manner since such an approach would necessarily require one to modify the embedding methods to preset their initialized embeddings. In this paper, we focus on designing a general-purpose framework to scale up embedding methods treating them as black boxes.
+
+# 3 Problem Formulation
+
+Let $\mathcal { G } = ( V , E )$ be the input graph where $V$ and $E$ are respectively the node set and edge set. Let $A$ be the adjacency matrix of the graph and we assume $\mathcal { G }$ is undirected, though our problem can be easily extended (Chung, 2005; Gleich, 2006; Satuluri & Parthasarathy, 2011) to directed graph. We first define graph embedding:
+
+Definition 3.1 Graph Embedding Given a graph $\mathcal { G } = ( V , E )$ and a dimensionality $d$ $d \ll | V | ,$ , the problem of graph embedding is to learn a $d$ -dimension vector representation for each node in $\mathcal { G }$ so that graph properties are best preserved.
+
+Following this, a graph embedding method is essentially a mapping function $f : \mathbb { R } ^ { | V | \times | V | } \mapsto$ $\mathbb { R } ^ { | V | \times d }$ , whose input is the adjacency matrix $A$ (or $\mathcal { G }$ ) and output is a lower dimension matrix. Motivated by the fact that the majority of graph embedding methods cannot scale to large datasets, we seek to speed up existing graph embedding methods without sacrificing quality. We formulate the problem as:
+
+![](images/fb482d9cba2073b13c63dc54c420991e712c4162a2c68a060e40243d93029567.jpg)  
+Figure 1: MILE framework
+
+Given a graph $\mathcal { G } = ( V , E )$ and a graph embedding method $f ( \cdot )$ , we aim to realize a strengthened graph embedding method ${ \hat { f } } ( \cdot )$ so that it is more scalable than $f ( \cdot )$ while generating embeddings of comparable or even better quality.
+
+# 4 Methodology
+
+MILE framework consists of three key phases: graph coarsening, base embedding, and embeddings refining. Figure 1a shows the overview.
+
+# 4.1 Graph Coarsening
+
+In this phase, the input graph $\mathcal { G }$ (or $\mathscr { G } _ { 0 }$ ) is repeatedly coarsened into a series of smaller graphs $\zeta _ { 1 }$ , $\zeta _ { 2 }$ , ..., $\mathcal { G } _ { m }$ such that $| V _ { 0 } | > | V _ { 1 } | > \ldots > | V _ { m } |$ . In order to coarsen a graph from $\beta _ { i }$ to $\mathscr { G } _ { i + 1 }$ , multiple nodes in $\beta _ { i }$ are collapsed to form super-nodes in $\mathscr { G } _ { i + 1 }$ , and the edges incident on a super-node are the union of the edges on the original nodes in $\beta _ { i }$ . Here the set of nodes forming a super-node is called a matching. We propose a hybrid matching technique containing two matching strategies that can efficiently coarsen the graph while retaining the global structure. An example is shared in Figure 2.
+
+![](images/867c4e4b65553fd72123e2f12005578fdaae6064a4b7aa009cda8ea1c403c8ab.jpg)  
+Figure 2: Toy example for illustrating graph coarsening. (a) shows the process of applying Structural Equivalence Matching (SEM) and Normalized Heavy Edge Matching (NHEM) for graph coarsening. (b) presents the adjacency matrix $A _ { 0 }$ of the input graph, the matching matrix $M _ { 0 , 1 }$ corresponding to the SEM and NHEM matchings, and the derivation of the adjacency matrix $A _ { 1 }$ of the coarsened graph using Eq. 2.
+
+Structural Equivalence Matching (SEM) : Given two vertices $u$ and $v$ in an unweighted graph $\mathcal { G }$ , we call they are structurally equivalent if they are incident on the same set of neighborhoods. In figure 2a, node D and E are structurally equivalent. The intuition of matching structually equivalent nodes is that if two vertices are structurally equivalent, then their node embeddings will be similar.
+
+Normalized Heavy Edge Matching (NHEM) : Heavy edge matching is a popular matching method for graph coarsening (Karypis & Kumar, 1998). For an unmatched node $u$ in $\beta _ { i }$ , its heavy edge matching is a pair of vertices $( u , v )$ such that the weight of the edge between $u$ and $v$ is the largest. In this paper, we propose to normalize the edge weights when applying heavy edge matching using the formula as follows
+
+$$
+W _ { i } ( u , v ) = \frac { A _ { i } ( u , v ) } { \sqrt { D _ { i } ( u , u ) \cdot D _ { i } ( v , v ) } } .
+$$
+
+Here, the weight of an edge is normalized by the degree of the two vertices on which the edge is incident. Intuitively, it penalizes the weights of edges connected with high-degree nodes. As we will show in Sec. 4.3, this normalization is tightly connected with the graph convolution kernel.
+
+Hybrid Matching Method : We use a hybrid of two matching methods above for graph coarsening. To construct $\mathscr { G } _ { i + 1 }$ from $\beta _ { i }$ , we first find out all the structural equivalence matching (SEM) $\mathcal { M } _ { 1 }$ , where $\beta _ { i }$ is treated as an unweighted graph. This is followed by the searching of the normalized heavy edge matching (NHEM) $\mathcal { M } _ { 2 }$ on $\mathcal { G } _ { i }$ . Nodes in each matching are then collapsed into a super-node in $\mathcal { G } _ { i + 1 }$ . Note that some nodes might not be matched at all and they will be directly copied to $\mathscr { G } _ { i + 1 }$ .
+
+Formally, we build the adjacency matrix $A _ { i + 1 }$ of $\mathscr { G } _ { i + 1 }$ through matrix operations. To this end, we define the matching matrix storing the matching information from graph $\mathscr { G } _ { i }$ to $\mathcal { G } _ { i + 1 }$ as a binary matrix $M _ { i , i + 1 } \in \{ 0 , 1 \} ^ { | V _ { i } | \times | V _ { i + 1 } | }$ . The $r$ -th row and $c$ -th column of $M _ { i , i + 1 }$ is set to $^ { 1 }$ if node $r$ in $\beta _ { i }$ will be collapsed to super-node $c$ in $\mathscr { G } _ { i + 1 }$ , and is set to $0$ if otherwise. Each column of $M _ { i , i + 1 }$ represents a matching with the 1s representing the nodes in it. Each unmatched vertex appears as an individual column in $M _ { i , i + 1 }$ with merely one entry set to 1. Following this formulation, we construct the adjacency matrix of $\mathscr { G } _ { i + 1 }$ by using
+
+$$
+A _ { i + 1 } = M _ { i , i + 1 } ^ { T } A _ { i } M _ { i , i + 1 } .
+$$
+
+# 4.2 Base Embedding on Coarsened Graph
+
+The size of the graph reduces drastically after each iteration of coarsening, halving the size of the graph in the best case. We coarsen the graph for $m$ iterations and apply the graph embedding method $f ( \cdot )$ on the coarsest graph $\mathcal { G } _ { m }$ . Denoting the embeddings on $\mathcal { G } _ { m }$ as ${ \mathcal { E } } _ { m }$ , we have ${ \mathcal { E } } _ { m } = f ( { \mathcal { G } } _ { m } \mathbf { \theta } )$ . Since our framework is agnostic to the adopted graph embedding method, we can use any graph embedding algorithm for base embedding.
+
+# 4.3 Refinement of Embeddings
+
+The final phase of MILE is the embeddings refinement phase. Given a series of coarsened graph $\mathcal { G } _ { 0 } , \mathcal { G } _ { 1 } , \mathcal { G } _ { 2 } , . . . , \mathcal { G } _ { m }$ , their corresponding matching matrix $M _ { 0 , 1 } , M _ { 1 , 2 } , . . . , M _ { m - 1 , m }$ , and the node embeddings ${ \mathcal { E } } _ { m }$ on $\mathcal { G } _ { m }$ , we seek to develop an approach to derive the node embeddings of $\mathscr { G } _ { 0 }$ from $\mathcal { G } _ { m }$ . To this end, we first study an easier subtask: given a graph $\beta _ { i }$ , its coarsened graph $\mathcal { G } _ { i + 1 }$ , the matching matrix $M _ { i , i + 1 }$ and the node embeddings $\mathcal { E } _ { i + 1 }$ on $\mathscr { G } _ { i + 1 }$ , how to infer the embeddings $\mathcal { E } _ { i }$ on graph $\beta _ { i }$ . Once we solved this subtask, we can then iteratively apply the technique on each pair of consecutive graphs from $\mathcal { G } _ { m }$ to $\mathscr { G } _ { 0 }$ and eventually derive the node embeddings on $\mathscr { G } _ { 0 }$ . In this work, we propose to use a graph-based neural network model to perform embeddings refinement.
+
+Graph Convolution Network for Refinement Learning : Since we know the matching information between the two consecutive graphs $\beta _ { i }$ and $\mathscr { G } _ { i + 1 }$ , we can easily project the node embeddings from the coarse-grained graph $\mathscr { G } _ { i + 1 }$ to the fine-grained graph $\beta _ { i }$ using
+
+$$
+\mathcal { E } _ { i } ^ { p } = M _ { i , i + 1 } \mathcal { E } _ { i + 1 }
+$$
+
+In this case, embedding of a super-node is directly copied to its original node(s). We call $\mathcal { E } _ { i } ^ { p }$ the projected embeddings from $\mathscr { G } _ { i + 1 }$ to $\beta _ { i }$ , or simply projected embeddings without ambiguity. While this way of simple projection maintains some information of node embeddings, it has obvious limitations that nodes will share the same embeddings if they are matched and collapsed into a super-node during the coarsening phase. This problem will be more serious when the embedding refinement is performed iteratively from $\mathcal { G } _ { m }$ , ..., $\mathscr { G } _ { 0 }$ . To address this issue, we propose to use a graph convolution network for embedding refinement. Specifically, we design a graph-based neural network model $\mathcal { E } _ { i } = \mathcal { R } ( \mathcal { E } _ { i } ^ { p } , A _ { i } )$ , which derives the embeddings $\mathcal { E } _ { i }$ on graph $\mathscr { G } _ { i }$ based on the projected embeddings $\mathcal { E } _ { i } ^ { p }$ and the graph adjacency matrix $A _ { i }$ .
+
+Given graph $G$ with adjacency matrix $A$ , we consider the fast approximation of graph convolution from (Kipf $\&$ Welling, 2017). The $k$ -th layer of this neural network model is
+
+$$
+H ^ { ( k ) } ( X , A ) = \sigma \left( \tilde { D } ^ { - \frac { 1 } { 2 } } \tilde { A } \tilde { D } ^ { - \frac { 1 } { 2 } } H ^ { ( k - 1 ) } ( X , A ) \Theta ^ { ( k ) } \right)
+$$
+
+where $\sigma ( \cdot )$ is an activation function, $\Theta ^ { ( k ) }$ is a layer-specific trainable weight matrix, and $H ^ { ( 0 ) } ( X , A ) = X$ . In this paper, we define our embedding refinement model as a $\it l$ -layer graph convolution model
+
+$$
+\mathcal { E } _ { i } = \mathcal { R } \left( \mathcal { E } _ { i } ^ { p } , A _ { i } \right) \equiv H ^ { ( l ) } \left( \mathcal { E } _ { i } ^ { p } , A _ { i } \right) .
+$$
+
+The architecture of the refinement model is shown in Figure 1b. The intuition behind this refinement model is to integrate the structural information of the current graph $\mathcal { G } _ { i }$ into the projected embedding $\mathcal { E } _ { i } ^ { p }$ by repeatedly performing the spectral graph convolution. Each layer of graph convolution network in Eq. 4 can be regarded as one iteration of embedding propagation in the graph following the re-normalized adjacency matrix $\tilde { D } ^ { - \frac { 1 } { 2 } } \tilde { A } \tilde { D } ^ { - \frac { 1 } { 2 } }$ . Note that this re-normalized matrix is well aligned with the way we conduct normalized heavy edge matching in Eq. 1. We next discuss how the weight matrix $\Theta ^ { ( k ) }$ is learned.
+
+Intricacies of Refinement Learning : The learning of the refinement model is essentially learning $\Theta ^ { ( k ) }$ for each $k \in \left[ 1 , l \right]$ according to Eq. 4. Here we study how to design the learning task and construct the loss function. Since the graph convolution model $H ^ { ( l ) } ( \cdot )$ aims to predict the embeddings $\mathcal { E } _ { i }$ on graph $\mathcal { G } _ { i }$ , we can directly run a base embedding on $\mathscr { G } _ { i }$ to generate the “ground-truth” embeddings and use the difference between these embeddings and the predicted ones as the loss function for training. We propose to learn $\Theta ^ { ( k ) }$ on the coarsest graph and reuse them across all the levels for refinement. Specifically, we can define the loss function as the mean square error as follows
+
+$$
+L = \frac { 1 } { | V _ { m } | } \left\| \mathcal { E } _ { m } - H ^ { ( l ) } ( M _ { m , m + 1 } \mathcal { E } _ { m + 1 } , A _ { m } ) \right\| ^ { 2 } .
+$$
+
+We refer to the learning task associated with the above loss function as double-base embedding learning. We point out, however, there are two key drawbacks to this method. First of all, the above loss function requires one more level of coarsening to construct $\mathcal { G } _ { m + 1 }$ and an extra base embedding on $\mathcal { G } _ { m + 1 }$ . These two steps, especially the latter, introduce nonnegligible overheads to the MILE framework, which contradicts our motivation of scaling up graph embedding. More importantly, ${ \mathcal { E } } _ { m }$ might not be a desirable “ground truth” for the refined embeddings. This is because most of the embedding methods are invariant to an orthogonal transformation of the embeddings, i.e., the embeddings can be rotated by an arbitrary orthogonal matrix (Hamilton et al., 2017). In other words, the embedding spaces of graph $\mathcal { G } _ { m }$ and $\mathcal { G } _ { m + 1 }$ can be totally different since the two base embeddings are learned independently. Even if we follow the paradigm in (Chen et al., 2018) and conduct base embedding on $\mathcal { G } _ { m }$ using the simple projected embeddings from $\mathcal { G } _ { m + 1 }$ ( $\mathcal { E } _ { m } ^ { p }$ ) as initialization, the embedding space does not naturally generalize and can drift during re-training. One possible solution is to use an alignment procedure to force the embeddings to be aligned between the two graphs (Hamilton et al., 2016). But it could be very expensive.
+
+In this paper, we propose a very simple method to address the above issues. Instead of conducting an additional level of coarsening, we construct a dummy coarsened graph by simply copying $\mathcal { G } _ { m }$ , i.e., $M _ { m , m + 1 } = I$ and $\mathcal { G } _ { m + 1 } = \mathcal { G } _ { m }$ . By doing this, we not only reduce one iteration of graph coarsening, but also avoid performing base embedding on $\mathcal { G } _ { m + 1 }$ simply because $\mathcal { E } _ { m + 1 } = \mathcal { E } _ { m }$ . Moreover, the embeddings of $\mathcal { G } _ { m }$ and $\mathcal { G } _ { m + 1 }$ are guaranteed to be in the same space in this case without any drift. With this strategy, we change the loss function for model learning as follows
+
+$$
+L = \frac { 1 } { | V _ { m } | } \left\| \mathcal { E } _ { m } - H ^ { ( l ) } ( \mathcal { E } _ { m } , A _ { m } ) \right\| ^ { 2 } .
+$$
+
+With the above loss function, we adopt gradient descent with back-propagation to learn the parameters $\Theta ^ { ( k ) }$ , $k \in [ 1 , l ]$ . In the subsequent refinement steps, we apply the same set of parameters $\Theta ^ { ( k ) }$ to infer the refined embeddings. We point out that the training of the refinement model is rather efficient as it is done on the coarsest graph. The embeddings refinement process involves merely sparse matrix multiplications using Eq. 5 and is relatively affordable compared to conducting embedding on the original graph. With these different components, we summarize the whole algorithm of our MILE framework in Algorithm 1. The appendix contains the time complexity of the algorithm in Section A.2
+
+# Algorithm 1 Multi-Level Algorithm for Graph Embedding
+
+Input: A input graph $\mathcal { G } _ { 0 } = ( V _ { 0 } , E _ { 0 } )$ , $\#$ coarsening levels $m$ , and a base embedding method $f ( \cdot )$ Output: Graph embeddings $\dot { \varepsilon _ { \mathrm { 0 } } }$ on $\mathscr { G } _ { 0 }$ .
+
+1: Coarsen $\mathscr { G } _ { 0 }$ into $\mathcal { G } _ { 1 } , \mathcal { G } _ { 2 } , . . . , \mathcal { G } _ { m }$ using proposed hybrid matching method.   
+2: Perform base embedding on the coarsest graph $\mathcal { G } _ { m }$ (See Section. 4.2).   
+3: Learn the weights $\Theta ^ { ( k ) }$ using the loss function in Eq. 7.   
+4: for $i = ( m - 1 ) . . . 0$ do   
+5: Compute the projected embeddings $\mathcal { E } _ { i } ^ { p }$ on $\beta _ { i }$ .   
+6: Use Eq. 4 and Eq. 5 to compute refined embeddings $\xi _ { i }$ .   
+7: Return graph embeddings $\dot { \varepsilon _ { \mathrm { 0 } } }$ on $\mathscr { G } _ { 0 }$ .
+
+# 5 Experiments and Analysis
+
+# 5.1 Experimental Configuration
+
+The datasets used in our experiments is shown in Table 1. Yelp dataset is preprocessed by us following similar procedures in (Huang et al., 2017)1. To demonstrate that MILE can work with different graph embedding methods , we explore several popular methods for graph embedding, mainly, DeepWalk (Perozzi et al., 2014), Node2vec (Grover & Leskovec, 2016), Line (Tang et al., 2015), GraRep (Cao et al., 2015) and NetMF (Qiu et al., 2018). To evaluate the quality of the embeddings, we follow the typical method in existing work to perform multi-label node classification (Perozzi et al., 2014; Grover & Leskovec, 2016).
+
+Table 1: Dataset Information   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>#Nodes</td><td rowspan=1 colspan=1># Edges</td><td rowspan=1 colspan=1># Classes</td></tr><tr><td rowspan=1 colspan=1>PPI</td><td rowspan=1 colspan=1>3,852</td><td rowspan=1 colspan=1>38,705</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>Blog</td><td rowspan=1 colspan=1>10,312</td><td rowspan=1 colspan=1>333,983</td><td rowspan=1 colspan=1>39</td></tr><tr><td rowspan=1 colspan=1>Flickr</td><td rowspan=1 colspan=1>80,513</td><td rowspan=1 colspan=1>5,899,882</td><td rowspan=1 colspan=1>195</td></tr><tr><td rowspan=1 colspan=1>YouTube</td><td rowspan=1 colspan=1>1,134,890</td><td rowspan=1 colspan=1>2,987,624</td><td rowspan=1 colspan=1>47</td></tr><tr><td rowspan=1 colspan=1>Yelp</td><td rowspan=1 colspan=1>8,938,630</td><td rowspan=1 colspan=1>39,821,123</td><td rowspan=1 colspan=1>22</td></tr></table>
+
+# 5.2 MILE Framework Performance
+
+We first evaluate the performance of our MILE framework when applied to different graph embedding methods. Figure 3 summarizes the performance of MILE on different datasets with various base embedding methods on various coarsening levels $^ 2$ (exact numbers can be seen in Table 3 of Appendix). Note that $m { = } 0$ corresponds to original embedding method. We make the following observations:
+
+• MILE is scalable. MILE greatly boosts the speed of the explored embedding methods. With a single level of coarsening ( $_ { \mathbf { \nabla } ^ { \prime } \mathbf { \nabla } ^ { \prime } } \psi _ { \mathbf { \nabla } ^ { \prime } }$ =1), we are able to achieve speedup ranging from $1 . 5 \times$ to 3.4 $\times$ (on PPI, Blog, and Flickr) while improving qualitative performance. Larger speedups are typically observed on GraRep and NetMF. Increasing the coarsening level $m$ to 2, the speedup increases further (up to $1 4 . 4 \times$ ), while the quality of the embeddings is comparable with the original methods reflected by Micro-F1. On YouTube, for the coarsening levels 6 and 8, we observe more than $1 0 \times$ speedup for DeepWalk, Node2Vec and LINE. For NetMF on YouTube, the speedup is even larger – original NetMF runs out of memory within 9.5 hours while MILE (NetMF) only takes around 20 minutes ( $m = 8$ ).
+
+• MILE improves quality. For the smaller coarsening levels across all the datasets and methods, MILE-enhanced embeddings almost always offer a qualitative improvement over the original embedding method as evaluated by the Micro-F1 score (as high as $2 4 . 2 \%$ while many others also show a $1 0 \% +$ increase). Examples include MILE (DeepWalk, $m = 1$ ) on Blog/PPI, MILE (Line, $m = 1$ ) on PPI and MILE (NetMF, $m = 1$ ) on PPI/Blog/Flickr. Even with higher number of coarsening level ( $m = 2$ for PPI/Blog/Flickr; $m = 6 , 8$ for YouTube), MILE in addition to being much faster can still improve, qualitatively, over the original methods on most of the datasets, e.g., MILE(NetMF, $m = 2$ ) $\gg$ NETMF on PPI, Blog, and Flickr. We conjecture the observed improvement on quality is because the embeddings begin to rely on a more holistic view of the graph.
+
+![](images/32fc83f891331a313c500f53e1f75d24574879de4c15de4e1a31f2ac5c190fbd.jpg)  
+Figure 3: Changes in performance as the number of coarsening levels in MILE increases (best viewed in color). Micro-F1 and running-time are reported in the first and second row respectively. Running time in minutes is shown in logarithm scale. Note that # level = 0 represents the original embedding method without using MILE. Lines/points are missing for algorithms that use over 128 GB of RAM.
+
+• MILE supports multiple embedding strategies. We make some embedding-specific observations here. We observe that MILE consistently improves both the quality and the efficiency of NetMF on all four datasets (for YouTube, NetMF runs out of memory). For the largest dataset, the speedups afforded exceed 30-fold. We observe that for GraRep, while speedups with MILE are consistently observed, the qualitative improvements, if any, are smaller (for both YouTube and Flickr, the base method runs out of memory). For Line, even though its time complexity is linear to the number of edges (Tang et al., 2015), applying MILE framework on top of it still generates significant speed-up (likely due to the fact that the complexity of Line contains a larger constant factor $k$ than MILE). On the other hand, MILE on top of Line generates better quality of embeddings on PPI and YouTube while falling a bit short on Blog and Flickr. For DeepWalk and Node2Vec, we again observe consistent improvements in scalability (up to 11-fold on the larger datasets) as well as quality using MILE with a few levels of coarsening. However, when the coarsening level is increased, the additional speedup afforded (up to 17-fold) comes at a mixed cost to quality (micro-F1 drops slightly).
+
+• Impact of varying coarsening levels on MILE. When coarsening level $m$ is small, MILE tends to significantly improve the quality of embeddings while taking much less time. From $m = 0$ to $m = 1$ , we see a clear jump of the Micro-F1 score on all the datasets across the base embedding methods. This observation is more evident on larger datasets (Flickr and YouTube). On YouTube, MILE (DeepWalk) with $m { = } 1$ increases the Micro-F1 score by $5 . 3 \%$ while only consuming half of the time compared to the original DeepWalk. MILE (DeepWalk) continues to generate embeddings of better quality than DeepWalk until $m = 7$ , where the speedup is $1 3 \times$ . As the coarsening level $m$ in MILE increases, the running time drops dramatically while the quality of embeddings only decreases slightly.
+
+Table 2: MILE vs. HARP   
+
+<table><tr><td></td><td colspan="2">PPI</td><td colspan="2">Blog</td></tr><tr><td>DeepWalk (DW)</td><td>Mi-F1 23.0</td><td>Time</td><td>Mi-F1</td><td>Time 8.0</td></tr><tr><td>MILE (DW) HARP (DW) Node2Vec (NV)</td><td>25.6 24.1 24.3</td><td>2.4 1.2 3.0 4.0</td><td>37.0 42.9 41.3 39.1</td><td>4.6 9.8 13.0</td></tr><tr><td>MILE (NV)</td><td>25.9</td><td>1.7</td><td>42.8</td><td>6.9</td></tr><tr><td>HARP (NV)</td><td>22.3</td><td>3.9</td><td>36.2</td><td>13.16</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td colspan="2">Flickr</td><td colspan="2">YouTube</td></tr><tr><td></td><td>Mi-F1</td><td>Time</td><td>Mi-F1</td><td>Time</td></tr><tr><td>DeepWalk</td><td>40.0</td><td>50.0</td><td>45.2</td><td>604.8</td></tr><tr><td>MILE (DW)</td><td>40.4</td><td>34.4</td><td>46.1</td><td>55.2</td></tr><tr><td>HARP (DW)</td><td>40.6</td><td>78.2</td><td>46.6</td><td>1727.7</td></tr><tr><td>Node2Vec</td><td>40.5</td><td>78.2</td><td>45.5</td><td>951.2</td></tr><tr><td>MILE (NV)</td><td>40.7</td><td>50.5</td><td>46.3</td><td>83.5</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>HARP (NV)</td><td>40.5</td><td>101.1</td><td>47.2</td><td>1981.3</td></tr></table>
+
+![](images/2d9163b7fb79dfc220ffdc2e72b13fae8923f7c2edaf53f52156470e2e60b225.jpg)  
+Figure 4: Running MILE on Yelp dataset.
+
+The running time decreases at an almost exponential rate (logarithm scale on the y-axis in the second row of Figure 3). On the other hand, the Micro-F1 score descends much more slowly (the first row of Figure 3). most of which are still better than the original methods. This shows that MILE can not only consolidates the existing embedding methods, but also provides nice trade-off between effectiveness and efficency.
+
+# 5.3 Comparing MILE with HARP
+
+HARP is a multi-level method primarily for improving the quality of graph embeddings. We compare HARP with our MILE framework using DeepWalk and Node2vec as the base embedding methods3. Table 2 shows the performance of these two methods on the four datasets (coarsening level is 1 on PPI/Blog/Flickr and 6 on YouTube). From the table we can observe that MILE generates embeddings of comparable quality with HARP. MILE performs much better than HARP on PPI and Blog, marginally better on Flickr and marginally worse on YouTube. However, MILE is significantly faster than HARP on all the four datasets (e.g. on YouTube, MILE affords a $3 1 \times$ speedup). This is because HARP requires running the whole embedding algorithm on each coarsened graph, which introduces a huge computational overhead. Note that for PPI and BLOG – MILE with NetMF (not shown) as its base embeddings produces the best micro-F1 of 26.9 and 43.8, respectively. This shows another advantage of MILE - agnostic to the base embedding when compared with HARP.
+
+# 5.4 MILE: Large Graph Embedding
+
+We now explore the scalability of MILE on the large Yelp dataset. None of the five graph embedding methods studied in this paper can successfully conduct graph embedding on Yelp within 60 hours on a modern machine with 28 cores and 128 GB RAM. Even extending the run-time deadline to 100 hours, we see DeepWalk and Line barely finish. Leveraging the proposed MILE framework now makes it much easier to perform graph embedding on this scale of datasets (see Figure 4 for the results). We observe that MILE significantly reduces the running time and improves the Micro-F1 score. For example, Micro-F1 score of original DeepWalk and Line are 0.640 and 0.625 respectively, which all take more than 80 hours. But using MILE with $m = 4$ , the micro-F1 score improves to 0.643 (DeepWalk) and 0.642 (Line) while achiving speedups of around $1 . 6 \times$ . Moreover, MILE reduces the running time of DeepWalk from 53 hours (coarsening level 4) to 2 hours (coarsening level 22) while reducing the Micro-F1 score just by $1 \%$ (from 0.643 to 0.634). Meanwhile, there is no change in the Micro-F1 score from coarsening level 4 to 10, where the running time is improved by a factor of two. These results affirm the power of the proposed MILE framework on scaling up graph embedding algorithms while generating quality embeddings.
+
+# 6 Conclusion
+
+In this work, we propose a novel multi-level embedding (MILE) framework to scale up graph embedding techniques, without modifying them. Our framework incorporates existing embedding techniques as black boxes, and significantly improves the scalability of extant methods by reducing both the running time and memory consumption. Additionally, MILE also provides a lift in the quality of node embeddings in most of the cases. A fundamental contribution of MILE is its ability to learn a refinement strategy that depends on both the underlying graph properties and the embedding method in use. In the future, we plan to generalize MILE for information-rich graphs and employing MILE for more applications.
+
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+
+# A Appendix
+
+A.1 Experimental Configuration Details
+
+A.1.1 Datasets
+
+The details about the datasets used in our experiments are :
+
+• PPI is a Protein-Protein Interaction graph constructed based on the interplay activity between proteins of Homo Sapiens, where the labels represent biological states.   
+• Blog is a network of social relationship of bloggers on BlogCatalog and the labels indicate interests of the bloggers.   
+• Flickr is a social network of the contacts between users on flickr.com with labels denoting the interest groups.   
+• YouTube is a social network between users on YouTube, where labels represent genres of groups subscribed by users.   
+• Yelp is a social network of friends on Yelp and labels indicate the business categories on which the users review.
+
+# A.1.2 Baseline Methods
+
+Baseline Methods: To demonstrate that MILE can work with different graph embedding methods, we explore several popular methods for graph embedding.
+
+• DeepWalk (DW) (Perozzi et al., 2014): Following the original work (Perozzi et al., 2014), we set the length of random walks as 80, number of walks per node as 10, and context windows size as 10.   
+• Node2Vec (NV) (Grover & Leskovec, 2016): We use the same setting as DeepWalk for those common hyper-parameters while setting $p = 4 . 0$ and $q = 1 . 0$ , which we found empirically to generate better results across all the datasets.   
+• Line (LN) (Tang et al., 2015): This method aims at preserving first-order and secondorder proximities and has been applied on large-scale graph. We learn the first-order and second-order embeddings respectively and concatenate them to a unified embedding.   
+• GraRep (GR) (Cao et al., 2015): This method considers different powers (up to $k$ ) of the adjacency matrix to preserve higher-order graph proximity for graph embedding. It uses SVD decomposition to generate the low-dimensional representation of nodes. We set $k = 4$ as suggested in the original work.   
+• NetMF (NM) (Qiu et al., 2018): It is a recent effort that supports graph embedding via matrix factorization. We set the window size to 10 and the rank $h$ to 1024, and lever the approximate version, as suggested and reported by the authors.
+
+# A.1.3 MILE-specific Settings
+
+For all the above base embedding methods, we set the embedding dimensionality $d$ as 128. When applying our MILE framework, we vary the coarsening levels $m$ from 1 to 10 whenever possible. For the graph convolution network model, the self-loop weight $\lambda$ is set to 0.05, the number of hidden layers $\it l$ is 2, and tanh $( \cdot )$ is used as the activation function, the learning rate is set to 0.001 and the number of training epochs is 200. The Adam Optimizer is used for model training.
+
+# A.1.4 System Specification
+
+The experiments were conducted on a machine running Linux with an Intel Xeon E5-2680 CPU (28 cores, 2.40GHz) and 128 GB of RAM. We implement our MILE framework in Python. Our code and data are will be available for the replicability purpose. For all the five base embedding methods, we adapt the original code from the authors4. We additionally use TensorFlow package for the embeddings refinement learning component. We lever the available parallelism (on 28 cores) for each method (e.g., the generation of random walks in DeepWalk and Node2Vec, the training of the refinement model in MILE, etc.).
+
+# A.1.5 Evaluation Metrics
+
+To evaluate the quality of the embeddings, we follow the typical method in existing work to perform multi-label node classification (Perozzi et al., 2014; Grover & Leskovec, 2016). Specifically, after the graph embeddings are learned for nodes (label is not used for this part), we run a 10-fold cross validation using the embeddings as features and report the average Micro-F1 and average Macro-F1. We also record the end-to-end wallclock time consumed by each method for scalability comparisons.
+
+# A.2 Time Complexity
+
+It is non-trivial to derive the exact time complexity of MILE as it is dependent on the graph structure, the chosen base embedding method, and the convergence rate of the GCN model training. Here, we provide a rough estimation of the time complexity. For simplicity, we assume the number of vertices and the number of edges are reduced by factor $\alpha$ and $\beta$ respectively at each step of coarsening ( $\alpha > 1 . 0$ and $\beta > 1 . 0$ ), i.e., $\begin{array} { r } { V _ { i } ~ = ~ \frac { 1 } { \alpha } V _ { i - 1 } } \end{array}$ and $\begin{array} { r } { E _ { i } = \frac { 1 } { \beta } E _ { i - 1 } } \end{array}$ . (we found $\alpha$ and $\beta$ in range [1.5, 2.0], empirically). With $m$ levels of coarsening, the coarsening complexity is approximately ${ \cal O } ( ( 1 - 1 / \beta ^ { \mathrm { m } } ) / ( 1 - 1 / \beta ) \times E ) \}$ and since $1 / \beta ^ { \mathrm { m } }$ is small, the complexity reduces to $\scriptstyle O ( { \frac { \beta } { \beta - 1 } } \times E )$ . For the base embedding phase, if the embeddiphase is m has time complexity of . For the refinement ph $T ( V , E )$ , the complexity of the base embedding time complexity can be divided into $\begin{array} { r } { T ( \frac { V } { \alpha ^ { m } } , \frac { E } { \beta ^ { m } } ) } \end{array}$   
+two parts, i.e. the GCN model training and the embedding inference applying the GCN $\begin{array} { r } { O ( k _ { 1 } * { \frac { E } { \beta ^ { m } } } ) } \end{array}$ e forme (Kipf $\&$ has similar complexity Welling, 2017), where $k _ { 1 }$ the original GCN and can be denoted asis a small constant related to embedding dimensionality and the number of training epochs. The embedding inference part is simply sparse matrix multiplication using Eq. 4 with time complexity $O ( k _ { 2 } * E _ { i } )$ when refining the embeddings on graph $\beta _ { i }$ , where $k _ { 2 }$ is an even smaller constant $( k _ { 2 } < k _ { 1 }$ ). As a result, the time complexity of the whole refinement phase is $\begin{array} { r } { O ( k _ { 1 } * \frac { E } { \beta ^ { m } } + k _ { 2 } * ( E + \frac { E } { \beta ^ { 1 } } . . . + \frac { E } { \beta ^ { m - 1 } } ) ) \approx O ( k _ { 3 } * E ) } \end{array}$ where $k _ { 3 }$ is a small constant.
+
+reduce it to be Overall, for an embedding algorithm of time complexity $\begin{array} { r } { T ( \frac { V } { \alpha ^ { m } } , \frac { E } { \beta ^ { m } } ) + O ( k * E ) } \end{array}$ . This is a significant improvement considering $T ( V , E )$ , the MILE framework can $T ( V , E )$ is usually very large. The reduction in time complexity is attributed to the fact that we run the embedding learning and refinement model training at the coarsest graph. In addition, the overhead introduced by the coarsening phase and recursive embedding refinement is relatively small (linear to the number of edges $E$ ). Note that the constant factor $k$ in the complexity term is usually small and we empirically found it to be in the scale of tens. Because of this, even when the complexity of the original embedding algorithm is linear to $E$ , our MILE framework could still potentially speed up the embedding process because the complexity of MILE contains a smaller constant factor $k$ (see Sec. 5.2 for the experiment of applying MILE on LINE).
+
+Furthermore, it is worth noting that many of the existing embedding strategies involve hyperparameters tunning for the best performance, especially for those methods based on neural networks (e.g., DeepWalk, Node2Vec, etc.). This in turn requires the algorithm to be run repeatedly – hence any savings in runtime by applying MILE are magnified across multiple runs of the algorithm with different hyper-parameter settings.
+
+# A.3 MILE Performance
+
+The detailed information about performance evaluation is available in Table 3.   
+(a) PPI Dataset   
+
+<table><tr><td>Method</td><td>Micro-F1</td><td>Macro-F1</td><td>Time (mins)</td></tr><tr><td>DeepWalk MILE (DeepWalk,m =1) MILE(DeepWalk,m =2)</td><td>23.0 25.6(11.3%↑) 25.5(10.9%↑)</td><td>18.6 20.4(9.7%1) 20.7(11.3%↑)</td><td>2.42 1.22(2.0×) 0.67(3.6x)</td></tr><tr><td>Node2Vec MILE (Node2Vec,m =1) MILE (Node2Vec,m=2)</td><td>24.3 25.9(6.6%↑) 26.0(7.0%↑)</td><td>19.6 20.6(5.1%↑) 21.1(7.7%↑)</td><td>4.01 1.77(2.3×) 0.98(4.1x)</td></tr><tr><td>Line MILE (Line, m = 1) MILE (Line, m = 2)</td><td>25.0 25.8 (3.2%↑) 24.7 (-1.2%↓)</td><td>19.5 19.8 (1.5%↑) 19.0 (-2.6%↓)</td><td>2.27 1.22 (1.9×) 0.68 (3.3x)</td></tr><tr><td>GraRep</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td>25.3(-0.8%↓)</td><td></td><td>1.11(2.7×)</td></tr><tr><td>MILE (GraRep, m = 1)</td><td>25.5</td><td>20.0</td><td>2.99</td></tr><tr><td></td><td>25.6(0.4%↑)</td><td>19.8(-1.0%↓)</td><td></td></tr><tr><td>MILE(GraRep,m=2)</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>19.5(-2.5%↓)</td><td>0.43(6.9x)</td></tr><tr><td>NetMF</td><td>24.6</td><td>20.1</td><td>0.65</td></tr><tr><td>MILE (NetMF, m = 1)</td><td>26.9(9.3%↑)</td><td></td><td></td></tr><tr><td></td><td></td><td>21.6(7.5%↑)</td><td>0.27(2.5×)</td></tr><tr><td>MILE (NetMF,m=2)</td><td>26.7(8.5%↑)</td><td>21.1(5.0%↑)</td><td>0.17(3.9×)</td></tr></table>
+
+<table><tr><td>Method</td><td>Micro-F1</td><td>Macro-F1</td><td>Time (mins)</td></tr><tr><td>DeepWalk MILE (DeepWalk,m =1)</td><td>37.0 42.9(15.9%↑)</td><td>21.0 27.0(28.6%↑)</td><td>8.02 4.69(1.7×)</td></tr><tr><td>MILE (DeepWalk,m= 2) Node2Vec MILE (Node2Vec, m = 1)</td><td>39.4(6.5%↑) 39.1 42.8(9.5%↑)</td><td>23.5(11.9%↑) 23.0 26.4(14.8%↑)</td><td>2.71(3.0×) 13.04 6.99(1.9×)</td></tr><tr><td>MILE(Node2Vec,m=2) Line</td><td>40.2(2.8%↑) 39.1</td><td>23.9(3.9%↑)</td><td>3.89(3.4×)</td></tr><tr><td></td><td></td><td>22.6</td><td>5.95</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>MILE (Line,m=1)</td><td>38.4 (-1.8%↓)</td><td>21.0 (-7.0%↓)</td><td></td></tr><tr><td></td><td></td><td></td><td>3.84 (1.55x)</td></tr><tr><td>MILE (Line,m=2)</td><td>37.3 (-4.6%↓)</td><td>19.6 (-13.2%↓)</td><td>2.58 (2.31x)</td></tr><tr><td>GraRep</td><td></td><td></td><td></td></tr><tr><td></td><td>40.6</td><td>23.3</td><td>28.76</td></tr><tr><td>MILE (GraRep,m =1)</td><td>41.7(2.7%↑)</td><td>24.0(3.0%↑)</td><td>12.25(2.3×)</td></tr><tr><td>MILE(GraRep,m=2)</td><td>38.3(-5.7%↓)</td><td>20.4(-12.4%↓)</td><td>4.22(6.8x)</td></tr><tr><td>NetMF</td><td>41.4</td><td>25.0</td><td>2.64</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>MILE (NetMF,m=1)</td><td>43.8(5.8%↑)</td><td>27.6(10.4%↑)</td><td>1.98(1.3×)</td></tr><tr><td>MILE(NetMF,m=2)</td><td>42.4(2.4%↑)</td><td>25.5(2.0%↑)</td><td>1.27(2.1×)</td></tr></table>
+
+(b) Blog Dataset
+
+<table><tr><td>Method</td><td>Micro-F1</td><td>Macro-F1</td><td>Time (mins)</td></tr><tr><td>DeepWalk MILE (DeepWalk, m = 1)</td><td>40.0 40.4(1.0%↑)</td><td>26.5 27.3(3.0%↑)</td><td>50.08 34.48(1.5×)</td></tr><tr><td>MILE (DeepWalk,m=2)</td><td>39.3(-1.8%↓)</td><td>26.1(-1.5%)</td><td>26.88(1.9x)</td></tr><tr><td>Node2Vec</td><td>40.5</td><td>27.3</td><td>78.21</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>MILE (Node2Vec, m = 1)</td><td>40.7(0.5%↑)</td><td>27.7(1.5%↑)</td><td>50.54(1.5x)</td></tr><tr><td>MILE(Node2Vec,m=2)</td><td>38.8(-4.2%↓)</td><td>25.8(-5.5%)</td><td>36.85(2.1x)</td></tr><tr><td>Line</td><td>34.0</td><td>14.5</td><td>60.42</td></tr><tr><td>MILE (Line, m =1)</td><td>33.9 (-0.3%↓)</td><td>13.8 (-4.8%↓)</td><td>30.24 (2.00×)</td></tr><tr><td>MILE (Line, m= 2)</td><td>33.3 (-2.1%↓)</td><td>12.9 (-11.0%↓)</td><td>19.05 (3.17×)</td></tr><tr><td>GraRep</td><td>N/A</td><td>N/A</td><td>&gt;2343.37</td></tr><tr><td>MILE (GraRep,m =1)</td><td>36.7</td><td>18.6</td><td>697.39(&gt;3.4×)</td></tr><tr><td>MILE (GraRep,m=2)</td><td>36.3</td><td>18.6</td><td>163.05(&gt;14.4×)</td></tr><tr><td>NetMF</td><td>31.8</td><td>14.0</td><td></td></tr><tr><td></td><td></td><td></td><td>69.72</td></tr><tr><td>MILE (NetMF, m=1)</td><td>39.3(23.6%↑)</td><td>24.5(75.0%↑)</td><td>24.03(2.9×)</td></tr><tr><td>MILE (NetMF,m= 2)</td><td>39.5(24.2%↑)</td><td>25.9(85.0%↑)</td><td>15.84(4.4×)</td></tr></table>
+
+(c) Flickr Dataset
+
+<table><tr><td>Method</td><td>Micro-F1</td><td>Macro-F1</td><td>Time (mins)</td></tr><tr><td>DeepWalk MILE (DeepWalk,m = 6)</td><td>45.2 46.1(2.0%↑)</td><td>34.7 38.5(11.0%↑)</td><td>604.83 55.20(11.0×)</td></tr><tr><td>MILE (DeepWalk,m = 8) Node2Vec</td><td>44.3(-2.0%↓)</td><td>35.3(1.7%↑)</td><td>37.35(16.2x)</td></tr><tr><td>MILE (Node2Vec, m = 6)</td><td>45.5</td><td>34.6</td><td>951.27</td></tr><tr><td></td><td>46.3(1.8%↑)</td><td>38.3(10.7%↑)</td><td>83.52(11.4×)</td></tr><tr><td>MILE (Node2Vec,m=8)</td><td></td><td>35.8(3.5%↑)</td><td></td></tr><tr><td></td><td>44.3(-2.6%↓)</td><td></td><td>55.55(17.1×)</td></tr><tr><td>Line</td><td>46.0</td><td>35.0</td><td>583.37</td></tr><tr><td>MILE (Line, m = 6)</td><td>46.2 (0.4%↑)</td><td>36.2 (3.4%↑)</td><td>53.97 (10.81x)</td></tr><tr><td>MILE (Line,m=8)</td><td>44.4 (-3.5%↓)</td><td>35.7 (2.0%↑)</td><td>33.41 (17.46×)</td></tr><tr><td>GraRep</td><td>N/A</td><td>N/A</td><td>&gt;3167.00</td></tr><tr><td></td><td>43.2</td><td>32.7</td><td></td></tr><tr><td>MILE(GraRep,m=6)</td><td></td><td></td><td>1644.89(&gt;1.9×)</td></tr><tr><td>MILE(GraRep,m=8)</td><td>42.3</td><td>30.9</td><td>673.95(&gt;4.7×)</td></tr><tr><td>NetMF</td><td>N/A</td><td>N/A</td><td>&gt; 574.75</td></tr><tr><td>MILE (NetMF,m=6)</td><td>40.9</td><td>27.8</td><td>35.22(&gt;16.3×)</td></tr><tr><td>MILE (NetMF,m=8)</td><td>39.2</td><td>25.5</td><td>19.22(&gt;29.9×)</td></tr></table>
+
+(d) YouTube Dataset
+
+Table 3: Performance of MILE. DeepWalk, Node2Vec, GraRep, and NetMF denotes the original method without using our MILE framework. $m$ is the number of coarsening levels. The numbers within the parenthesis by the reported Micro-F1 and Macro-F1 scores are the relative percentage of change compared to the original method Numbers along with “ $\times$ ” is the speedup compared to the original method. “N/A” indicates the method runs out of memory and we show the amount of running time spent when it happens.
+
+# A.4 MILE Drilldown: Design Choices
+
+We now study the role of the design choices we make within the MILE framework related to the coarsening and refinement procedures described. To this end, we examine alternative design choices and systematically examine their performance. The alternatives we consider are:
+
+• Random Matching (MILE-rm): For each iteration of coarsening, we repeatedly pick a random pair of connected nodes as a match and merge them into a super-node until no more matching can be found. The rest of the algorithm is the same as our MILE.   
+• Simple Projection (MILE-proj): We replace our embedding refinement model with a simple projection method. In other words, we directly copy the embedding of a super-node to its original node(s) without any refinement (see Eq. 3).   
+• Averaging Neighborhoods (MILE-avg): For this baseline method, the refined embedding of each node is a weighted average node embeddings of its neighborhoods (weighted by the edge weights). This can be regarded as an embeddings propagation method. We add self-loop to each node $_ 6$ and conduct the embeddings propagation for two rounds.   
+• Untrained Refinement Model (MILE-untr): Instead of training the refinement model to minimize the loss defined in Eq. 7, this baseline merely uses a fixed set of values for parameters $\Theta ^ { ( k ) }$ without training (values are randomly generated; other parts of the model in Eq. 4 are the same, including $\tilde { A }$ and $\tilde { D }$ ).
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>PPI</td><td rowspan=1 colspan=3>Blog</td><td rowspan=1 colspan=2>Flickr</td><td rowspan=1 colspan=2>YouTube</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Mi-F1</td><td rowspan=1 colspan=1>Time</td><td rowspan=1 colspan=2>Mi-F1</td><td rowspan=1 colspan=1>Time</td><td rowspan=1 colspan=1>Mi-F1</td><td rowspan=1 colspan=1>Time</td><td rowspan=1 colspan=1>Mi-F1</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=3 colspan=1>DeepWalkMILE (DW)MILE-rm (DW)</td><td rowspan=1 colspan=1>23.0</td><td rowspan=1 colspan=1>2.42</td><td rowspan=1 colspan=2>37.0</td><td rowspan=1 colspan=1>8.02</td><td rowspan=1 colspan=1>40.0</td><td rowspan=1 colspan=1>50.08</td><td rowspan=1 colspan=1>45.2</td><td rowspan=1 colspan=1>604.83</td></tr><tr><td rowspan=1 colspan=1>25.6</td><td rowspan=1 colspan=1>1.22</td><td rowspan=1 colspan=2>42.9</td><td rowspan=1 colspan=1>4.69</td><td rowspan=1 colspan=1>40.4</td><td rowspan=1 colspan=1>34.48</td><td rowspan=1 colspan=1>46.1</td><td rowspan=1 colspan=1>55.20</td></tr><tr><td rowspan=1 colspan=1>25.3</td><td rowspan=1 colspan=1>1.01</td><td rowspan=1 colspan=2>40.4</td><td rowspan=1 colspan=1>3.62</td><td rowspan=1 colspan=1>38.9</td><td rowspan=1 colspan=1>26.67</td><td rowspan=1 colspan=1>44.9</td><td rowspan=1 colspan=1>55.10</td></tr><tr><td rowspan=2 colspan=1>MILE-proj (DW)MILE-avg (DW)</td><td rowspan=1 colspan=1>20.9</td><td rowspan=1 colspan=1>1.12</td><td rowspan=1 colspan=2>34.5</td><td rowspan=1 colspan=1>3.92</td><td rowspan=1 colspan=1>35.5</td><td rowspan=1 colspan=1>25.99</td><td rowspan=1 colspan=1>40.7</td><td rowspan=1 colspan=1>53.97</td></tr><tr><td rowspan=1 colspan=1>23.5</td><td rowspan=1 colspan=1>1.07</td><td rowspan=1 colspan=2>37.7</td><td rowspan=1 colspan=1>3.86</td><td rowspan=1 colspan=1>37.2</td><td rowspan=1 colspan=1>25.99</td><td rowspan=1 colspan=1>41.4</td><td rowspan=4 colspan=1>55.2654.5294.74394.72</td></tr><tr><td rowspan=3 colspan=1>MILE-untr (DW)MILE-2base (DW)MILE-gs (DW)</td><td rowspan=1 colspan=1>23.5</td><td rowspan=1 colspan=1>1.08</td><td rowspan=1 colspan=2>35.5</td><td rowspan=1 colspan=1>3.96</td><td rowspan=1 colspan=1>37.6</td><td rowspan=1 colspan=1>26.02</td><td rowspan=3 colspan=1>41.841.643.6</td></tr><tr><td rowspan=2 colspan=1>25.422.4</td><td rowspan=2 colspan=1>2.222.03</td><td rowspan=2 colspan=2>35.635.3</td><td rowspan=1 colspan=1>.6</td><td rowspan=1 colspan=1>6.74</td><td rowspan=2 colspan=1>37.736.4</td><td rowspan=2 colspan=1>53.3244.81</td></tr><tr><td rowspan=1 colspan=1>6.44</td></tr><tr><td rowspan=8 colspan=1>NetMFMILE (NM)MILE-rm (NM)MILE-proj (NM)MILE-avg (NM)MILE-untr (NM)MILE-2base (NM)MILE-gs(NM)</td><td rowspan=1 colspan=1>24.6</td><td rowspan=1 colspan=1>0.65</td><td rowspan=1 colspan=2>41.4</td><td rowspan=1 colspan=1>2.64</td><td rowspan=1 colspan=1>31.8</td><td rowspan=1 colspan=1>69.72</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>&gt;574</td></tr><tr><td rowspan=1 colspan=1>26.9</td><td rowspan=1 colspan=1>0.27</td><td rowspan=1 colspan=2>43.8</td><td rowspan=1 colspan=1>1.98</td><td rowspan=1 colspan=1>39.3</td><td rowspan=1 colspan=1>24.03</td><td rowspan=1 colspan=1>40.9</td><td rowspan=1 colspan=1>35.22</td></tr><tr><td rowspan=2 colspan=1>25.223.5</td><td rowspan=2 colspan=1>0.220.12</td><td rowspan=2 colspan=2>41.038.7</td><td rowspan=1 colspan=1>1.69</td><td rowspan=1 colspan=1>37.6</td><td rowspan=1 colspan=1>20.00</td><td rowspan=2 colspan=1>39.626.4</td><td rowspan=2 colspan=1>33.5226.48</td></tr><tr><td rowspan=1 colspan=1>1.06</td><td rowspan=1 colspan=1>34.5</td><td rowspan=1 colspan=1>15.10</td></tr><tr><td rowspan=1 colspan=1>24.5</td><td rowspan=1 colspan=1>0.13</td><td rowspan=1 colspan=2>39.9</td><td rowspan=1 colspan=1>1.05</td><td rowspan=1 colspan=1>36.4</td><td rowspan=1 colspan=1>14.86</td><td rowspan=2 colspan=1>26.430.2</td><td rowspan=3 colspan=1>27.7127.2055.18</td></tr><tr><td rowspan=1 colspan=1>24.8</td><td rowspan=1 colspan=1>0.13</td><td rowspan=1 colspan=2>39.4</td><td rowspan=1 colspan=1>1.08</td><td rowspan=1 colspan=1>36.4</td><td rowspan=1 colspan=1>15.23</td><td rowspan=1 colspan=1>30.2</td></tr><tr><td rowspan=2 colspan=1>26.624.8</td><td rowspan=1 colspan=1>0.29</td><td rowspan=1 colspan=2>41.3</td><td rowspan=1 colspan=1>2.33</td><td rowspan=1 colspan=1>37.7</td><td rowspan=1 colspan=1>31.65</td><td rowspan=1 colspan=1>34.7</td></tr><tr><td rowspan=1 colspan=1>1.08</td><td rowspan=1 colspan=2>40.0</td><td rowspan=1 colspan=1>3.70</td><td rowspan=1 colspan=1>35.1</td><td rowspan=1 colspan=1>34.25</td><td rowspan=1 colspan=1>36.4</td><td rowspan=1 colspan=1>345.28</td></tr></table>
+
+Table 4: Comparisons of graph embeddings between MILE and its variants. Except for the original methods (DeepWalk and NetMF), the number of coarsening level $m$ is set to 1 on PPI/Blog/Flickr and 6 on YouTube. Mi-F1 is the Micro-F1 score in $1 0 ^ { - 2 }$ scale while Time column shows the running time of the method in minutes. “N/A” denotes the method consumes more than 128 GB RAM.
+
+• Double-base Embedding for Refinement Training (MILE-2base): This method replaces the loss function in Eq. 7 with the alternative one in Eq. 6 for model training. It conducts one more layer of coarsening and base embedding (level $m + 1$ ), from which the embeddings are projected to level $m$ and used as the input for model training.
+
+• GraphSAGE as Refinement Model (MILE-gs): It replaces the graph convolution network in our refinement method with GraphSAGE (Hamilton et al., 2017)7. We choose max-pooling for aggregation and set the number of sampled neighbors as 100, as suggested by the authors. Also, concatenation is conducted instead of replacement during the process of propagation.
+
+Table 4 shows the comparison of performance on these methods across the four datasets. Here, we focus on using DeepWalk and NetMF for base embedding with a smaller coarsening level ( $m = 1$ for PPI, Blog, and Flickr; $m = 6$ for YouTube). Results are similar for the other embedding options we consider. We hereby summarize the key information derived from Table 4 as follows:
+
+• The matching methods used within MILE offer a qualitative benefit at a minimal cost to execution time. Comparing MILE with MILE-rm for all the datasets, we can see that MILE generates better embeddings than MILE-rm using either DeepWalk or NetMF as the base embedding method. Though MILE-rm is slightly faster than MILE due to its random matching, its Micro-F1 score and Macro-F1 score are consistently lower than of MILE.
+
+• The graph convolution based refinement learning methodology in MILE is particularly effective. Simple projection-based MILE-proj, performs significantly worse than MILE. The other two variants (MILE-avg and MILE-untr) which do not train the refinement model at all, also perform much worse than the proposed method. Note MILE-untr is the same as MILE except it uses a default set of parameters instead of learning those parameters. Clearly, the model learning part of our refinement method is a fundamental contributing factor to the effectiveness of MILE. Through training, the refinement model is tailored to the specific graph under the base embedding method in use. The overhead cost of this learning (comparing MILE with MILE-untr), can vary depending on the base embedding employed (for instance on the YouTube dataset, it is
+
+![](images/bcfee66886bcfe9d33102d08de76353eaf807d04b701b4e600c21bca732fe6a8.jpg)
+
+Figure 5: Memory consumption of MILE (GraRep) and MILE (NetMF) on Blog with varied coarsening levels.
+
+an insignificant $1 . 2 \%$ on DeepWalk - while being up to $2 0 \%$ on NetMF) but is still worth it due to qualitative benefits (Micro-F1 up from 30.2 to 40.9 with NetMF on YouTube).
+
+• Graph convolution refinement learning outperforms GraphSAGE. Replacing the graph convolution network with GraphSAGE for embeddings refinement, MILE-gs does not perform as well as MILE. It is also computationally more expensive, partially due to its reliance on embeddings concatenation, instead of replacement, during the process the embeddings propagation (higher model complexity).
+
+• Double-base embedding learning is not effective. In Sec. 4.3, we discuss the issues with unaligned embeddings of the double-base embedding method for the refinement model learning. The performance gap between MILE and MILE-2base in Table 4 provides empirical evidence supporting our argument. This gap is likely caused by the fact that the base embeddings of level $m$ and level $m + 1$ might not lie in the same embedding space (rotated by some orthogonal matrix) (Hamilton et al., 2017). As a result, using the projected embeddings $\mathcal { E } _ { m } ^ { p }$ as input for model training (MILE-2base) is not as good as directly using ${ \mathcal { E } } _ { m }$ (MILE). Moreover, Table 4 shows that the additional round of base embedding in MILE-2base introduces a non-trivial overhead. On YouTube, the running time of MILE-2base is 1.6 times as much as MILE.
+
+# A.5 MILE Drilldown: Memory Consumption
+
+We also study the impact of MILE on reducing memory consumption. For this purpose, we focus on MILE (GraRep) and MILE (NetMF), with GraRep and NetMF as base embedding methods respectively. Both of these are embedding methods based on matrix factorization, which possibly involves a dense objective matrix and could be rather memory expensive. We do not explore DeepWalk and Node2Vec here since their embedding learning methods generate truncated random walks (training data) on the fly with almost negligible memory consumption (compared to the space storing the graph and the embeddings). Figure 5 shows the memory consumption of MILE (GraRep) and MILE(NetMF) as the coarsening level increases on Blog (results on other datasets are similar). We observe that MILE significantly reduces the memory consumption as the coarsening level increases. Even with one level of coarsening, the memory consumption of GraRep and NetMF reduces by $6 4 \%$ and $4 2 \%$ respectively. The dramatic reduction continues as the coarsening level increases until it reaches 4, where the memory consumption is mainly contributed by the storage of the graph and the embeddings. This memory reduction is consistent with our intuition, since both # rows and # columns in the objective matrix reduce almost by half with one level of coarsening.
+
+# A.6 MILE Drilldown: Discussion on reusing $\Theta ^ { ( k ) }$ across all levels
+
+Similar to GCN, $\Theta ^ { ( k ) }$ is a matrix of filter parameters and is of size $d * d$ (where $d$ is the embedding dimensionality). Eq. 4 in this paper defines how the embeddings are propagated during embedding refinements, parameterized by $\Theta ^ { ( k ) }$ . Intuitively, $\Theta ^ { ( k ) }$ defines how different embedding dimensions interact with each other during the embedding propagation. This interaction is dependent on graph structure and base embedding method, which can be learned from the coarsest level. Ideally, we would like to learn this parameter $\Theta ^ { ( k ) }$ on every two consecutive levels. But this is not practical since this could be expensive as the graph get more fine-grained (and defeat our purpose of scaling up graph embedding). This trick of “sharing” parameters across different levels is the trade-off between efficiency and effectiveness. To some extent, it is similar to the original GCN (Kipf & Welling, 2017), where the authors share the same filter parameters $\Theta ^ { ( k ) }$ over the whole graph (as opposed to using different $\Theta ^ { ( k ) }$ for different nodes; see Eq (6) and (7) in(Kipf & Welling, 2017)). Moreover, we empirically found this works good enough and much more efficient. Table 4 shows that if we do not share $\Theta ^ { ( k ) }$ values and use random values for $\Theta ^ { ( k ) }$ during refinements, the quality of embedding is much worse (see baseline MILE-untr).
+
+# A.7 MILE Drilldown: Discussion on choice of embedding methods
+
+We wish to point out that we chose the base embedding methods as they are either recently proposed NetMF (introduced in 2018) or are widely used (DeepWalk, Node2Vec, LINE). By showing the performance gain of using MILE on top of these methods, we want to ensure the contribution of this work is of broad interest to the community. We also want to reiterate that these methods are quite different in nature:
+
+• DeepWalk (DW) and Node2vec (N2V) rely on the use of random walks for latent representation of features.   
+• LINE learns an embedding that directly optimizes a carefully constructed objective function that preserves both first/second order proximity among nodes in the embedding space. GraRep constructs multiple objective matrices based on high orders of random walk laplacians, factories each objective matrix to generate embeddings and then concatenates the generated embeddings to form final embedding.   
+• NetMF constructs an objective matrix based on random walk Laplacian and factorizes the objective matrix in order to generate the embeddings.
+
+Indeed NetMF (Qiu et al., 2018; Levy & Goldberg, 2014) with an appropriately constructed objective matrix has been shown to approximate DW, N2V and LINE allowing such be conducting implicit matrix factorization of approximated matrices. There are limitations to such approximations (shown in a related context by (Arora et al., 2016)) - the most important one is the requirement of a sufficiently large embedding dimensionality. Additionally, we note that while unification is possible under such a scenario, the methods based on matrix factorization are quite different from the original methods and do place a much larger premium on space (memory consumption) - in fact this is observed by the fact we are unable to run NetMF and GraRep in many cases without incorporating them within MILE.
+
+A.8 MILE Drilldown: Discussion on extending MILE to directed graphs
+
+Note that as pointed out by (Chung, 2005), one can construct random-walk Laplacians for a directed graph thus incorporating approaches like NetMF to accommodate such solutions. Another simple solution is to symmetrize the graph while accounting for directionality. Once the graph is symmetrized, any of the embedding strategies we discuss can be employed within the MILE framework (including the coarsening technique). There are many ideas for symmetrization of directed graphs (see for example work described by (Gleich, 2006) or (Satuluri & Parthasarathy, 2011).
+
+# A.9 MILE Drilldown: Discussion on effectiveness of SEM
+
+The effectiveness of structurally equivalent matching (SEM) is highly dependent on graph structure but in general 5% - $2 0 \%$ of nodes are structurally equivalent (most of which are low-degree nodes). For example, during the first level of coarsening, YouTube has 172,906 nodes (or 86,453 pairs) out of 1,134,890 nodes that are found to be SEM (around $1 5 \%$ ); Yelp has 875,236 nodes (or 437,618 pairs) out of 8,938,630 nodes are SEM (around $1 0 \%$ ). In fact, more nodes are involved in SEM as SEM is run iteratively at each coarsening level.

parse/train/HJflg30qKX/HJflg30qKX.md

@@ -0,0 +1,860 @@
+# GRADIENT DESCENT ALIGNS THE LAYERS OF DEEP LINEAR NETWORKS
+
+Ziwei Ji & Matus Telgarsky Department of Computer Science University of Ilinois at Urbana-Champaign {ziweiji2,mjt}@illinois.edu
+
+# ABSTRACT
+
+This paper establishes risk convergence and asymptotic weight matrix alignment —a form of implicit regularization — of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to O; (ii) the normalized $\bar { i } ^ { \mathrm { { t h } } }$ weight matrix asymptotically equals its rank-1 approximation $u _ { i } v _ { i } ^ { \top }$ ； (ii) these rank-1 matrices are aligned across layers, meaning $| v _ { i + 1 } ^ { \top } u _ { i } |  1$ . In the case of the logistic loss (binary cross entropy), more can be said: the linear function induced by the network — the product of its weight matrices — converges to the same direction as the maximum margin solution. This last property was identified in prior work, but only under assumptions on gradient descent which here are implied by the alignment phenomenon.
+
+# 1INTRODUCTION
+
+Efforts to explain the effectiveness of gradient descent in deep learning have uncovered an exciting possibility: it not only finds solutions with low error, but also biases the search for low complexity solutions which generalize well(Zhang et al.,2O17; Bartlett et al., 2O17; Soudry et al., 2017; Gunasekar et al., 2018).
+
+This paper analyzes the implicit regularization of gradient descent and gradient flow on deep linear networks and linearly separable data. For strictly decreasing losses,the optimum is at infinity,and we establish various alignment phenomena:
+
+· For each weight matrix $W _ { i }$ , the corresponding normalized weight matrix $W _ { i } / | | W _ { i } | | _ { F }$ asymptotically equals its rank-1 approximation $u _ { i } v _ { i } ^ { \top }$ ,where the Frobenius norm $\| W _ { i } \| _ { F }$ satisfies $\| \boldsymbol { W } _ { i } \| _ { F }  \infty$ .In other words, $\| W _ { i } \| _ { 2 } / \| W _ { i } \| _ { F } ~ \to ~ 1$ ，and asymptotically only the rank-1 approximation of $W _ { i }$ contributes to the final predictor,a form of implicit regularization. · Adjacent rank-1 weight matrix approximations are aligned: $| v _ { i + 1 } ^ { \top } u _ { i } |  1$ For the logistic loss,the first right singular vector $v _ { 1 }$ of $W _ { 1 }$ is aligned with the data, meaning $v _ { 1 }$ converges to the unique maximum margin predictor $\bar { u }$ defined by the data.Moreover, the linear predictor induced by the network, $w _ { \mathrm { p r o d } } : = W _ { L } \cdot \cdot \cdot W _ { 1 }$ ,is also aligned with the data, meaning $w _ { \mathrm { p r o d } } / \| w _ { \mathrm { p r o d } } \|  \bar { u }$ ：
+
+Simultaneously, this work proves that the risk is globally optimized: it asymptotes to 0. Alignment and risk convergence are proved simultaneously; the phenomena are coupled within the proofs.
+
+Since the layers align, they can be viewed as a minimum norm solution: they do not“waste norm” on components which are killed off when the layers are multiplied together. Said another way, given data $( ( x _ { i } , y _ { i } ) ) _ { i = 1 } ^ { n }$ , the normalized matrices $\left( W _ { 1 } \middle / \Vert W _ { 1 } \Vert _ { F } , \dotsc , W _ { L } \middle / \Vert W _ { L } \Vert _ { F } \right)$ asymptotically solve a maximum margin problem which demands all weight matrices be small, not merely their product:
+
+$$
+\operatorname* { m a x } _ { \substack { W _ { L } \in \mathbb { R } ^ { 1 \times d _ { L - 1 } } } } \cdots \operatorname* { m a x } _ { \substack { W _ { 1 } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } } } \operatorname* { m i n } _ { i } y _ { i } ( W _ { L } \cdot \cdot \cdot W _ { 1 } ) x _ { i } .
+$$
+
+![](images/367f51f4bcdd867972bf66665fd502e617af7dd1438fcd9089801b8e2c2755b9.jpg)  
+Figure 1: Visualization of margin maximization and self-regularization of layers on synthetic data with a 4-layer linear network compared to a 1-layer network (a linear predictor). Figure la shows the convergence of 1-layer and 4-layer networks to the same margin-maximizing linear predictor on positive (blue) and negative (red) separable data. Figure 1b shows the convergence of $\| W _ { i } \| _ { 2 } / \| W _ { i } \| _ { F }$ to 1 on each layer, plotted against the risk.
+
+The paper is organized as follows.This introduction continues with related work,notation,and assumptions in Sections 1.1 and 1.2. The analysis of gradient flow is in Section 2,and gradient descent is analyzed in Section 3. The paper closes with future directions in Section 4; a particular highlight is a preliminary experiment on CIFAR-1O which establishes empirically that a form of the alignment phenomenon occurs on the standard nonlinear network AlexNet.
+
+# 1.1RELATED WORK
+
+On the implicit regularization of gradient descent, Soudry et al. (2O17) show that for linear predictors and linearly separable data, the gradient descent iterates converge to the same direction as the maximum margin solution. Ji & Telgarsky (2O18) further characterize such an implicit bias for general nonseparable data. Gunasekar et al. (2O18) consider gradient descent on fully connected linear networks and linear convolutional networks. In particular, for the exponential loss,assuming the risk is minimized to O and the gradients converge in direction,they show that the whole network converges in direction to the maximum margin solution. These two assumptions are on the gradient descent process itself,and specifically the second one might be hard to interpret and justify. Compared with Gunasekar et al. (2O18), this paper proves that the risk converges to O and the weight matrices align; moreover the proof here proves the properties simultaneously, rather than assuming one and deriving the other. Lastly,Arora et al. (2O18) show for deep linear networks (and later Du et al. (2018) for ReLU networks) that gradient flow does not change the difference between squared Frobenius norms of any two layers. We use a few of these tools in our proofs; please see Sections 2 and 3 for details.
+
+For a smooth (nonconvex) function,Lee et al. (2O16) show that any strict saddle can be avoided almost surely with small step sizes.If there are only countably many saddle points and they are all strict,and if gradient descent iterates converge, then this implies (almost surely) they converge to a local minimum. In the present work, since there is no finite local minimum, gradient descent will go to infinity and never converge,and thus these results of Lee et al. (2Ol6)do not show that the risk converges to 0.
+
+There has been a rich literature on linear networks. Saxe et al. (2O13) analyze the learning dynamics of deep linear networks,showing that they exhibit some learning paterns similar to nonlinear networks,such as a long plateau followed by a rapid risk drop. Arora et al. (2O18) show that depth can help accelerate optimization. On the landscape properties of deep linear networks,Lu & Kawaguchi (2017); Laurent & von Brecht (2O17) show that under various structural assumptions, all local optima are global. Zhou & Liang (2O18) give a necessary and sufficient characterization of critical points for deep linear networks.
+
+![](images/83729ba78f720ba3735072c4b0d203f2e2227bee8c612ce82670bf8073388b0a.jpg)  
+Figure 2: A visualization of inter-layer alignment on data consisting of two well-separated circles with a 3-layer linear network. Figure 2a depicts,as in Figure la, that optimizing $1 \mathrm { - }$ and 3-layer linear networks finds the same maximum margin solution.The other three plots show the data as it is mapped through progressively more and more layers. Due to alignment, the product $W _ { i } \cdots W _ { 1 }$ becomes $u _ { i } \bar { u } ^ { \top }$ ，where $u _ { i }$ is the top left singular vector of $W _ { i }$ , which means that asymptotically the mapped data wil be well separated and lie along the span of $u _ { i }$ ， as depicted by the flattening in Figures 2b to 2d. Additionally, these three subfigures show that the top right singular vector $v _ { i + 1 }$ of the subsequent layer is aligned with this $u _ { i }$ ,which in these plots (with principal component axes) corresponds to following a horizontal line.
+
+# 1.2NOTATION,SETTING,AND ASSUMPTIONS
+
+Consider a data set $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ ， where $x _ { i } \in \mathbb { R } ^ { d }$ ， $\| x _ { i } \| \leq 1$ ,and $y _ { i } \in \{ - 1 , + 1 \}$ . The data set is assumed to be linearly separable,i.e.,there exists a unit vector $u$ which correctly classifies every data point: for any $1 \leq i \leq n$ $y _ { i } \langle u , x _ { i } \rangle > 0$ .Furthermore, let $\begin{array} { r } { \gamma : = \operatorname* { m a x } _ { \| u \| = 1 } \operatorname* { m i n } _ { 1 \leq i \leq n } y _ { i } \langle u , x _ { i } \rangle > 0 } \end{array}$ denote the maximum margin, and $\begin{array} { r } { \bar { u } : = \mathrm { \ a r g { m a x } } _ { \| u \| = 1 } \operatorname* { m i n } _ { 1 \leq i \leq n } { y _ { i } \langle u , x _ { i } \rangle } } \end{array}$ denote the maximum margin solution (the solution to the hard-margin SVM).
+
+A linear network of depth $L$ is parameterized by weight matrices $W _ { L } , \ldots , W _ { 1 }$ ，where $W _ { k } \in \mathsf { \Gamma }$ $\mathbb { R } ^ { d _ { k } \times d _ { k - 1 } }$ ， $d _ { 0 } = d$ ,and $d _ { L } = 1$ Let $W = ( W _ { L } , \dots , W _ { 1 } )$ denote all parameters of the network. The (empirical) risk induced by the network is given by
+
+$$
+{ \mathcal R } ( W ) = { \mathcal R } \left( W _ { L } , \dots , W _ { 1 } \right) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell \left( y _ { i } W _ { L } \cdots W _ { 1 } x _ { i } \right) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) ,
+$$
+
+where $w _ { \mathrm { p r o d } } : = ( W _ { L } \cdot \cdot \cdot W _ { 1 } ) ^ { \top }$ ,and $z _ { i } : = y _ { i } x _ { i }$
+
+The loss $\ell$ is assumed to be continuously differentiable,unbounded, and strictly decreasing to 0. Examples include the exponential loss $\dot { \ell _ { \mathrm { e x p } } ( x ) } = e ^ { - x }$ and the logistic loss $\ell _ { \log } ( x ) = \ln \left( 1 + \bar { e } ^ { - x } \right)$ ：
+
+Assumption 1. $\ell ^ { \prime } < 0$ is continuous, $\begin{array} { r } { \operatorname* { l i m } _ { x  - \infty } \ell ( x ) = \infty } \end{array}$ and $\scriptstyle \operatorname* { l i m } _ { x \to \infty } \ell ( x ) = 0$
+
+This paper considers gradient flow and gradient descent, where gradient flow $\{ W ( t ) | t \geq 0 , t \in \mathbb { R } \}$ can be interpreted as gradient descent with infinitesimal step sizes. It starts from some $W ( 0 )$ at
+
+$t = 0$ , and proceeds as
+
+$$
+\frac { \mathrm { d } W ( t ) } { \mathrm { d } t } = - \nabla \mathcal { R } \left( W ( t ) \right) .
+$$
+
+By contrast, gradient descent $\{ W ( t ) | t \geq 0 , t \in \mathbb { Z } \}$ is a discrete-time process given by
+
+$$
+\begin{array} { r } { W ( t + 1 ) = W ( t ) - \eta _ { t } \nabla \mathcal { R } \left( W ( t ) \right) , } \end{array}
+$$
+
+where $\eta _ { t }$ is the step size at time $t$
+
+We assume that the initialization of the network is not a critical point and induces a risk no larger than the risk of the trivial linear predictor O.
+
+Assumption 2. The initialization $W ( 0 )$ satisfies $\nabla \mathcal { R } \left( W ( 0 ) \right) \neq 0$ and $\begin{array} { r } { \mathcal { R } \left( W ( 0 ) \right) \leq \mathcal { R } ( 0 ) = \ell ( 0 ) . } \end{array}$
+
+It is natural to require that the initialization is not a critical point, since otherwise gradient flow/descent will never make a progress. The requirement $\mathcal { R } \left( \hat { W ( 0 ) } \right) \leq \mathcal { R } ( 0 )$ can be easily satisfied, for example,by making $W _ { 1 } ( 0 ) = 0$ and $W _ { L } ( 0 ) \cdot \cdot \cdot W _ { 2 } ( 0 ) \not = 0$ .On the other hand,if $\mathcal { R } \left( W ( 0 ) \right) > \mathcal { R } ( 0 )$ , gradient flow/descent may never minimize the risk to O. Proofs of those claims are given in Appendix A.
+
+# 2RESULTS FOR GRADIENT FLOW
+
+In this section,we consider gradient flow. Although impractical when compared with gradient descent, gradient flow can simplify the analysis and highlight proof ideas. For convenience,we usually use $W$ $W _ { k }$ ,and $w _ { \mathrm { p r o d } }$ ,but they all change with (the continuous time) $t$ . Only proof sketches are given here; detailed proofs are deferred to Appendix B.
+
+# 2.1RISK CONVERGENCE
+
+One key property of gradient flow is that it never increases the risk:
+
+$$
+\frac { \mathrm { d } \mathcal { R } ( W ) } { \mathrm { d } t } = \left. \nabla \mathcal { R } ( W ) , \frac { \mathrm { d } W } { \mathrm { d } t } \right. = - \| \nabla \mathcal { R } ( W ) \| ^ { 2 } = - \sum _ { k = 1 } ^ { L } \biggl \| \frac { \partial \mathcal { R } } { \partial W _ { k } } \biggr \| _ { F } ^ { 2 } \leq 0 .
+$$
+
+We now state the main result: under Assumptions 1 and 2, gradient flow minimizes the risk, $W _ { k }$ and $w _ { \mathrm { p r o d } }$ all go to infinity,and the alignment phenomenon occurs.
+
+Theorem 1. Under Assumptions 1 and 2, gradient flow iterates satisfy the following properties:
+
+$$
+\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \mathcal { R } ( W ) = 0 . } \end{array}
+$$
+
+· For any $1 \le k \le L$ $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| W _ { k } \| _ { F } = \infty . } \end{array}$
+
+· For any $1 \leq k \leq L$ letting $( u _ { k } , v _ { k } )$ denote the first left and right singular vectors of $W _ { k }$ ，
+
+$$
+\operatorname* { l i m } _ { t \to \infty } \left\| \frac { W _ { k } } { \| W _ { k } \| _ { F } } - u _ { k } v _ { k } ^ { \top } \right\| _ { F } = 0 .
+$$
+
+Moreover, for any $\begin{array} { r } { 1 \leq k < L , \operatorname* { l i m } _ { t  \infty } | \langle v _ { k + 1 } , u _ { k } \rangle | = 1 } \end{array}$ As a result,
+
+$$
+\operatorname* { l i m } _ { t \to \infty } \left. \left. \frac { w _ { \mathrm { p r o d } } } { \prod _ { k = 1 } ^ { L } \| W _ { k } \| _ { F } } , v _ { 1 } \right. \right. = 1 ,
+$$
+
+and thus $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| w _ { \mathrm { p r o d } } \| = \infty } \end{array}$
+
+Theorem 1 is proved using two lemmas,which may be of independent interest. To show the ideas, let us first introduce a little more notation. Recall that $\mathcal { R } ( W )$ denotes the empirical risk induced by the deep linear network $W$ . Abusing the notation a little, for any linear predictor $w \in \mathbb { R } ^ { d }$ , we also use $\mathcal { R } ( w )$ to denote the risk induced by $w$ . With this notation, $\begin{array} { r } { \dot { \mathcal { R } } ( W ) = \mathbf { \bar { \mathcal { R } } } ( w _ { \mathrm { p r o d } } ) } \end{array}$ ,while
+
+$$
+\nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) z _ { i } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( W _ { L } \cdot \cdot \cdot W _ { 1 } z _ { i } \right) z _ { i }
+$$
+
+is in $\mathbb { R } ^ { d }$ and different from $\nabla \mathcal { R } ( W )$ , which has $\scriptstyle \sum _ { k = 1 } ^ { L } d _ { k } d _ { k - 1 }$ entries, as given below:
+
+$$
+\frac { \partial \mathcal { R } } { \partial W _ { k } } = W _ { k + 1 } ^ { \top } \cdot \cdot \cdot W _ { L } ^ { \top } \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) ^ { \top } W _ { 1 } ^ { \top } \cdot \cdot \cdot W _ { k - 1 } ^ { \top } .
+$$
+
+Furthermore, for any $R > 0$ ,let
+
+$$
+B ( R ) = \left\{ W \bigg | \operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq R \right\} .
+$$
+
+The first lemma shows that for any $R > 0$ , the time spent by gradient flow in $B ( R )$ is finite.
+
+Lemma 1. Under Assumption $^ { l }$ and 2, for any $R > 0$ ,there exists a constant $\epsilon ( R ) > 0$ ，such that for any $t \geq 1$ and any $W \in B ( R )$ ， $\| \partial \mathcal { R } / \partial W _ { 1 } \| _ { F } \geq \epsilon ( R )$ 、As a result, gradient flow spends a finite amount of time in $B ( R )$ for any $R > 0$ ,and $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F }$ is unbounded.
+
+Here is the proof sketch. If $\| W _ { k } \| _ { F }$ are bounded, then $\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \|$ will be lower bounded by a positive constant, therefore if $\| \partial \mathcal { R } / \partial W _ { 1 } \| _ { F } = \| W _ { L } \cdot \cdot \cdot W _ { 2 } \| \| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \|$ can be arbitrarily small, then $\| W _ { L } \cdot \cdot \cdot W _ { 2 } \|$ and $\lVert w _ { \mathrm { p r o d } } \rVert$ can also be arbitrarily small, and thus $\mathcal { R } ( W )$ can be arbitrarily close to $\mathcal { R } ( 0 )$ . This cannot happen after $t = 1$ ,otherwise it will contradict Assumption 2 and eq. (1).
+
+To proceed, we need the folowing properties of linear networks from prior work (Arora et al., 2018; Du et al.,2018). For any time $t \geq 0$ and any $1 \leq k < L$
+
+$$
+\begin{array} { r } { W _ { k + 1 } ^ { \top } ( t ) W _ { k + 1 } ( t ) - W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) = W _ { k } ( t ) W _ { k } ^ { \top } ( t ) - W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) . } \end{array}
+$$
+
+To see this, just notice that
+
+$$
+\boldsymbol { W } _ { k + 1 } ^ { \top } \frac { \partial \boldsymbol { \mathcal { R } } } { \partial \boldsymbol { W } _ { k + 1 } } = \boldsymbol { W } _ { k + 1 } ^ { \top } \cdot \cdot \cdot \boldsymbol { W } _ { L } ^ { \top } \nabla \boldsymbol { \mathcal { R } } ( \boldsymbol { w } _ { \mathrm { p r o d } } ) ^ { \top } \boldsymbol { W } _ { 1 } ^ { \top } \cdot \cdot \cdot \boldsymbol { W } _ { k } ^ { \top } = \frac { \partial \boldsymbol { \mathcal { R } } } { \partial \boldsymbol { W } _ { k } } \boldsymbol { W } _ { k } ^ { \top } .
+$$
+
+Taking the trace on both sides of eq. (2), we have
+
+$$
+\left\| W _ { k + 1 } ( t ) \right\| _ { F } ^ { 2 } - \left\| W _ { k + 1 } ( 0 ) \right\| _ { F } ^ { 2 } = \left\| W _ { k } ( t ) \right\| _ { F } ^ { 2 } - \left\| W _ { k } ( 0 ) \right\| _ { F } ^ { 2 } .
+$$
+
+In other words, the diference between the squares of Frobenius norms of any two layers remains a constant. Together with Lemma 1,it implies that all $\| W _ { k } \| _ { F }$ are unbounded.
+
+However, even if $\| W _ { k } \| _ { F }$ are large, it does not follow necessarily that $\lVert w _ { \mathrm { p r o d } } \rVert$ is also large. Lemma 2 shows that this is indeed true: for gradient flow, as $\| W _ { k } \| _ { F }$ get larger,adjacent layers also get more aligned to each other, which ensures that their product also has a large norm.
+
+For $1 \le k \le L$ ,let $\sigma _ { k } , u _ { k }$ ,and $v _ { k }$ denote the first singular value (the 2-norm), the first left singular vector, and the first right singular vector of $W _ { k }$ ,respectively. Furthermore, define
+
+$$
+D : = \bigg ( \operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } ( 0 ) \| _ { F } ^ { 2 } \bigg ) - \| W _ { L } ( 0 ) \| _ { F } ^ { 2 } + \sum _ { k = 1 } ^ { L - 1 } \Big \| W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) - W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) \Big \| _ { 2 } ,
+$$
+
+which depends onlyonthe initialization.Ifforany $\begin{array} { r } { 1 \leq k < L , W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) = W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) , } \end{array}$ then $D = 0$
+
+Lemma 2. The gradient flow iterates satisfy the following properties:
+
+$$
+\begin{array} { r } { \bullet \bullet \ s u p p o s e \ \operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F }  \infty , t h e n \Big | \Big \langle w _ { \mathrm { p r o d } } \big / \prod _ { k = 1 } ^ { L } \| W _ { k } \| _ { F } , v _ { 1 } \Big \rangle \Big |  1 . } \end{array}
+$$
+
+The profis based oneq.(2)andeq.(3).If $W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) = W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 )$ , then eq.(2) gives that $W _ { k + 1 }$ and $W _ { k }$ have the same singular values, and $W _ { k + 1 }$ 's right singular vectors and $W _ { k }$ 's left singular vectors are the same. If it is true for any two adjacent layers, since $W _ { L }$ is a row vector, all layers have rank 1.With general initialization,we have similar results when $\| W _ { k } \| _ { F }$ is large enough so that the initialization is negligible. Careful calculations give the exact results in Lemma 2.
+
+An interesting point is that the implicit regularization result in Lemma 2 helps establish risk convergence in Theorem 1. Specifically,by Lemma 2, if all layers have large norms, $\| W _ { L } \cdot \cdot \cdot W _ { 2 } \|$ will be large. If the risk is not minimized to $0$ ， $\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \|$ will be lower bounded by a positive constant, and thus $\| \partial \mathcal { R } / \partial W _ { 1 } \| _ { F } = \| W _ { L } \cdot \cdot \cdot W _ { 2 } \| \| \nabla \mathcal { R } ( \dot { w } _ { \mathrm { p r o d } } ) \|$ will be large. Invoking eq. (1),Lemma l and eq. (3)gives a contradiction. Since the risk has no finite optimum, $\| W _ { k } \| _ { F } \to \infty$
+
+Here we focus on the exponential loss $\ell _ { \mathrm { e x p } } ( x ) = e ^ { - x }$ and the logistic loss $\ell _ { \log } ( x ) = \ln ( 1 + e ^ { - x } )$ In addition to risk convergence, these two losses also enable gradient descent to find the maximum margin solution.
+
+To get such a strong convergence, we need one more assumption on the data set. Recall that $\gamma =$ $\begin{array} { r } { \operatorname* { m a x } _ { \| u \| = 1 } \operatorname* { m i n } _ { 1 \leq i \leq n } \langle u , z _ { i } \rangle > 0 } \end{array}$ denotes the maximum margin,and $\bar { u }$ denotes the unique maximum margin predictor which attains this margin $\gamma$ . Those data points $z _ { i }$ for which $\langle \bar { u } , z _ { i } \rangle = \gamma$ are called support vectors.
+
+Assumption 3. The support vectors span the whole space $\mathbb { R } ^ { d }$
+
+Assumption 3 appears in prior work Soudry et al. (2O17),and can be satisfied in many cases: for example, it is almost surely true if the number of support vectors is larger than or equal to $d$ and the data set is sampled from some density w.r.t. the Lebesgue measure. It can also be relaxed to the situation that the support vectors and the whole data set span the same space; in this case $\nabla \mathcal { R } ( w _ { \mathrm { p r o d } } )$ will never leave this space,and we can always restrict our attention to this space.
+
+With Assumption 3,we can state the main theorem.
+
+Theorem 2. Under Assumptions 2 and 3, for almost all data and for losses $\ell _ { \mathrm { e x p } }$ and $\ell _ { \mathrm { l o g } }$ then $\mathrm { l i m } _ { t  \infty } | \langle v _ { 1 } , \bar { u } \rangle | ~ = ~ 1$ ，where $v _ { 1 }$ is the first right singular vector of $W _ { 1 }$ As a result, $\begin{array} { r } { \operatorname* { l i m } _ { t \to \infty } w _ { \mathrm { p r o d } } \Big / \prod _ { k = 1 } ^ { L } \| W _ { k } \| _ { F } = \bar { u } . } \end{array}$
+
+Before summarizing the proof, we can simplify both theorems into the following minimum norm property mentioned in the introduction.
+
+Corollary 1. Under Assumptions 2 and 3,for almost alldata and for losses $\ell _ { \mathrm { e x p } }$ and $\ell _ { \mathrm { l o g } ; }$
+
+$$
+\operatorname* { m i n } _ { i } y _ { i } \left( \frac { W _ { L } } { \| W _ { L } \| _ { F } } \cdots \frac { W _ { 1 } } { \| W _ { 1 } \| _ { F } } \right) x _ { i } \ \quad \xrightarrow [ t \to \infty ] { } \quad \operatorname* { m a x } _ { \begin{array} { c } { A _ { L } \in \mathbb { R } ^ { 1 \times d _ { L - 1 } } } \cdots \operatorname* { m a x } _ { \begin{array} { c } { A _ { 1 } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } } \end{array} } \operatorname* { m i n } _ { i } y _ { i } \left( A _ { L } \cdots A _ { 1 } \right) x _ { i } . } \\ { \left\| A _ { L } \right\| _ { F } = 1 \ \qquad \left\| A _ { 1 } \right\| _ { F } = 1 } \end{array} 
+$$
+
+Theorem 2 relies on two structural lemmas.The first one is based on a similar almost-all argument due to Soudry et al. (2017, Lemma 12). Let $S \subset \{ 1 , \ldots , n \}$ denote the set of indices of support vectors.
+
+Lemma 3. Under Assumption $^ 3$ ， if the data set is sampled from some density w.r.t. the Lebesgue measure,then with probability 1,
+
+$$
+\alpha : = \operatorname* { m i n } _ { | \xi | = 1 , \xi \perp \bar { u } } \operatorname* { m a x } _ { i \in S } \langle \xi , z _ { i } \rangle > 0 .
+$$
+
+Let $\bar { u } ^ { \perp }$ denote the orthogonal complement of $\mathrm { s p a n } ( \bar { u } )$ ,and let $\Pi _ { \perp }$ denote the projection onto $\bar { u } ^ { \perp }$ We prove that if $\Vert \Pi _ { \perp } w \Vert$ is large enough, gradient flow starting from $w$ will tend to decrease $\Vert \Pi _ { \perp } w \Vert$ ： Lemma 4. Under Assumption 3, for almost all data, $\ell _ { \mathrm { e x p } }$ and $\ell _ { \mathrm { l o g } }$ ,and any $w \in \mathbb { R } ^ { d }$ ， $i f \left. w , \bar { u } \right. \geq 0$ and $\Vert \Pi _ { \perp } w \Vert$ is larger than $1 + \ln ( n ) / \alpha$ for $\ell _ { \mathrm { e x p } } o r ^ { 2 n } / e \alpha$ for $\ell _ { \mathrm { l o g } }$ then $\langle \Pi _ { \perp } w , \nabla \mathcal { R } ( w ) \rangle \geq 0$
+
+With Lemma 3 and Lemma 4 in hand, we can prove Theorem 2. Let $\Pi _ { \perp } W _ { 1 }$ denote the projection of rows of $W _ { 1 }$ onto $\bar { u } ^ { \perp }$ . Notice that
+
+$$
+\Pi _ { \perp } w _ { \mathrm { p r o d } } = \big ( W _ { L } \ldots W _ { 2 } ( \Pi _ { \perp } W _ { 1 } ) \big ) ^ { \top } \quad \mathrm { a n d } \quad \frac { \mathrm { d } \| \Pi _ { \perp } W _ { 1 } \| _ { F } ^ { 2 } } { \mathrm { d } t } = - 2 \langle \Pi _ { \perp } w _ { \mathrm { p r o d } } , \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \rangle .
+$$
+
+If $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F }$ is large compared with $\| W _ { 1 } \| _ { F }$ ， since layers become aligned, $\| \Pi _ { \perp } w _ { \mathrm { p r o d } } \|$ will also be large,and then Lemma 4 implies that $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F }$ will not increase. At the same time, $\| W _ { 1 } \| _ { F } \to \infty$ ， and thus for large enough $t$ ， $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F }$ must be very small compared with $\| W _ { 1 } \| _ { F }$ . Many details need to be handled to make this intuition exact; the proof is given in Appendix B.
+
+# 3RESULTS FOR GRADIENT DESCENT
+
+One key property of gradient flow which is used in the previous proofs is that it never increases the risk, which is not necessarily true for gradient descent. However, for smooth losses (i.e, with
+
+Lipschitz continuous derivatives),we can design some decaying step sizes, with which gradient descent never increases the risk, and basically the same results hold as in the gradient flow case. Deferred proofs are given in Appendix C.
+
+We make the following aditionalassumption on the loss which is satisfiedbythe logistic loss $\ell _ { \mathrm { l o g } }$ Assumption 4. $\ell ^ { \prime }$ is $\beta$ -Lipschitz (i.e, $\ell$ is $\beta$ -smooth), and $| \ell ^ { \prime } | \leq G$ (i.e., $\ell$ is $G$ -Lipschitz).
+
+Under Assumption 4, the risk is also a smooth function of $W$ , if all layers are bounded.
+
+Lemma 5. Suppose $\ell$ is $\beta$ -smooth. If $R \geq 1$ ,then $\beta ( R ) = 2 L ^ { 2 } R ^ { 2 L - 2 } ( \beta + G )$ and $\mathcal { R } ( W )$ is $a$ $\beta ( R )$ -smooth function on the set $B ( R ) = \big \{ W \big | \| W _ { k } \| _ { F } \leq R , 1 \leq k \leq L \big \}$ ：
+
+Smoothness ensures that for any $W , V ~ \in ~ B ( R )$ ， $\mathcal { R } ( W ) \mathrm { ~ - ~ } \mathcal { R } ( V ) \leq \langle \nabla \mathcal { R } ( V ) , W \mathrm { ~ - ~ } V \rangle \mathrm { ~ + ~ }$ $\beta ( R ) \| W { - } V \| ^ { 2 } \big / 2$ (see Bubeck et al. (2015) Lemma 3.4). In particular, if we choose some $R$ and set a constant step size $\eta _ { t } = 1 / \beta ( R )$ , then as long as $W ( t + 1 )$ and $W ( t )$ are both in $B ( R )$
+
+$$
+\begin{array} { r l } & { \mathcal { R } \left( W ( t + 1 ) \right) - \mathcal { R } \left( W ( t ) \right) \leq \left. \nabla \mathcal { R } \left( W ( t ) \right) , - \eta _ { t } \nabla \mathcal { R } \left( W ( t ) \right) \right. + \frac { \beta ( R ) \eta _ { t } ^ { 2 } } { 2 } \Big \Vert \nabla \mathcal { R } \left( W ( t ) \right) \Big \Vert ^ { 2 } } \\ & { \quad \quad \quad \quad \quad = - \frac { 1 } { 2 \beta ( R ) } \Big \Vert \nabla \mathcal { R } \left( W ( t ) \right) \Big \Vert ^ { 2 } = - \frac { \eta _ { t } } { 2 } \Big \Vert \nabla \mathcal { R } \left( W ( t ) \right) \Big \Vert ^ { 2 } . } \end{array}
+$$
+
+In other words,the risk does not increase at this step. However, similar to gradient flow, the gradient descent iterate will eventually escape $B ( R )$ ,which may increase the risk.
+
+Lemma 6. Under Assumption $^ { l }$ , 2 and 4, suppose gradient descent is run with a constant step size $1 / \beta ( R )$ . Then there exists a time $t$ when $W ( t ) \not \in B ( R )$ , in other words, $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } ( t ) \| _ { F } ^ { - } > R$
+
+Fortunately,this issue can be handled by adaptively increasing $R$ and correspondingly decreasing the step sizes,formalized in the following assumption.
+
+Assumption 5. The step size $\eta _ { t } = \operatorname* { m i n } \{ 1 / \beta ( R _ { t } ) , 1 \}$ , where $R _ { t }$ satisfies $W ( t ) \in B ( R _ { t } - 1 )$ ,and if$W ( t + \bar { 1 } ) \in B ( R _ { t } - 1 )$ ， $R _ { t + 1 } = R _ { t }$ ：
+
+Assumption 5 can be satisfied by a line search, which ensures that the gradient descent update is not too aggressive and the boundary $R$ is increased properly.
+
+With the additional Assumptions 4 and 5,exactly the same theorems can be proved for gradient descent. We restate them briefly here.
+
+Theorem 3. Under Assumption 1, 2, 4, and 5, gradient descent satisfies
+
+· $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \mathcal { R } ( W ( t ) ) = 0 . } \end{array}$   
+· For any $\begin{array} { r } { 1 \leq k \leq L , \operatorname* { l i m } _ { t  \infty } \| W _ { k } ( t ) \| _ { F } = \infty . } \end{array}$ $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \Big | \Big \langle w _ { \mathrm { p r o d } } ( t ) \big / \prod _ { k = 1 } ^ { L } \| W _ { k } ( t ) \| _ { F } , v _ { 1 } ( t ) \big \rangle \Big | = 1 } \end{array}$ where $v _ { 1 } ( t )$ isthe first right singuar vector of $W _ { 1 } ( t )$
+
+Theorem 4. Under Assumptions 2， 3 and 5, for the logistic loss $\ell _ { \mathrm { l o g } }$ and almost all data, $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \big \lvert \langle v _ { 1 } ( t ) , \bar { u } \rangle \big \rvert = 1 } \end{array}$ ,and $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } w _ { \mathrm { p r o d } } ( t ) \big / \prod _ { k = 1 } ^ { L } \| W _ { k } ( t ) \| _ { F } = \bar { u } . } \end{array}$
+
+Corollary 2. Under Assumptions 2, 3 and $5$ for the logistic loss $\ell _ { \mathrm { l o g } }$ and almost all data,
+
+$$
+\operatorname* { m i n } _ { i } \ y _ { i } \left( \frac { W _ { L } } { \| W _ { L } \| _ { F } } \cdots \frac { W _ { 1 } } { \| W _ { 1 } \| _ { F } } \right) x _ { i } \ \quad \xrightarrow [ t \to \infty ] { } \quad \operatorname* { m a x } _ { \begin{array} { c } { A _ { L } \in \mathbb { R } ^ { 1 \times d _ { L - 1 } } } \\ { \| A _ { L } \| _ { F } = 1 } \end{array} } \cdots \operatorname* { m a x } _ { \begin{array} { c } { A _ { 1 } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } } \\ { \| A _ { 1 } \| _ { F } = 1 } \end{array} } \operatorname* { m i n } _ { \begin{array} { c } { j _ { i } \left( A _ { L } \cdot \cdot \cdot A _ { 1 } \right) x _ { i } } \\ { i } \end{array} } .
+$$
+
+Proofs of Theorem 3 and 4 are given in Appendix C,and are basically the same as the gradient flow profsTheeydienceistrf $\begin{array} { r } { \sum _ { t = 0 } ^ { \infty } \eta _ { t } ^ { 2 } \| \nabla \mathcal { R } ( W ( t ) ) \| ^ { 2 } } \end{array}$ Wwil be introduced in many parts of the proof. However, it is bounded in light of eq. (4):
+
+$$
+\sum _ { t = 0 } ^ { \infty } \eta _ { t } ^ { 2 } \Big \| \nabla \mathcal { R } \left( W ( t ) \right) \Big \| ^ { 2 } \leq \sum _ { t = 0 } ^ { \infty } \eta _ { t } \Big \| \nabla \mathcal { R } \left( W ( t ) \right) \Big \| ^ { 2 } \leq 2 \mathcal { R } \left( W ( 0 ) \right) .
+$$
+
+Since all weight matrices go to infinity, such a bounded error does not mater asymptotically, and thus proofs still go through.
+
+![](images/26aaf95d1fd63dff5e198c20487648a4304b76728717c54c21123ed63c3bd719.jpg)  
+Figure 3: Risk and alignment of dense layers (the ratio $\| W _ { i } \| _ { 2 } / \| W _ { i } \| _ { F } )$ of (nonlinear!) AlexNet on CIFAR-10.Figure 3a uses default PyTorch initialization,while Figure 3b forces initial Frobenius norms to be equal amongst dense layers.
+
+# 4SUMMARY AND FUTURE DIRECTIONS
+
+This paper rigorously proves that, for deep linear networks on linearly separable data, gradient flow and gradient descent minimize the risk to O,align adjacent weight matrices,and align the first right singular vector of the first layer to the maximum margin solution determined by the data. There are many potential future directions;a few are as follows.
+
+Convergence rates and practical step sizes. This paper only proves asymptotic convergence, moreover for adaptive step sizes depending on the current weight matrix norms. A refined analysis with rates for practical step sizes (e.g. constant step sizes) would allow the algorithm to be compared to other methods which also globally optimize this objective, would suggest ways to improve step sizes and initialization,and ideally even exhibit a sensitivity to the network architecture and suggest how it could be improved.
+
+Nonseparable data and nonlinear networks. Real-world data is generally not linearly separable, but nonlinear deep networks can reliably decrease the risk to O,even with random labels (Zhang et al.,2017). This seems to suggest that a nonlinear notion of separability is at play; is there some way to adapt the present analysis?
+
+The present analysis is crucially tied to the alignment of weight matrices: alignment and risk are analyzed simultaneously. Motivated by this, consider a preliminary experiment, presented in Figure 3, where stochastic gradient descent was used to minimize the risk of a standard AlexNet on CIFAR-10 (Krizhevsky et al., 2012; Krizhevsky & Hinton, 2009).
+
+Even though there are ReLUs,max-pooling layers,and convolutional layers,the alignment phenomenon is occurring in a reduced form on the dense layers (the last three layers of the network). Specificall, despite these weight matrices having shape (1024,4096), (4096,4096),and $( 4 0 9 6 , 1 0 )$ the key alignment ratios $\| \bar { W } _ { i } \| _ { 2 } / \| W _ { i } \| _ { F }$ are much larger than their respective lower bounds $( \mathrm { { 1 0 2 4 ^ { - 1 / 2 } , 4 0 9 6 ^ { - 1 / 2 } , 1 0 ^ { - 1 / 2 } } } )$ Two initializations were tried: default PyTorch initialization,and a Gaussan initialization forcing all initial Frobenius norms to be just 4, which is suggested by the norm preservation property in the analysis and removes noise in the weights.
+
+# ACKNOWLEDGEMENTS
+
+The authors are grateful for support from the NSF under grant IIS-175OO51. This grant allowed them to focus on research,and when combined with an NVIDIA GPU grant, led to the creation of their beloved GPU machine DUTCHCRUNCH.
+
+# REFERENCES
+
+Sanjeev Arora,Nadav Cohen,and Elad Hazan. On the optimization of deep networks: Implicit acceleration by overparameterization. arXiv preprint arXiv:1802.06509,2018.   
+Peter Bartlett, Dylan Foster,and Matus Telgarsky. Spectrally-normalized margin bounds for neural networks.NIPS,2017.   
+Sebastien Bubeck et al. Convex optimization:_ Algorithms and complexity. Foundations and Trends $\textsuperscript { \textregistered }$ in Machine Learning, 8(3-4):231-357,2015.   
+Simon S Du, Wei Hu, and Jason D Lee. Algorithmic regularization in learning deep homogeneous models: Layers are automatically balanced. arXiv preprint arXiv:1806.00900,2018.   
+Suriya Gunasekar, Jason Lee, Daniel Soudry, and Nathan Srebro. Implicit bias of gradient descent on linear convolutional networks. arXiv preprint arXiv:1806.00468, 2018.   
+Ziwei Ji and Matus Telgarsky. Risk and parameter convergence of logistic regression. arXiv preprint arXiv:1803.07300,2018.   
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.   
+Alex Krizhevsky, Ilya Sutskever, and Geoffery Hinton. Imagenet classification with deep convolutional neural networks. In NIPS,2012.   
+Thomas Laurent and James von Brecht. Deep linear neural networks with arbitrary loss: All local minima are global. arXiv preprint arXiv:1712.01473,2017.   
+Jason D Lee, Max Simchowitz, Michael I Jordan, and Benjamin Recht. Gradient descent converges to minimizers. arXiv preprint arXiv:1602.04915,2016.   
+Haihao Lu and Kenji Kawaguchi.Depth creates no bad local minima.arXiv preprint arXiv:1702.08580,2017.   
+Andrew M Saxe, James L McClelland,and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. arXiv preprint arXiv:1312.6l20, 2013.   
+Daniel Soudry,Elad Hofer, and Nathan Srebro.The implicit bias of gradient descent on separable data. arXiv preprint arXiv:1710.10345v3,2017.   
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017.   
+Yi Zhou and Yingbin Liang. Critical points of linear neural networks: Analytical forms and landscape properties.2018.
+
+# AREGARDING ASSUMPTION 2
+
+Suppose $W _ { 1 } ( 0 ) = 0$ while $W _ { L } ( 0 ) \cdot \cdot \cdot W _ { 2 } ( 0 ) \neq 0$ First of all, $W _ { L } ( 0 ) \cdot \cdot \cdot W _ { 1 } ( 0 ) = 0$ and thus $\mathcal { R } \left( W ( 0 ) \right) = \mathcal { R } ( 0 )$ . Moreover,
+
+$$
+\left. \nabla \mathcal { R } \left( w _ { \mathrm { p r o d } } ( 0 ) \right) , \bar { u } \right. = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } ( 0 ) \langle z _ { i } , \bar { u } \rangle \leq \ell ^ { \prime } ( 0 ) \gamma < 0 ,
+$$
+
+which implies $\nabla \mathcal { R } \left( w _ { \mathrm { p r o d } } ( 0 ) \right) \neq 0$ and $\begin{array} { r } { \partial \mathcal { R } / \partial W _ { 1 } = \left( W _ { L } ( 0 ) \cdots W _ { 2 } ( 0 ) \right) ^ { \top } \nabla \mathcal { R } \left( w _ { \mathrm { p r o d } } ( 0 ) \right) ^ { \top } \neq 0 . } \end{array}$
+
+On the other hand, if $\mathcal { R } \left( W ( 0 ) \right) > \mathcal { R } ( 0 )$ ， gradient flow/descent may never minimize the risk to 0. For example,suppose the network has two layers, and both the input and output have dimension 1； the network just computes the dot product of two vectors $w _ { 1 }$ and $w _ { 2 }$ .Consider minimizing $\mathcal { R } ( w _ { 1 } , w _ { 2 } ) \stackrel { - } { = } \exp \left( - \langle w _ { 1 } , w _ { 2 } \rangle \right)$ .If $w _ { 1 } ( 0 ) = - w _ { 2 } ( 0 ) \neq 0$ ，then $\mathcal { R } \left( w _ { 1 } ( 0 ) , w _ { 2 } ( 0 ) \right) \ =$ $\exp \left( \Vert w _ { 1 } \Vert ^ { 2 } \right) > \exp ( 0 )$ . It is easy to verify that for any $t$ ， $w _ { 1 } ( t ) = - w _ { 2 } ( t )$ ,and $\mathcal { R } \left( w _ { 1 } ( t ) , w _ { 2 } ( t ) \right) \geq$ $\exp ( 0 ) > 0$ 、：
+
+# BOMITTED PROOFS FROM SECTION 2
+
+Proof of Lemma $^ { l }$ Fix an arbitrary $R > 0$ . If the claim is not true, then for any $\epsilon > 0$ , there exists some $t \geq 1$ such that $\| W _ { k } \| _ { F } \leq R$ for all $k$ while $\left\| \partial \mathcal { R } / \partial W _ { 1 } \right\| _ { F } ^ { 2 } \leq \epsilon ^ { 2 }$ , which means
+
+$$
+\left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } \right\| _ { F } ^ { 2 } = \left\| W _ { 2 } ^ { \top } \cdot \cdot \cdot W _ { L } ^ { \top } \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) ^ { \top } \right\| _ { F } ^ { 2 } = \left\| W _ { L } \cdot \cdot \cdot W _ { 2 } \right\| ^ { 2 } \left\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \right\| ^ { 2 } \leq \epsilon ^ { 2 } .
+$$
+
+Since $\| w _ { \mathrm { p r o d } } \| \leq R ^ { L }$ , we have
+
+$$
+\left. \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) , \bar { u } \right. = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) \left. z _ { i } , \bar { u } \right. \leq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) \gamma \leq - M \gamma ,
+$$
+
+where $- M = \mathrm { m a x } _ { - R ^ { L } \leq x \leq R ^ { L } } \ell ^ { \prime } ( x )$ Since $\ell ^ { \prime }$ is continuous and the domain is bounded, the maximum is attained and negative, and thus $M \ > \ 0$ .Therefore $\left\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \right\| \ \ge \ M \gamma$ ，and thus $\| W _ { L } \cdot \cdot \cdot W _ { 2 } \| \le \epsilon / M \gamma$ .Since $\| W _ { 1 } \| _ { F } \leq R$ ，we further have $\lVert w _ { \mathrm { p r o d } } \rVert \leq \epsilon R / M \gamma$ . In other words, after $t = 1 , \| w _ { \mathrm { p r o d } } \|$ may be arbitrarily small, which implies $\mathcal { R } \left( w _ { \mathrm { p r o d } } \right)$ can be arbitrarily close to $\mathcal { R } \left( 0 \right)$ ：
+
+On the other hand, by Assumption 2, $\mathrm { d } \mathscr { R } ( W ) / \mathrm { d } t = - \| \nabla \mathscr { R } ( W ) \| ^ { 2 } < 0$ at $t = 0$ . This implies that $\mathcal { R } \left( W ( 1 ) \right) < \mathcal { R } \left( W ( 0 ) \right)$ , and for any $t \geq 1$ $\mathcal { R } \left( W ( t ) \right) \leq \mathcal { R } \left( W ( 1 ) \right) < \mathcal { R } \left( W ( 0 ) \right) \leq \mathcal { R } ( 0 )$ , which is a contradiction.
+
+Since the risk is always positive, we have
+
+$$
+\begin{array} { r l } { \mathcal { R } \left( W ( 0 ) \right) \geq \displaystyle \int _ { t = 0 } ^ { \infty } \displaystyle \sum _ { k = 1 } ^ { L } \left\| \frac { \partial \mathcal { R } } { \partial W _ { k } } \right\| _ { F } ^ { 2 } \mathrm { d } t } & { } \\ { \geq \displaystyle \int _ { t = 0 } ^ { \infty } \left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } \right\| _ { F } ^ { 2 } \mathrm { d } t } \\ { \geq \displaystyle \int _ { t = 0 } ^ { \infty } \left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } \right\| _ { T } ^ { 2 } \displaystyle \mathbf { 1 } \left[ \frac { \operatorname* { m a x } } { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq R \right] \mathrm { d } t } & { } \\ { \geq \displaystyle \int _ { t = 1 } ^ { \infty } \left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } \right\| _ { T } ^ { 2 } \displaystyle \left[ \frac { \operatorname* { m a x } } { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq R \right] \mathrm { d } t } & { } \\ { \geq \epsilon ( R ) ^ { 2 } \displaystyle \int _ { t = 1 } ^ { \infty } \mathbb { 1 } \left[ \frac { \operatorname* { m a x } } { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq R \right] \mathrm { d } t , } & { } \end{array}
+$$
+
+which implies gradient flow only spends a fnite amount of time in $\left\{ W \middle | \operatorname* { m a x } _ { 1 \leq k \leq L } \Vert W _ { k } \Vert _ { F } \leq R \right\}$ This directly implies that $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F }$ is unbounded. □
+
+Proof of Lemma 2. The first claim is true for $k = L$ since $W _ { L }$ is a row vector. For any $1 \le k < L$ recall that Arora et al. (2018); Du et al. (2018) give the following relation:
+
+$$
+\begin{array} { r } { W _ { k + 1 } ^ { \top } ( t ) W _ { k + 1 } ( t ) - W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) = W _ { k } ( t ) W _ { k } ^ { \top } ( t ) - W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) . } \end{array}
+$$
+
+Let $A _ { k , k + 1 } = W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) - W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 )$ . By eq. (5)and the definition of singular vectors and singular values, we have
+
+$$
+\begin{array} { r l } & { \sigma _ { k } ^ { 2 } \geq v _ { k + 1 } ^ { \top } W _ { k } W _ { k } ^ { \top } v _ { k + 1 } } \\ & { \quad = v _ { k + 1 } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } v _ { k + 1 } + v _ { k + 1 } ^ { \top } A _ { k , k + 1 } v _ { k + 1 } } \\ & { \quad = \sigma _ { k + 1 } ^ { 2 } + v _ { k + 1 } ^ { \top } A _ { k , k + 1 } v _ { k + 1 } } \\ & { \quad \geq \sigma _ { k + 1 } ^ { 2 } - \| A _ { k , k + 1 } \| _ { 2 } . } \end{array}
+$$
+
+Moreover, by taking the trace on both sides of eq. (5), we have
+
+$$
+\begin{array} { r l } & { \| \boldsymbol { W } _ { k } \| _ { F } ^ { 2 } = \operatorname { t r } \Big ( \boldsymbol { W } _ { k } \boldsymbol { W } _ { k } ^ { \top } \Big ) = \operatorname { t r } \Big ( \boldsymbol { W } _ { k + 1 } ^ { \top } \boldsymbol { W } _ { k + 1 } \Big ) + \operatorname { t r } \Big ( \boldsymbol { W } _ { k } ( 0 ) \boldsymbol { W } _ { k } ^ { \top } ( 0 ) \Big ) - \operatorname { t r } \Big ( \boldsymbol { W } _ { k + 1 } ^ { \top } ( 0 ) \boldsymbol { W } _ { k + 1 } ( 0 ) \Big ) } \\ & { \qquad = \| \boldsymbol { W } _ { k + 1 } \| _ { F } ^ { 2 } + \| \boldsymbol { W } _ { k } ( 0 ) \| _ { F } ^ { 2 } - \| \boldsymbol { W } _ { k + 1 } ( 0 ) \| _ { F } ^ { 2 } . } \end{array}
+$$
+
+Summing eq. (6) and eq. (7) from $k$ to $L - 1$ , we get
+
+$$
+\| W _ { k } \| _ { F } ^ { 2 } - \| W _ { k } \| _ { 2 } ^ { 2 } \leq \| W _ { k } ( 0 ) \| _ { F } ^ { 2 } - \| W _ { L } ( 0 ) \| _ { F } ^ { 2 } + \sum _ { k ^ { \prime } = k } ^ { L - 1 } \| A _ { k ^ { \prime } , k ^ { \prime } + 1 } \| _ { 2 } \leq D .
+$$
+
+Next we prove that singular vectors get aligned. Consider $u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } u _ { k }$ . On one hand, similar to eq. (6), we can get that
+
+$$
+\begin{array} { r l } & { u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } u _ { k } = u _ { k } ^ { \top } W _ { k } W _ { k } ^ { \top } u _ { k } - u _ { k } ^ { \top } W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) u _ { k } + u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) u _ { k } } \\ & { \qquad \geq u _ { k } ^ { \top } W _ { k } W _ { k } ^ { \top } u _ { k } - u _ { k } ^ { \top } W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) u _ { k } } \\ & { \qquad \geq \sigma _ { k } ^ { 2 } - \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+On the other hand, it follows from the definition of singular vectors and eq. (8) that
+
+$$
+\begin{array} { r l } & { u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } u _ { k } = \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + u _ { k } ^ { \top } \left( W _ { k + 1 } ^ { \top } W _ { k + 1 } - v _ { k + 1 } \sigma _ { k + 1 } ^ { 2 } v _ { k + 1 } ^ { \top } \right) u _ { k } } \\ & { \qquad \leq \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + \| W _ { k + 1 } \| _ { F } ^ { 2 } - \| W _ { k + 1 } \| _ { 2 } ^ { 2 } } \\ & { \qquad \leq \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + D . } \end{array}
+$$
+
+Combining eq. (9) and eq. (10), we get
+
+$$
+\sigma _ { k } ^ { 2 } \le \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + D + \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } .
+$$
+
+Similar to eq. (9), we can get
+
+$$
+\begin{array} { r } { \sigma _ { k } ^ { 2 } \geq v _ { k + 1 } ^ { \top } W _ { k } W _ { k } ^ { \top } v _ { k + 1 } \geq \sigma _ { k + 1 } ^ { 2 } - \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Therefore
+
+$$
+\frac { \sigma _ { k } ^ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } \geq 1 - \frac { \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } .
+$$
+
+Combining eq. (11) and eq. (12), we finally get
+
+$$
+\langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \geq 1 - \frac { D + \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } + \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } .
+$$
+
+Regarding the last claim, first recall that since the difference between the squares of Frobenius norms of any two layers is a constant, $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } \to \infty$ implies $\| W _ { k } \| _ { F } \to \infty$ for any $k$ ： We further have the following.
+
+· Since $\| W _ { k } \| _ { 2 } \to \infty$ $| \langle u _ { k } , v _ { k + 1 } \rangle |  1$
+
+As a result,
+
+$$
+\begin{array} { r l r } & { } & {  \frac { w _ { \mathrm { p r o d } } } { \prod _ { k = 1 } ^ { L }  W _ { k }  _ { F } } , v _ { 1 }   =   \displaystyle \prod _ { k = 1 } ^ { L } \frac { W _ { k } } {  W _ { k }  _ { F } } , v _ { 1 }  } \\ & { } & {    \displaystyle \prod _ { k = 1 } ^ { L } u _ { i } v _ { i } ^ { \top } , v _ { 1 }   } \\ & { } & {  1 . } \end{array}
+$$
+
+Proof of Theorem $^ { l }$ . Suppose for some $\epsilon > 0$ ， $\mathcal { R } \left( W \right) \geq \epsilon$ for any $t$ Then there exists some $1 \leq j \leq n$ such that $\ell \left( \langle w _ { \mathrm { p r o d } } , z _ { j } \rangle \right) \geq \epsilon$ ， and thus $\langle w _ { \mathrm { p r o d } } , z _ { j } \rangle \leq \ell ^ { - 1 } ( \epsilon )$ .On the other hand, since $\mathcal { R } ( W ) \leq \mathcal { R } ( 0 ) = \ell ( 0 )$ ， $\ell \left( \langle w _ { \mathrm { p r o d } } , z _ { j } \rangle \right) \leq n \ell ( 0 )$ ,and thus $\langle w _ { \mathrm { p r o d } } , z _ { j } \rangle \geq \ell ^ { - 1 } \left( n \ell ( 0 ) \right)$ . Let $\begin{array} { r } { - M = \operatorname* { m a x } _ { \ell ^ { - 1 } \left( n \ell ( 0 ) \right) \leq x \leq \ell ^ { - 1 } \left( \epsilon / n \right) } \ell ^ { \prime } ( x ) < 0 } \end{array}$ , we have for any $t$ ，
+
+$$
+\begin{array} { r l } & { \langle \nabla { \mathcal R } ( w _ { \mathrm { p r o d } } ) , { \bar { u } } \rangle = \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) \left. z _ { i } , { \bar { u } } \right. } \\ & { \quad \quad \quad \quad \quad \quad \quad \leq \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { i } \right. \right) \gamma } \\ & { \quad \quad \quad \quad \quad \quad \leq \displaystyle \frac { 1 } { n } \ell ^ { \prime } \left( \left. w _ { \mathrm { p r o d } } , z _ { j } \right. \right) \gamma } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \leq \displaystyle \frac { - M \gamma } { n } < 0 , } \end{array}
+$$
+
+and thus $\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \| \ge M \gamma / n$
+
+Similar to the proof of Lemma 2, we can show that if $\| W _ { k } \| _ { F } \to \infty$ ，
+
+$$
+|  \frac { ( W _ { L } \cdot \cdot \cdot W _ { 2 } ) ^ { \top } } { \| W _ { k } \| _ { F } \cdot \cdot \cdot \| W _ { 2 } \| _ { F } } , v _ { 2 }  |  1 .
+$$
+
+In other words, there exists some $C > 0$ ， such that when $\begin{array} { r } { \operatorname* { m i n } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } > C , \| W _ { L } \cdot \cdot \cdot W _ { 2 } \| \geq } \end{array}$ $\| W _ { k } \| _ { F } \cdot \cdot \cdot \| W _ { 2 } \| _ { F } / 2 > C ^ { L } / 2$
+
+Lemma 1 shows that gradient flow spends a finite amount of time in $\left\{ W \big | \operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq R \right\}$ for any $R > 0$ .Since the difference between the squares of Frobenius norms of any two layers is a constant, gradient flow also spends a finite amount of time in $\left\{ W \big | \operatorname* { m i n } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } \leq \dot { C } \right\}$ . Now we have
+
+$$
+\begin{array} { r l } { \mathcal { R } ( W ( 0 ) ) \geq \displaystyle \int _ { t = 0 } ^ { \infty } \displaystyle \sum _ { k = 1 } ^ { L } \| \frac { \partial \mathcal { R } } { \partial W _ { k } } \| _ { \Gamma } ^ { 2 } \mathrm { d } t } \\ { \geq \displaystyle \int _ { t = 0 } ^ { \infty } \| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } \| _ { \Gamma } ^ { 2 } \mathrm { d } t } \\ { } & { = \displaystyle \int _ { t = 0 } ^ { \infty } \| W _ { L } \cdots W _ { 2 } \| ^ { 2 } \| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \| ^ { 2 } \mathrm { d } t } \\ { } & { \geq \displaystyle \int _ { t = 0 } ^ { \infty } \| W _ { L } \cdots W _ { 2 } \| ^ { 2 } \| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \| ^ { 2 } [ W \displaystyle | \operatorname* { m i n } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } > C ] \mathrm { d } t } \\ { } & { \geq ( \frac { \mathcal { M } \gamma } { n } ) ^ { 2 } ( \frac { C ^ { L } } { 2 } ) ^ { 2 } \displaystyle \int _ { t = 0 } ^ { \infty } \Im [ W \displaystyle | \operatorname* { m i n } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F } > C ] \mathrm { d } t } \\ { } & { = \displaystyle \infty , } \end{array}
+$$
+
+which is a contradiction. Therefore $\mathcal { R } ( \epsilon )  0$ . This further implies $\| W _ { k } \| _ { F } \to \infty$ ,since $\mathcal { R } ( W )$ has no finite optimum. Finally, invoking Lemma 2 proves the final claim of Theorem 1. □
+
+Proof of Lemma 3. Soudry et al. (2017) Lemma 12 proves that, with probability 1,there are at most $d$ support vectors,and moreover, the $i$ -th support vector $z _ { i }$ has a positive dual variable $\alpha _ { i }$ ,such that $\textstyle \sum _ { i \in S } \alpha _ { i } z _ { i } = { \bar { u } }$ ，
+
+Suppose there exists some $\xi \perp \bar { u }$ , such that $\operatorname* { m a x } _ { i \in S } \langle \xi , z _ { i } \rangle \leq 0$ Since
+
+$$
+\sum _ { i \in S } \alpha _ { i } \langle \xi , z _ { i } \rangle = \left. \xi , \sum _ { i \in S } \alpha _ { i } z _ { i } \right. = \langle \xi , \bar { u } \rangle = 0 ,
+$$
+
+we actually have $\langle \xi , z _ { i } \rangle = 0$ for all $i \in S$ . This is impossible under Assumption 3, since the support vectors span the whole space. □
+
+Proof of Lemma 4. For the sake of presentation, we leave out the subscript in $z _ { i }$ and denote a data point by $z$ generally. For any data point $z$ and predictor $w$ ,let $z _ { \bot }$ and $w _ { \perp }$ denote their projection onto $\bar { u } ^ { \perp }$ . Let $z ^ { \prime } \in \arg \operatorname* { m a x } _ { i \in S } \langle - w _ { \bot } , z \rangle$ , and thus $\langle - w _ { \perp } , z ^ { \prime } \rangle \geq \alpha \Vert w _ { \perp } \Vert$ ：
+
+For $\ell _ { \mathrm { e x p } }$ , we have
+
+$$
+\begin{array} { r l } & { \langle w _ { \perp } , \nabla { \mathcal R } ( w ) \rangle = \displaystyle \sum _ { z } \frac { 1 } { n } \left[ - \exp \left( - \langle w , z \rangle \right) \right] \langle w _ { \perp } , z _ { \perp } \rangle } \\ & { \quad \quad \quad = \displaystyle \sum _ { z } \frac { 1 } { n } \left[ \exp \left( \langle - w , z \rangle \right) \right] \langle - w _ { \perp } , z _ { \perp } \rangle } \\ & { \quad \quad \quad \geq \displaystyle \frac { 1 } { n } \exp \left( \langle - w , z ^ { \prime } \rangle \right) \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle + \displaystyle \sum _ { \langle z _ { \perp } , w _ { \perp } \rangle \geq 0 } \frac { 1 } { n } \exp \left( \langle - w , z \rangle \right) \langle - w _ { \perp } , z _ { \perp } \rangle . } \end{array}
+$$
+
+The first part can be lower bounded as below (recall that $\langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle = \langle - w _ { \perp } , z ^ { \prime } \rangle \geq \alpha \| w _ { \perp } \| )$
+
+$$
+\begin{array} { r l } & { \frac { 1 } { n } \exp \left( \left. - w , z ^ { \prime } \right. \right) \left. - w _ { \perp } , z _ { \perp } ^ { \prime } \right. = \displaystyle \frac { 1 } { n } \exp \left( \left. - w , \gamma \bar { u } \right. \right) \exp \left( \left. - w _ { \perp } , z _ { \perp } ^ { \prime } \right. \right) \left. - w _ { \perp } , z _ { \perp } ^ { \prime } \right. } \\ & { \phantom { \frac { 1 } { n } } \geq \displaystyle \frac { 1 } { n } \exp \left( - \left. w , \gamma \bar { u } \right. \right) \exp \left( \alpha \| w _ { \perp } \| \right) \alpha \| w _ { \perp } \| . } \end{array}
+$$
+
+To bound the second part,frst notice that since we assume $\langle w , \bar { u } \rangle \geq 0$ ,for any $z$ ，
+
+$$
+\langle w , z - \gamma \bar { u } \rangle = \langle w , z _ { \perp } \rangle + \langle w , z - \gamma \bar { u } - z _ { \perp } \rangle \ge \langle w , z _ { \perp } \rangle = \langle w _ { \perp } , z _ { \perp } \rangle .
+$$
+
+The reason is that every data point has margin at least $\gamma$ ,and thus $z - \gamma \bar { u } - z _ { \bot } = c \bar { u }$ for some $c \geq 0$ Using eq. (15), we can bound the second part of eq. (13).
+
+$$
+\begin{array} { r l } & { \underset { ( \varepsilon _ { \perp } , w , \lambda ) \geq 0 } { \sum } \frac { 1 } { n } \exp ( ( - w , z ) ) \langle - w _ { \perp } , z _ { \perp } \rangle } \\ & { = \underset { ( \varepsilon _ { \perp } , w , \lambda ) \geq 0 } { \sum } \frac { 1 } { n } \exp ( ( - w , \gamma \tilde { u } ) ) \exp ( ( - w , z - \gamma \tilde { u } ) ) \langle - w _ { \perp } , z _ { \perp } \rangle } \\ & { \underset { ( \varepsilon _ { \perp } , w , \lambda ) \geq 0 } { \geq } \frac { 1 } { n } \exp ( ( - w , \gamma \tilde { u } ) ) \exp ( ( - w _ { \perp } , z _ { \perp } ) \big \langle - w _ { \perp } , z _ { \perp } \big \rangle  } \\ & {  \sum _ { ( \varepsilon _ { \perp } , w , \lambda ) \geq 0 } \frac { 1 } { n } \exp ( ( - w , \gamma \tilde { u } ) ) ( - \frac { 1 } { e } ) } \\ & { \underset { ( \varepsilon _ { \perp } , w , \lambda ) \geq 0 } { \geq } \frac { 1 } { n } \exp ( ( - w , \gamma \tilde { u } ) ) ( - \frac { 1 } { e } ) } \\ & { \leq \ \exp ( ( - w , \gamma \tilde { u } ) ) ( - \frac { 1 } { e } ) . } \end{array}
+$$
+
+On the third line eq. (15) is applied. The fourth line applies the property that $f ( x ) = - x e ^ { - x } \geq - 1 / e$ when $x \geq 0$ ，
+
+Combining eq. (13), eq. (14) and eq. (16), we get
+
+$$
+\langle w _ { \perp } , \nabla { \mathcal R } ( w ) \rangle \geq \exp \left( \langle - w , \gamma { \bar { u } } \rangle \right) \left( { \frac { 1 } { n } } \exp \left( \alpha \| w _ { \perp } \| \right) \alpha \| w _ { \perp } \| - { \frac { 1 } { e } } \right) .
+$$
+
+As long as $\| w _ { \perp } \| \ge ( 1 + \ln ( n ) ) / \alpha , \langle w _ { \perp } , \nabla { \mathcal { R } } ( w ) \rangle \ge 0$
+
+For $\ell _ { \mathrm { l o g } }$ , similar to eq. (13), we have
+
+$$
+\begin{array} { r l } & { \langle w _ { \perp } , \nabla { \mathcal { R } } ( w ) \rangle \geq \displaystyle \frac { 1 } { n } \displaystyle \frac { \exp \left( \langle - w , z ^ { \prime } \rangle \right) } { 1 + \exp \left( \langle - w , z ^ { \prime } \rangle \right) } \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle + \displaystyle \sum _ { \langle z _ { \perp } , w _ { \perp } \rangle \geq 0 } \frac { 1 } { n } \displaystyle \frac { \exp \left( \langle - w , z \rangle \right) } { 1 + \exp \left( \langle - w , z \rangle \right) } \langle - w _ { \perp } , z _ { \perp } \rangle } \\ & { \qquad \geq \displaystyle \frac { 1 } { n } \displaystyle \frac { \exp \left( \langle - w , z ^ { \prime } \rangle \right) } { 1 + \exp \left( \langle - w , z ^ { \prime } \rangle \right) } \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle + \displaystyle \sum _ { \langle z _ { \perp } , w _ { \perp } \rangle \geq 0 } \frac { 1 } { n } \exp \left( \langle - w , z \rangle \right) \langle - w _ { \perp } , z _ { \perp } \rangle . } \end{array}
+$$
+
+The second part of eq. (17) can be bounded again by eq. (16). To bound the first part of eq. (17), first notice that (recali $\langle w , \bar { u } \rangle \geq 0 \mathrm { ~ }$
+
+$$
+\begin{array} { r } { \exp \left( \langle - w , z ^ { \prime } \rangle \right) = \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \exp \left( \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle \right) \leq \exp \left( \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle \right) . } \end{array}
+$$
+
+Using eq. (18), and recall that $\begin{array} { r } { \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle = \langle - w _ { \perp } , z ^ { \prime } \rangle \geq \alpha \| w _ { \perp } \| \geq 0 , } \end{array}$ ， we can bound the first part of eq.(17) as below.
+
+$$
+\begin{array} { r l } & { \frac { 1 } { n } \displaystyle \frac { \exp \left( \langle - w , z ^ { \prime } \rangle \right) } { 1 + \exp \left( \langle - w , z ^ { \prime } \rangle \right) } \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle = \displaystyle \frac { 1 } { n } \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \displaystyle \frac { \exp \left( \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle \right) } { 1 + \exp \left( \langle - w , z ^ { \prime } \rangle \right) } \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle } \\ & { \quad \quad \quad \geq \displaystyle \frac { 1 } { n } \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \displaystyle \frac { \exp \left( \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle \right) } { 1 + \exp \left( \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle \right) } \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle } \\ & { \quad \quad \quad \geq \displaystyle \frac { 1 } { 2 n } \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \langle - w _ { \perp } , z _ { \perp } ^ { \prime } \rangle } \\ & { \quad \quad \quad \geq \displaystyle \frac { 1 } { 2 n } \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \alpha \| w _ { \perp } \| . } \end{array}
+$$
+
+Combining eq. (17),eq. (19) and eq. (16), we get
+
+$$
+\langle w _ { \perp } , \nabla { \mathcal R } ( w ) \rangle \geq \exp \left( \langle - w , \gamma \bar { u } \rangle \right) \left( \frac { 1 } { 2 n } \alpha \| w _ { \perp } \| - \frac { 1 } { e } \right) .
+$$
+
+As long as $\| w _ { \perp } \| \ge 2 n / e \alpha , \langle w _ { \perp } , \nabla \mathcal { R } ( w ) \rangle \ge 0 .$
+
+Proof of Theorem 2. Recall that
+
+$$
+\frac { \mathrm d W _ { 1 } } { \mathrm d t } = - \frac { \partial \mathcal R } { \partial W _ { 1 } } = - W _ { 2 } ^ { \top } \cdot \cdot \cdot W _ { L } ^ { \top } \nabla \mathcal R ( w _ { \mathrm { p r o d } } ) ^ { \top } ,
+$$
+
+and thus
+
+$$
+\frac { \mathrm { d } \| \boldsymbol { W } _ { 1 } \| _ { F } ^ { 2 } } { \mathrm { d } t } = \left. \boldsymbol { W } _ { 1 } , \frac { \mathrm { d } \boldsymbol { W } _ { 1 } } { \mathrm { d } t } \right. = - 2 \langle \boldsymbol { w } _ { \mathrm { p r o d } } , \nabla \mathcal { R } ( \boldsymbol { w } _ { \mathrm { p r o d } } ) \rangle .
+$$
+
+Let $\Pi _ { \bar { u } }$ denote the projection onto $\mathrm { s p a n } ( \bar { u } )$ ，and let $\Pi _ { \perp }$ denote the projection onto $\bar { u } ^ { \perp }$ .Also let $\Pi _ { \bar { u } } W _ { 1 }$ and $\Pi _ { \perp } W _ { 1 }$ denote the projection of rows of $W _ { 1 }$ onto span $( \bar { u } )$ and $\bar { u } ^ { \perp }$ , respectively. Notice that
+
+$$
+\Pi _ { \bar { u } } w _ { \mathrm { p r o d } } = \left( W _ { L } \cdot \cdot \cdot W _ { 2 } ( \Pi _ { \bar { u } } W _ { 1 } ) \right) ^ { \top } , \quad \mathrm { a n d } \quad \Pi _ { \perp } w _ { \mathrm { p r o d } } = \left( W _ { L } \cdot \cdot \cdot W _ { 2 } ( \Pi _ { \perp } W _ { 1 } ) \right) ^ { \top } .
+$$
+
+We further have
+
+$$
+\frac { \mathrm { d } \| \Pi _ { \perp } W _ { 1 } \| _ { F } ^ { 2 } } { \mathrm { d } t } = - 2 \langle \Pi _ { \perp } w _ { \mathrm { p r o d } } , \nabla { \mathcal { R } } ( w _ { \mathrm { p r o d } } ) \rangle .
+$$
+
+Let $W _ { 1 } = u _ { 1 } \sigma _ { 1 } v _ { 1 } ^ { \top } + S$ .We have $\| S \| _ { 2 } \leq \sigma _ { 1 , 2 } \leq \sqrt { \sigma _ { 1 , 2 } ^ { 2 } } \leq \sqrt { \| W _ { 1 } \| _ { F } ^ { 2 } - \| W _ { 1 } \| _ { 2 } ^ { 2 } } \leq \sqrt { D }$ where $\sigma _ { 1 , 2 }$ is the second singular value of $W _ { 1 }$ and $D$ is the constant introduced in Lemma 2. Then
+
+$$
+\Pi _ { \perp } W _ { 1 } = u _ { 1 } \sigma _ { 1 } \left( \Pi _ { \perp } v _ { 1 } \right) ^ { \top } + \Pi _ { \perp } S ,
+$$
+
+and
+
+$$
+\begin{array} { r } { \| \Pi _ { \bot } W _ { 1 } \| _ { F } \leq \left\| u _ { 1 } \sigma _ { 1 } \left( \Pi _ { \bot } v _ { 1 } \right) ^ { \top } \right\| _ { F } + \| \Pi _ { \bot } S \| _ { F } = \sigma _ { 1 } \| \Pi _ { \bot } v _ { 1 } \| + \| \Pi _ { \bot } S \| _ { F } \leq \sigma _ { 1 } \| \Pi _ { \bot } v _ { 1 } \| + \sqrt { d D } . } \end{array}
+$$
+
+It follows that
+
+$$
+\| \Pi _ { \bot } v _ { 1 } \| \ge \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \sigma _ { 1 } } - \frac { \sqrt { d D } } { \sigma _ { 1 } } \ge \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \| W _ { 1 } \| _ { F } } - \frac { \sqrt { d D } } { \| W _ { 1 } \| _ { 2 } } .
+$$
+
+Fix an arbitrary $\epsilon > 0$ . By Theorem 1, we can find some $t _ { 0 }$ large enough such that for any $t \geq t _ { 0 }$
+
+1. $\sqrt { d D } / \| W _ { 1 } \| _ { 2 } \leq \epsilon / 3$   
+2. $\begin{array} { r } { \| w _ { \mathrm { p r o d } } \ / \| W _ { L } \| _ { F } \cdots \| W _ { 1 } \| _ { F } - v _ { 1 } \| \le \epsilon / 3 , \mathrm { o r } \| | w _ { \mathrm { p r o d } } \ / \| W _ { L } \| _ { F } \cdots \| W _ { 1 } \| _ { F } + v _ { 1 } \| \le \epsilon / 3 . } \end{array}$   
+3. $\| W _ { L } \| _ { F } \cdot \cdot \cdot \| W _ { 1 } \| _ { F } \geq 3 K / \epsilon$ where $K$ is the threshold given in Lemma 4, i.e., $1 + \ln ( n ) / \alpha$ for $\ell _ { \mathrm { e x p } } , { 2 n } / { e \alpha }$ for $\ell _ { \mathrm { l o g } }$   
+4. $\mathcal { R } ( W ) \leq \ell ( 0 ) / n$ ，which implies $\langle w _ { \mathrm { p r o d } } , z _ { i } \rangle \geq 0$ for all $1 \leq i \leq n$ . By Lemma 3, there always exists a support vector $z$ for which $\langle \Pi _ { \perp } w _ { \mathrm { p r o d } } , z \rangle \le 0$ ,and therefore $\langle w _ { \mathrm { p r o d } } , \bar { u } \rangle \geq 0$ ：
+
+Suppose for some $t \geq t _ { 0 }$ ， $\| \Pi _ { \perp } W _ { 1 } \| _ { F } / \| W _ { 1 } \| _ { F } \ge \epsilon$ By eq. (22) and bullet 1 above, $\| \Pi _ { \perp } v _ { 1 } \| \ge$ $2 \epsilon / 3$ Bullet 2 above then gives $\left\| \Pi _ { \perp } w _ { \mathrm { p r o d } } \middle / \left\| W _ { L } \right\| _ { F } \cdots \left\| W _ { 1 } \right\| _ { F } \right\| \ \geq \ \epsilon / 3$ ，which together with bulet 3 above implies $\| \Pi _ { \bot } w _ { \mathrm { p r o d } } \| \ge K$ . Since also $\langle w _ { \mathrm { p r o d } } , \bar { u } \rangle \geq 0$ ，we can apply Lemma 4 and get that $\langle \Pi _ { \perp } w _ { \mathrm { p r o d } } , \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \rangle \geq 0 $ In light of eq. (21), $\mathrm { d } \| \Pi _ { \perp } W _ { 1 } \| _ { F } ^ { 2 } / \mathrm { d } t \leq 0$ ：
+
+On the other hand, since after $t \geq t _ { 0 }$ ， $\langle w _ { \mathrm { p r o d } } , z _ { i } \rangle \geq 0$ ，we have $\mathrm { d } \| W _ { 1 } \| _ { F } ^ { 2 } / \mathrm { d } t \geq 0$ by eq. (20). Therefore $\| \Pi _ { \perp } W _ { 1 } \| _ { F } / \| W _ { 1 } \| _ { F }$ will not increase, and since $\| W _ { 1 } \| _ { F } \stackrel { \cdot \cdot } {  } \infty$ , it will eventually drop below $\epsilon$ ,and will never exceed $\epsilon$ again. Therefore,
+
+$$
+\operatorname* { l i m } _ { t \to \infty } { \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \| W _ { 1 } \| _ { F } } } \leq \epsilon .
+$$
+
+Since $\epsilon$ is arbitrary, we have
+
+$$
+\operatorname* { l i m } _ { t \to \infty } { \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \| W _ { 1 } \| _ { F } } } = 0 ,
+$$
+
+and thus $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } | \langle v _ { 1 } , \bar { u } \rangle | = 1 } \end{array}$ . An application of Theorem 1 gives the other part of Theorem 2.[
+
+Proof of Corollary $^ { l }$ By Theorem 2,
+
+$$
+\operatorname* { m i n } _ { i } y _ { i } \left( \frac { W _ { L } } { \| W _ { L } \| _ { F } } \cdots \frac { W _ { 1 } } { \| W _ { 1 } \| _ { F } } \right) x _ { i } \quad \longrightarrow \quad \operatorname* { m i n } _ { i } ^ { } y _ { i } \bar { u } ^ { \top } x _ { i } .
+$$
+
+Next, pick arbitrary unit vectors $a _ { i } \in \mathbb { R } ^ { d _ { i } }$ , and note
+
+$$
+\begin{array} { r l } & { \displaystyle \operatorname* { m a x } _ { 1 _ { 1 } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } } \cdots \cdot \operatorname* { m a x } _ { A _ { L } \in \mathbb { R } ^ { 1 \times d _ { L - 1 } } } \operatorname* { m i n } _ { i } y _ { i } \left( A _ { L } \cdot \cdot \cdot A _ { 1 } \right) x _ { i } \geq \operatorname* { m i n } _ { i } y _ { i } \left( ( 1 a _ { L - 1 } ^ { \top } ) ( a _ { L - 1 } a _ { L - 2 } ^ { \top } ) \cdot \cdot \cdot ( a _ { 2 } a _ { 1 } ^ { \top } ) ( a _ { 1 } \bar { u } ^ { \top } ) \right) } \\ & { \| A _ { 1 } \| _ { F } = 1 \qquad \| A _ { L } \| _ { F } = 1 } \\ & { \qquad = \displaystyle \operatorname* { m i n } _ { i } y _ { i } \bar { u } ^ { \top } x _ { i } . } \end{array}
+$$
+
+On the other hand, if matrices $( A _ { L } , \ldots , A _ { 1 } )$ are feasible, then
+
+$$
+\begin{array} { r } { \| A _ { L } \cdot \cdot \cdot A _ { 1 } \| _ { 2 } \leq \| A _ { L } \| _ { 2 } \cdot \cdot \cdot \| A _ { 2 } \| _ { 2 } \| A _ { 1 } \| _ { F } \leq \| A _ { L } \| _ { F } \cdot \cdot \cdot \| A _ { 1 } \| _ { F } \leq 1 , } \end{array}
+$$
+
+whereby
+
+$$
+\begin{array} { r l } & { \underset { i } { \operatorname* { m i n } } y _ { i } ( A _ { L } \cdot \cdot \cdot A _ { 1 } ) x _ { i } \leq \| A _ { L } \cdot \cdot \cdot A _ { 1 } \| _ { 2 } \underset { i } { \operatorname* { m i n } } y _ { i } \left( \frac { A _ { L } \cdot \cdot \cdot A _ { 1 } } { \| A _ { L } \cdot \cdot \cdot A _ { 1 } \| _ { 2 } } \right) x _ { i } } \\ & { \qquad \leq 1 \cdot \underset { \| w \| _ { 2 } = 1 } { \operatorname* { m a x } } \underset { i } { \operatorname* { m i n } } y _ { i } w ^ { \top } x _ { i } } \\ & { \qquad = \underset { i } { \operatorname* { m i n } } y _ { i } \bar { u } ^ { \top } x _ { i } . } \end{array}
+$$
+
+# COMITTED PROOFS FROM SECTION 3
+
+Proof of Lemma 5. Given $W , V \in B ( R )$ , we need to show that $\begin{array} { r } { \| \nabla \mathcal { R } ( W ) - \nabla \mathcal { R } ( V ) \| \leq \beta ( R ) \| W - } \end{array}$ $V \|$ for some $\beta ( R )$
+
+Consider $k \ = \ 1$ first.Let $w ~ = ~ ( W _ { L } \cdot \cdot \cdot W _ { 1 } ) ^ { \top }$ ，and $v ~ = ~ ( V _ { L } \cdot \cdot \cdot V _ { 1 } ) ^ { \top }$ .Since $| \ell ^ { \prime } | \ \leq \ G$ $\| \nabla \mathcal { R } ( w ) \| , \| \nabla \mathcal { R } ( v ) \| \le G$ . We have
+
+$$
+\begin{array} { r l } { \displaystyle \left\| \frac { \partial { \mathcal R } } { \partial W _ { 1 } } - \frac { \partial { \mathcal R } } { \partial V _ { 1 } } \right\| = } & { \displaystyle \left\| W _ { 2 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) - V _ { 2 } ^ { \top } \cdots V _ { L } ^ { \top } \nabla { \mathcal R } ( v ) \right\| } \\ & { \leq \left\| W _ { 2 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) - V _ { 2 } ^ { \top } W _ { 3 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) \right\| } \\ & { \quad + \displaystyle \left\| V _ { 2 } ^ { \top } W _ { 3 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) - V _ { 2 } ^ { \top } \cdots V _ { L } ^ { \top } \nabla { \mathcal R } ( v ) \right\| } \\ & { \leq R ^ { L - 2 } G \| W _ { 2 } - V _ { 2 } \| } \\ & { \quad + \left\| V _ { 2 } ^ { \top } W _ { 3 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) - V _ { 2 } ^ { \top } \cdots V _ { L } ^ { \top } \nabla { \mathcal R } ( v ) \right\| } \\ & { \leq R ^ { L - 2 } G \| W - V \| } \\ & { \quad + \displaystyle \left\| V _ { 2 } ^ { \top } W _ { 3 } ^ { \top } \cdots W _ { L } ^ { \top } \nabla { \mathcal R } ( w ) - V _ { 2 } ^ { \top } \cdots V _ { L } ^ { \top } \nabla { \mathcal R } ( v ) \right\| . } \end{array}
+$$
+
+Proceeding in this way, we can get
+
+$$
+\left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } - \frac { \partial \mathcal { R } } { \partial V _ { 1 } } \right\| \leq ( L - 1 ) R ^ { L - 2 } G \| W - V \| + R ^ { L - 1 } \| \nabla \mathcal { R } ( w ) - \nabla \mathcal { R } ( v ) \| .
+$$
+
+Since $\| z _ { i } \| \leq 1 , \ell ^ { \prime }$ is $\beta$ -Lipschitz, we have
+
+$$
+\| \nabla \mathcal { R } ( w ) - \nabla \mathcal { R } ( v ) \| \leq \beta \| w - v \| \leq \beta L R ^ { L - 1 } \| W - V \| ,
+$$
+
+where the last inequality follows from a similar one-by-one replacement procedure as in eq. (23). Combining eq. (24) and eq. (25), we get for $R \geq 1$ ，
+
+$$
+\left\| \frac { \partial \mathcal { R } } { \partial W _ { 1 } } - \frac { \partial \mathcal { R } } { \partial V _ { 1 } } \right\| \leq \left( ( L - 1 ) R ^ { L - 2 } G + \beta L R ^ { 2 L - 2 } \right) \| W - V \| \leq 2 L R ^ { 2 L - 2 } ( \beta + G ) \| W - V \| .
+$$
+
+The same procedure can be done for other layers, and together
+
+$$
+\begin{array} { r } { \| \nabla \mathcal { R } ( W ) - \nabla \mathcal { R } ( V ) \| \le 2 L ^ { 2 } R ^ { 2 L - 2 } ( \beta + G ) \| W - V \| . } \end{array}
+$$
+
+Proof of Lemma $6$ Recall that if $W ( t ) , W ( t + 1 ) \in B ( R )$ and $\eta _ { t } = 1 / \beta ( R )$ ，
+
+$$
+\begin{array} { r l } & { \mathcal { R } \left( W ( t + 1 ) \right) - \mathcal { R } \left( W ( t ) \right) \leq \left. \nabla \mathcal { R } \left( W ( t ) \right) , - \eta _ { t } \nabla \mathcal { R } \left( W ( t ) \right) \right. + \frac { \beta ( R ) \eta _ { t } ^ { 2 } } { 2 } \left. \nabla \mathcal { R } \left( W ( t ) \right) \right. ^ { 2 } } \\ & { \quad \quad \quad = - \frac { 1 } { 2 \beta ( R ) } \left. \nabla \mathcal { R } \left( W ( t ) \right) \right. ^ { 2 } } \\ & { \quad \quad \quad = - \frac { \eta _ { t } } { 2 } \left. \nabla \mathcal { R } \left( W ( t ) \right) \right. ^ { 2 } . } \end{array}
+$$
+
+Suppose $W ( t ) \in B ( R )$ for all $t$ By Assumption 2 and eq. (26),
+
+$$
+\mathcal { R } \left( W ( 1 ) \right) \leq \mathcal { R } \left( W ( 0 ) \right) - \frac { 1 } { 2 \beta ( R ) } \Big \| \nabla \mathcal { R } \left( W ( 0 ) \right) \Big \| ^ { 2 } < \mathcal { R } \left( W ( 0 ) \right) .
+$$
+
+By eq.(26)，gradient descent never increases the risk,and thus for all $t ~ \geq ~ 1$ ， $\mathcal { R } \left( W ( t ) \right) \ \leq$ $\mathcal { R } \left( W ( 1 ) \right) \ < \ \mathcal { R } \left( W ( 0 ) \right)$ . In exactly the same way as in the proof of Lemma 1, one can show that there exists some constant $\epsilon ( R ) > 0$ ，so that $\Vert \partial \mathcal { R } / \partial W _ { 1 } ( t ) \Vert _ { F } \geq \epsilon ( R )$ for all $t$ Invoking eq. (26) again, we will get
+
+$$
+\mathcal { R } \left( W ( 0 ) \right) \geq \sum _ { t = 0 } ^ { \infty } \frac { 1 } { 2 \beta ( R ) } \epsilon ( R ) ^ { 2 } = \infty ,
+$$
+
+which is a contradiction. Therefore $W ( t )$ must go out of $B ( R )$ at some time.
+
+Next we prove Theorem 3 and 4. The proofs depend on several lemmas which are similar to the gradient flow ones. The following Lemma 7 is similar to Lemma 1.
+
+Lemma 7. Under Assumption 1,2, 4, and 5, gradient descent ensures that
+
+· $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } ( t ) \| _ { F }$ is unbounded. · $\textstyle \sum _ { t = 0 } ^ { \infty } \eta _ { t } = \infty .$ · For any $R > 0$ $\sum _ { t : W ( t ) \in B ( R ) } \eta _ { t } < \infty .$
+
+Proof. By Assumption5, wealways have that $W ( t ) \in B ( R _ { t } )$ . Since $\beta ( R _ { t } ) = 2 L ^ { 2 } R _ { t } ^ { 2 L - 2 } ( \beta + G ) \geq$ $R _ { t } ^ { L - 1 } G$ , we have for any $1 \le k \le L$ ，
+
+$$
+\begin{array} { r l } { \| W _ { k } ( t + 1 ) \| _ { F } \leq \| W _ { k } ( t ) \| _ { F } + \eta _ { t } \bigg \| \frac { \partial \mathcal { R } } { \partial W _ { k } ( t ) } \bigg \| _ { F } } & { } \\ { \leq \| W _ { k } ( t ) \| _ { F } + \frac { 1 } { \beta ( R _ { t } ) } \bigg \| \frac { \partial \mathcal { R } } { \partial W _ { k } ( t ) } \bigg \| _ { F } } & { } \\ { \leq \| W _ { k } ( t ) \| _ { F } + \frac { 1 } { \beta ( R _ { t } ) } R _ { t } ^ { L - 1 } G } & { } \\ { \leq \| W _ { k } ( t ) \| _ { F } + 1 . } \end{array}
+$$
+
+Moreover, Lemma 6 shows that $R _ { t } \to \infty$ . Since $R _ { t + 1 } = R _ { t }$ as long as $W ( t + 1 ) \in B ( R _ { t } - 1 )$ ， $\operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } ( t ) \| _ { F }$ is unbounded.
+
+It then follows that for any $t$ , by Cauchy-Schwarz,
+
+$$
+( \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ) ( \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } \Big \Vert \nabla \mathcal { R } ( W ( \tau ) ) \Big \Vert ^ { 2 } ) \geq ( \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } \Big \Vert \nabla \mathcal { R } ( W ( \tau ) ) \Big \Vert ) ^ { 2 }  \infty .
+$$
+
+since by eq. (26),
+
+$$
+\sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } \Big \lVert \nabla \mathcal { R } \left( W ( \tau ) \right) \Big \rVert ^ { 2 } \leq 2 \mathcal { R } \left( W ( 0 ) \right) - 2 \mathcal { R } \left( W ( t ) \right) \leq 2 \mathcal { R } \left( W ( 0 ) \right) ,
+$$
+
+we have $\textstyle \sum _ { t = 0 } ^ { \infty } \eta _ { t } = \infty$
+
+Since under Assumptions 4 and 5 gradient descent never increases the risk,it can be shown in exactly the same as in the proof of Lemma 1 that, for $W ( t ) \in B ( R )$ ， $\Vert \partial \mathcal { R } / \partial W _ { 1 } ( t ) \Vert _ { F } \geq \epsilon ( R )$ for some constant $\epsilon ( R ) > 0$ . Invoking eq. (26)again,we get that $\sum _ { t : W ( t ) \in B ( R ) } \eta _ { t } < \infty$ ， □
+
+The next lemma is an analogy to Lemma 2.
+
+Lemma 8. Under Assumption l and 4, the gradient descent iterates satisfy the following properties:
+
+$$
+\begin{array} { r } { \bullet \bullet \ s u p p o s e \ \operatorname* { m a x } _ { 1 \leq k \leq L } \| W _ { k } \| _ { F }  \infty , t h e n \Big | \Big \langle w _ { \mathrm { p r o d } } \big / \prod _ { k = 1 } ^ { L } \| W _ { k } \| _ { F } , v _ { 1 } \Big \rangle \Big |  1 . } \end{array}
+$$
+
+Proof. Recall that for any $W$ ，
+
+$$
+\boldsymbol { W } _ { k + 1 } ^ { \top } \frac { \partial \boldsymbol { \mathcal { R } } } { \partial \boldsymbol { W } _ { k + 1 } } = \boldsymbol { W } _ { k + 1 } ^ { \top } \cdot \cdot \cdot \boldsymbol { W } _ { L } ^ { \top } \nabla \boldsymbol { \mathcal { R } } ( \boldsymbol { w } _ { \mathrm { p r o d } } ) ^ { \top } \boldsymbol { W } _ { 1 } ^ { \top } \cdot \cdot \cdot \boldsymbol { W } _ { k } ^ { \top } = \frac { \partial \boldsymbol { \mathcal { R } } } { \partial \boldsymbol { W } _ { k } } \boldsymbol { W } _ { k } ^ { \top } .
+$$
+
+For gradient descent iterates, summing eq. (28) from O to $t - 1$ , we get
+
+$$
+\begin{array} { r l } & { W _ { k + 1 } ^ { \top } ( t ) W _ { k + 1 } ( t ) - W _ { k + 1 } ^ { \top } ( 0 ) W _ { k + 1 } ( 0 ) + \displaystyle \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \left( \frac { \partial \mathcal { R } } { \partial W _ { k + 1 } ( \tau ) } \right) ^ { \top } \left( \frac { \partial \mathcal { R } } { \partial W _ { k + 1 } ( \tau ) } \right) } \\ { = } & { W _ { k } ( t ) W _ { k } ^ { \top } ( t ) - W _ { k } ( 0 ) W _ { k } ^ { \top } ( 0 ) + \displaystyle \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) ^ { \top } . } \end{array}
+$$
+
+For any $1 \le k \le L$ and any $t$ let
+
+$$
+P _ { k } ( t ) = \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) ^ { \top } ,
+$$
+
+and
+
+$$
+Q _ { k } ( t ) = \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) ^ { \top } \left( \frac { \partial \mathcal { R } } { \partial W _ { k } ( \tau ) } \right) .
+$$
+
+We have $\| P _ { k } ( t ) \| _ { 2 } = \| Q _ { k } ( t ) \| _ { 2 } \leq \operatorname { t r } \big ( Q _ { k } ( t ) \big ) = \operatorname { t r } \big ( P _ { k } ( t ) \big )$ . Moreover, invoking eq. (26),
+
+$$
+\begin{array} { r l r } {  { \sum _ { k = 1 } ^ { L } \operatorname { t r } ( P _ { k } ( t ) ) = \sum _ { k = 1 } ^ { L } \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \bigg \Vert \frac { \partial R } { \partial W _ { k } ( \tau ) } \bigg \Vert _ { F } ^ { 2 } } } \\ & { } & { = \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } ^ { 2 } \Big \Vert \nabla \cdot ( W ( \tau ) ) \Big \Vert ^ { 2 } } \\ & { } & { \leq \sum _ { \tau = 0 } ^ { t - 1 } \eta _ { \tau } \Big \Vert \nabla \cdot ( W ( \tau ) ) \Big \Vert ^ { 2 } } \\ & { } & { \leq 2 \mathcal { R } ( W ( 0 ) ) - 2 \mathcal { R } ( W ( t ) ) } \\ & { } & { \leq 2 \mathcal { R } ( W ( 0 ) ) . } \end{array}
+$$
+
+Still let $\sigma _ { k } ( t ) , u _ { k } ( t )$ and $v _ { k } ( t )$ denote the first singular value,left singular vector and right singular vector of $W _ { k } ( t )$ . We can then proceed basically in the same way as in the proof of Lemma 2. For example,eq.(6) becomes
+
+$$
+\sigma _ { k } ^ { 2 } ( t ) \geq \sigma _ { k + 1 } ^ { 2 } ( t ) - \| A _ { k , k + 1 } ( t ) \| _ { 2 } - \| P _ { k } ( t ) \| _ { 2 } \geq \sigma _ { k + 1 } ^ { 2 } ( t ) - \| A _ { k , k + 1 } ( t ) \| _ { 2 } - \operatorname { t r } \left( P _ { k } ( t ) \right) ,
+$$
+
+while eq. (7) becomes
+
+$$
+\begin{array} { r } { \| W _ { k } ( t ) \| _ { F } ^ { 2 } = \| W _ { k + 1 } ( t ) \| _ { F } ^ { 2 } + \| W _ { k } ( 0 ) \| _ { F } ^ { 2 } - \| W _ { k + 1 } ( 0 ) \| _ { F } ^ { 2 } - \operatorname { t r } \big ( P _ { k } ( t ) \big ) + \operatorname { t r } \big ( Q _ { k + 1 } ( t ) \big ) . } \end{array}
+$$
+
+Summing eq. (31) and eq. (32) from $k$ t0 $L - 1$ , and invoke eq. (30), we get
+
+$$
+\| W _ { k } ( t ) \| _ { F } ^ { 2 } - \| W _ { k } ( t ) \| _ { 2 } ^ { 2 } \leq D - \operatorname { t r } \left( P _ { k } ( t ) \right) + \operatorname { t r } \left( Q _ { L } ( t ) \right) + \sum _ { k ^ { \prime } = k } ^ { L - 1 } \operatorname { t r } \left( P _ { k ^ { \prime } } ( t ) \right) \leq D + 2 \mathcal { R } \left( W ( 0 ) \right) .
+$$
+
+To prove singular vectors get aligned, we can stillproceed in nearly the same way as in the proof of Lemma 2.eq. (9) becomes
+
+$$
+u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } u _ { k } \geq \sigma _ { k } ^ { 2 } - \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } - \| Q _ { k + 1 } ( t ) \| _ { 2 } ,
+$$
+
+while eq. (10) becomes
+
+$$
+u _ { k } ^ { \top } W _ { k + 1 } ^ { \top } W _ { k + 1 } u _ { k } \leq \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + D + 2 \mathcal { R } \left( W ( 0 ) \right) .
+$$
+
+Combining eq. (33) and eq. (34)
+
+$$
+\sigma _ { k } ^ { 2 } \leq \langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \sigma _ { k + 1 } ^ { 2 } + D + 2 \mathcal { R } \left( W ( 0 ) \right) + \| Q _ { k + 1 } ( t ) \| _ { 2 } + \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } .
+$$
+
+Similar to eq. (33), we can get
+
+$$
+\sigma _ { k } ^ { 2 } \geq v _ { k + 1 } ^ { \top } W _ { k } W _ { k } ^ { \top } v _ { k + 1 } \geq \sigma _ { k + 1 } ^ { 2 } - \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } - \| P _ { k } ( t ) \| _ { 2 } ,
+$$
+
+and thus eq.(12) becomes
+
+$$
+\frac { \sigma _ { k } ^ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } \geq 1 - \frac { \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } + \| P _ { k } ( t ) \| _ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } .
+$$
+
+Combining eq. (35) and eq. (36), we get
+
+$$
+\langle u _ { k } , v _ { k + 1 } \rangle ^ { 2 } \geq 1 - \frac { D + \| W _ { k } ( 0 ) \| _ { 2 } ^ { 2 } + \| W _ { k + 1 } ( 0 ) \| _ { 2 } ^ { 2 } + 3 \mathcal { R } \left( W ( 0 ) \right) } { \sigma _ { k + 1 } ^ { 2 } } .
+$$
+
+The final claim of Lemma 8 can be proved in exactly the same way as Lemma 2.
+
+Proof of Theorem 3. Summing eq. (32), we know that for any two different layers $j > k$ ，
+
+$$
+\begin{array} { r } { \| W _ { k } ( t ) \| _ { F } ^ { 2 } - \| W _ { j } ( t ) \| _ { F } ^ { 2 } = \| W _ { k } ( 0 ) \| _ { F } ^ { 2 } - \| W _ { j } ( 0 ) \| _ { F } ^ { 2 } - \operatorname { t r } \big ( P _ { k } ( t ) \big ) + \operatorname { t r } \big ( Q _ { j } ( t ) \big ) . } \end{array}
+$$
+
+Recall eq. (30), we know that
+
+$$
+\left| \left( \| W _ { k } ( t ) \| _ { F } ^ { 2 } - \| W _ { j } ( t ) \| _ { F } ^ { 2 } \right) - \left( \| W _ { k } ( 0 ) \| _ { F } ^ { 2 } - \| W _ { j } ( 0 ) \| _ { F } ^ { 2 } \right) \right| \le 2 \mathcal { R } \left( W ( 0 ) \right) .
+$$
+
+In other words,the difference between the squares of Frobenius norms of any two layers is still bounded.
+
+The proof then goes in the same way as the proof of Theorem 1. Suppose the risk is always above $\epsilon > 0$ . Then there exists some $c ( \epsilon ) > 0$ such that $\| \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) \| \ge c ( \epsilon )$ . By Lemma 8, there exists some $C$ such that if $\mathrm { m i n } _ { 1 \leq k \leq L } \| W _ { k } ( t ) \| _ { F } > C$ ， $\| W _ { L } ( t ) \cdot \cdot \cdot W _ { 2 } ( t ) \| \ge C ^ { L } / 2$ By eq. (37) and Lemma 7, $\sum _ { t : \| W _ { k } ( t ) \| _ { F } \leq C }$ for some $ { k } ^ { \eta _ { t } }$ isfinite.Ontheoternd,y7 $\textstyle \sum _ { i = 0 } ^ { \infty } \eta _ { t } = \infty$ ,and thus $\sum _ { t : \| W _ { k } ( t ) \| _ { F } > C }$ for all $_ k \eta _ { t } = \infty$ Therefore wehave,byinvokingeq.(26),
+
+$$
+\begin{array} { r l r } {  { 2 \mathcal { R } ( W ( 0 ) ) \geq \sum _ { t = 0 } ^ { \infty } \eta _ { t } \Big \Vert \mathcal { R } ( W ( t ) ) \Big \Vert ^ { 2 } } } \\ & { } & { \geq \displaystyle \sum _ { t = 0 } ^ { \infty } \eta _ { t } \bigg \Vert \frac { \partial \mathcal { R } } { \partial W _ { 1 } ( t ) } \bigg \Vert ^ { 2 } } \\ & { } & { \geq c ( \epsilon ) \frac { C ^ { L } } { 2 } \sum _ { t : \| W _ { k } ( t ) \| _ { F } > C \mathrm { f o r a l l } \ k } \eta _ { t } } \\ & { } & { = \infty , } \end{array}
+$$
+
+which is a contradiction. Therefore $\mathcal { R } ( W ( t ) )  0$ ,and since it has no finite optimum, $\Vert W _ { k } \Vert _ { F } $ $\infty$ . The other results follow from Lemma 7. □
+
+Proof of Theorem 4. Recall that
+
+$$
+\frac { \partial \mathcal { R } } { \partial W _ { 1 } } = W _ { 2 } ^ { \top } \cdot \cdot \cdot W _ { L } ^ { \top } \nabla \mathcal { R } ( w _ { \mathrm { p r o d } } ) ,
+$$
+
+and thus
+
+$$
+\begin{array} { l } { \displaystyle \| W _ { 1 } ( t + 1 ) \| _ { F } ^ { 2 } = \| W _ { 1 } ( t ) \| _ { F } ^ { 2 } - 2 \eta _ { t } \left. W _ { 1 } ( t ) , \frac { \partial { \mathcal R } } { \partial W _ { 1 } ( t ) } \right. + \eta _ { t } ^ { 2 } \bigg \| \frac { \partial { \mathcal R } } { \partial W _ { 1 } ( t ) } \bigg \| _ { F } ^ { 2 } } \\ { = \| W _ { 1 } ( t ) \| _ { F } ^ { 2 } - 2 \eta _ { t } \left. w _ { \mathrm { p r o d } } ( t ) , \nabla { \mathcal R } \left( w _ { \mathrm { p r o d } } ( t ) \right) \right. + \eta _ { t } ^ { 2 } \bigg \| \frac { \partial { \mathcal R } } { \partial W _ { 1 } ( t ) } \bigg \| _ { F } ^ { 2 } . } \end{array}
+$$
+
+If $\langle w _ { \mathrm { p r o d } } , z _ { i } \rangle \geq 0$ for all $i$ ,then $\| W _ { 1 } ( t + 1 ) \| _ { F } \geq \| W _ { 1 } ( t ) \| _ { F }$
+
+Also recall that $\Pi _ { \perp } W _ { 1 } ( t )$ denote the projection of rows of $W _ { 1 } ( t )$ onto $\bar { u } ^ { \perp }$ , the orthogonal complement of $\mathrm { s p a n } ( \bar { u } )$ . We have
+
+$$
+\begin{array} { r l } & { \| \Pi _ { \bot } W _ { 1 } ( t + 1 ) \| _ { F } ^ { 2 } \leq \| \Pi _ { \bot } W _ { 1 } ( t ) \| _ { F } ^ { 2 } - 2 \eta _ { t } \left. \Pi _ { \bot } W _ { 1 } ( t ) , \frac { \partial { \mathcal { R } } } { \partial W _ { 1 } ( t ) } \right. + \eta _ { t } ^ { 2 } \bigg \| \frac { \partial { \mathcal { R } } } { \partial W _ { 1 } ( t ) } \bigg \| _ { F } ^ { 2 } } \\ & { \qquad = \| \Pi _ { \bot } W _ { 1 } ( t ) \| _ { F } ^ { 2 } - 2 \eta _ { t } \left. \Pi _ { \bot } w _ { \mathrm { p r o d } } ( t ) , \nabla { \mathcal { R } } \left( w _ { \mathrm { p r o d } } ( t ) \right) \right. + \eta _ { t } ^ { 2 } \bigg \| \frac { \partial { \mathcal { R } } } { \partial W _ { 1 } ( t ) } \bigg \| _ { F } ^ { 2 } . } \end{array}
+$$
+
+Invoking eq. (26) again gives
+
+$$
+\eta _ { t } ^ { 2 } \left. \frac { \partial \mathcal { R } } { \partial W _ { 1 } ( t ) } \right. _ { F } ^ { 2 } \le \eta _ { t } \left. \nabla \mathcal { R } \left( W ( t ) \right) \right. ^ { 2 } \le 2 \left( \mathcal { R } \left( W ( t ) \right) - \mathcal { R } \left( W ( t + 1 ) \right) \right) .
+$$
+
+The proof then goes in almost the same way as the proof of Theorem 2. For any $\epsilon > 0$ , we can find some large enough time $t _ { 0 }$ ,such that for any $t \geq t _ { 0 }$ ，
+
+1 $\| \Pi _ { \bot } W _ { 1 } ( t ) \| _ { F } / \| W _ { 1 } ( t ) \| F \geq \epsilon$ implies hat $\left. \Pi _ { \perp } w _ { \mathrm { p r o d } } ( t ) , \nabla \mathcal { R } \left( w _ { \mathrm { p r o d } } ( t ) \right) \right. \geq 0 .$
+
+2. $\langle w _ { \mathrm { p r o d } } ( t ) , z _ { i } \rangle \geq 0$ for all $i$ , and thus $\| W _ { 1 } ( t + 1 ) \| _ { F } \geq \| W _ { 1 } ( t ) \| _ { F }$
+
+3. $\| W _ { 1 } ( t ) \| _ { F } \ge 1 + \sqrt { 2 \mathcal { R } ( W ( 0 ) ) } \big / \epsilon .$
+
+Suppose at some time $t _ { 1 } ~ \geq ~ t _ { 0 }$ ， $\| \Pi _ { \perp } W _ { 1 } ( t _ { 1 } ) \| _ { F } / \| W _ { 1 } ( t _ { 1 } ) \| _ { F } ~ \ge ~ \epsilon \qquad $ ，As long as this still holds,in light of bullet (1) above, eq. (38)and eq. (39), $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F } ^ { 2 }$ will increase by at most $2 \mathcal { R } \left( W ( t _ { 1 } ) \right) \leq$ $2 \mathcal { R } \left( W ( 0 ) \right)$ .On the other hand, $\| W _ { 1 } \| _ { F } \to \infty$ ，and thus there exists some $t _ { 2 } ~ > ~ t _ { 1 }$ such that $\| \Pi _ { \perp } \mathbf { \dot { \boldsymbol { W } } } _ { 1 } ( t _ { 2 } ) \| _ { F } / \| \boldsymbol { W } _ { 1 } ( t _ { 2 } ) \| F < \epsilon$
+
+Let $t _ { 3 }$ denote the smallest time after $t _ { 2 }$ such that $\| \Pi _ { \perp } W _ { 1 } ( t _ { 3 } ) \| _ { F } / \| W _ { 1 } ( t _ { 3 } ) \| _ { F } \ge \epsilon$ (if it exists). Recall that $\| W _ { 1 } ( t + 1 ) \| _ { F } \leq \| W _ { 1 } ( t ) \| _ { F } + 1$ for any $t \geq 0$ ,and $\| W _ { 1 } ( t + 1 ) \| _ { F } \geq \| W _ { 1 } ( t ) \| _ { F }$ for any $t \geq t _ { 0 }$ , we have
+
+$$
+\frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 3 } ) \| _ { F } } { \| W _ { 1 } ( t _ { 3 } ) \| _ { F } } \leq \frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 3 } ) \| _ { F } } { \| W _ { 1 } ( t _ { 3 } - 1 ) \| _ { F } } \leq \frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 3 } - 1 ) \| _ { F } + 1 } { \| W _ { 1 } ( t _ { 3 } - 1 ) \| _ { F } } < \epsilon + \frac { 1 } { \| W _ { 1 } ( t _ { 3 } - 1 ) \| _ { F } } .
+$$
+
+After $t _ { 3 }$ ， $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F } ^ { 2 }$ will increase by at most $2 \mathcal { R } \left( W ( 0 ) \right)$ , and thus $\Vert \Pi _ { \perp } W _ { 1 } \Vert _ { F }$ will increase by at most $\sqrt { 2 \mathcal { R } \left( W ( 0 ) \right) }$ . Therefore, for any $t _ { 4 } \geq t _ { 3 }$ , as long as $\| \Pi _ { \bot } W _ { 1 } ( t _ { 4 } ) \| _ { F } / \| W _ { 1 } ( t _ { 4 } ) \| _ { F } \ge \epsilon$ we have
+
+$$
+\begin{array} { r l } { \displaystyle \frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 4 } ) \| _ { F } } { \| W _ { 1 } ( t _ { 4 } ) \| _ { F } } \leq \frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 4 } ) \| _ { F } } { \| W _ { 1 } ( t _ { 3 } ) \| _ { F } } } & { } \\ & { \leq \frac { \| \Pi _ { \bot } W _ { 1 } ( t _ { 3 } ) \| _ { F } + \sqrt { 2 \mathcal { R } \left( W ( 0 ) \right) } } { \| W _ { 1 } ( t _ { 3 } ) \| _ { F } } } \\ & { \leq \epsilon + \frac { 1 } { \| W _ { 1 } ( t _ { 3 } - 1 ) \| _ { F } } + \frac { \sqrt { 2 \mathcal { R } \left( W ( 0 ) \right) } } { \| W _ { 1 } ( t _ { 3 } ) \| _ { F } } \leq 2 \epsilon , } \end{array}
+$$
+
+since $\| W _ { 1 } ( t ) \| _ { F } \ge 1 + \sqrt { 2 \mathcal { R } ( W ( 0 ) ) } \big / \epsilon$ after $t _ { 0 }$ . In other words,
+
+$$
+\operatorname* { l i m } _ { t \to \infty } { \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \| W _ { 1 } \| _ { F } } } \leq 2 \epsilon .
+$$
+
+Since $\epsilon$ is arbitrary, we have
+
+$$
+\operatorname* { l i m } _ { t \to \infty } { \frac { \| \Pi _ { \bot } W _ { 1 } \| _ { F } } { \| W _ { 1 } \| _ { F } } } = 0 ,
+$$
+
+and thus $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } | \langle v _ { 1 } , \bar { u } \rangle | = 1 } \end{array}$
+
+Proof of Corollary 2. The proof is analogous to that of Corollary 1, except using Theorem 4 in place of Theorem 2. □

parse/train/HJgEMpVFwB/HJgEMpVFwB.md

@@ -0,0 +1,274 @@
+# ADVERSARIAL POLICIES: ATTACKING DEEP REINFORCEMENT LEARNING
+
+Adam Gleave1 Michael Dennis Cody Wild Neel Kant Sergey Levine Stuart Russell University of California, Berkeley
+
+ABSTRACT
+
+Deep reinforcement learning (RL) policies are known to be vulnerable to adversarial perturbations to their observations, similar to adversarial examples for classifiers. However, an attacker is not usually able to directly modify another agent’s observations. This might lead one to wonder: is it possible to attack an RL agent simply by choosing an adversarial policy acting in a multi-agent environment so as to create natural observations that are adversarial? We demonstrate the existence of adversarial policies in zero-sum games between simulated humanoid robots with proprioceptive observations, against state-of-the-art victims trained via self-play to be robust to opponents. The adversarial policies reliably win against the victims but generate seemingly random and uncoordinated behavior. We find that these policies are more successful in high-dimensional environments, and induce substantially different activations in the victim policy network than when the victim plays against a normal opponent. Fine-tuning protects a victim against a specific adversary, but the attack method can be successfully reapplied to find a new adversarial policy. Videos are available at https://adversarialpolicies.github.io/.
+
+# 1 INTRODUCTION
+
+The discovery of adversarial examples for image classifiers prompted a new field of research into adversarial attacks and defenses (Szegedy et al., 2014). Recent work has shown that deep RL policies are also vulnerable to adversarial perturbations of image observations (Huang et al., 2017; Kos and Song, 2017). However, real-world RL agents inhabit natural environments populated by other agents, including humans, who can only modify another agent’s observations via their actions. We explore whether it’s possible to attack a victim policy by building an adversarial policy that takes actions in a shared environment, inducing natural observations which have adversarial effects on the victim.
+
+RL has been applied in settings as varied as autonomous driving (Dosovitskiy et al., 2017), negotiation (Lewis et al., 2017) and automated trading (Noonan, 2017). In domains such as these, an attacker cannot usually directly modify the victim policy’s input. For example, in autonomous driving pedestrians and other drivers can take actions in the world that affect the camera image, but only in a physically realistic fashion. They cannot add noise to arbitrary pixels, or make a building disappear. Similarly, in financial trading an attacker can send orders to an exchange which will appear in the victim’s market data feed, but the attacker cannot modify observations of a third party’s orders.
+
+As a proof of concept, we show the existence of adversarial policies in zero-sum simulated robotics games with proprioceptive observations (Bansal et al., 2018a). The state-of-the-art victim policies were trained via self-play to be robust to opponents. We train each adversarial policy using model-free RL against a fixed black-box victim. We find the adversarial policies reliably beat their victim, despite training for less than $3 \%$ of the timesteps initially used to train the victim policies.
+
+Critically, we find the adversaries win by creating natural observations that are adversarial, and not by becoming generally strong opponents. Qualitatively, the adversaries fall to the ground in contorted positions, as illustrated in Figure 1, rather than learning to run, kick or block like normal opponents. This strategy does not work when the victim is ‘masked’ and cannot see the adversary’s position, suggesting that the adversary succeeds by manipulating a victim’s observations through its actions.
+
+Having observed these results, we wanted to understand the sensitivity of the attack to the dimensionality of the victim’s observations. We find that victim policies in higher-dimensional Humanoid environments are substantially more vulnerable to adversarial policies than in lower-dimensional Ant environments. To gain insight into why adversarial policies succeed, we analyze the activations of the victim’s policy network using a Gaussian Mixture Model and t-SNE (Maaten and Hinton, 2008). We find adversarial policies induce significantly different activations than normal opponents, and that the adversarial activations are typically more widely dispersed between timesteps than normal activations.
+
+![](images/bed7c6d84ef604ab225fc38240a0734aa665e13269680754a2d339a48b73f268.jpg)  
+Figure 1: Illustrative snapshots of a victim (in blue) against normal and adversarial opponents (in red). The victim wins if it crosses the finish line; otherwise, the opponent wins. Despite never standing up, the adversarial opponent wins $86 \%$ of episodes, far above the normal opponent’s $47 \%$ win rate.
+
+A natural defense is to fine-tune the victim against the adversary. We find this protects against that particular adversary, but that repeating the attack method finds a new adversary the fine-tuned victim is vulnerable to. However, this new adversary differs qualitatively by physically interfering with the victim. This suggests repeated fine-tuning might provide protection against a range of adversaries.
+
+Our paper makes three contributions. First, we propose a novel, physically realistic threat model for adversarial examples in RL. Second, we demonstrate the existence of adversarial policies in this threat model for several simulated robotics games. Our adversarial policies reliably beat the victim, despite training with less than $3 \%$ as many timesteps and generating seemingly random behavior. Third, we conduct a detailed analysis of why the adversarial policies work. We show they create natural observations that are adversarial to the victim and push the activations of the victim’s policy network off-distribution. Additionally, we find policies are easier to attack in high-dimensional environments.
+
+As deep RL is increasingly deployed in environments with potential adversaries, we believe it is important that practitioners are aware of this previously unrecognized threat model. Moreover, even in benign settings, we believe adversarial policies can be a useful tool for uncovering unexpected policy failure modes. Finally, we are excited by the potential of adversarial training using adversarial policies, which could improve robustness relative to conventional self-play by training against adversaries that exploit weaknesses undiscovered by the distribution of similar opponents present during self-play.
+
+# 2 RELATED WORK
+
+Most study of adversarial examples has focused on small $\ell _ { p }$ norm perturbations to images, which Szegedy et al. (2014) discovered cause a variety of models to confidently misclassify the image, even though the changes are visually imperceptible to a human. Gilmer et al. (2018a) argued that attackers are not limited to small perturbations, and can instead construct new images or search for naturally misclassified images. Similarly, Uesato et al. (2018) argue that the near-ubiquitous $\ell _ { p }$ model is merely a convenient local approximation for the true worst-case risk. We follow Goodfellow et al. (2017) in viewing adversarial examples more broadly, as “inputs to machine learning models that an attacker has intentionally designed to cause the model to make a mistake.”
+
+The little prior work studying adversarial examples in RL has assumed an $\ell _ { p }$ -norm threat model. Huang et al. (2017) and Kos and Song (2017) showed that deep RL policies are vulnerable to small perturbations in image observations. Recent work by Lin et al. (2017) generates a sequence of perturbations guiding the victim to a target state. Our work differs from these previous approaches by using a physically realistic threat model that disallows direct modification of the victim’s observations.
+
+Lanctot et al. (2017) showed agents may become tightly coupled to the agents they were trained with. Like adversarial policies, this results in seemingly strong polices failing against new opponents. However, the victims we attack win against a range of opponents, and so are not coupled in this way.
+
+Adversarial training is a common defense to adversarial examples, achieving state-of-the-art robustness in image classification (Xie et al., 2019). Prior work has also applied adversarial training to improve the robustness of deep RL policies, where the adversary exerts a force vector on the victim or varies dynamics parameters such as friction (Pinto et al., 2017; Mandlekar et al., 2017; Pattanaik et al., 2018). Our defense of fine-tuning the victim against the adversary is inspired by this work.
+
+This work follows a rich tradition of worst-case analysis in RL. In robust MDPs, the transition function is chosen adversarially from an uncertainty set (Bagnell et al., 2001; Tamar et al., 2014). Doyle et al. (1996) solve the converse problem: finding the set of transition functions for which a policy is optimal. Methods also exist to verify controllers or find a counterexample to a specification. Bastani et al. (2018) verify decision trees distilled from RL policies, while Ghosh et al. (2018) test black-box closedloop simulations. Ravanbakhsh et al (2016) can even synthesize controllers robust to adversarial disturbances. Unfortunately, these techniques are only practical in simple environments with lowdimensional adversarial disturbances. By contrast, while our method lacks formal guarantees, it can test policies in complex multi-agent tasks and naturally scales with improvements in RL algorithms.
+
+# 3 FRAMEWORK
+
+We model the victim as playing against an opponent in a two-player Markov game (Shapley, 1953). Our threat model assumes the attacker can control the opponent, in which case we call the opponent an adversary. We denote the adversary and victim by subscript $\alpha$ and $\nu$ respectively. The game $M = ( S , ( \bar { A } _ { \alpha } , A _ { \nu } ) , T , ( R _ { \alpha } , R _ { \nu } ) )$ consists of state set $S$ , action sets $A _ { \alpha }$ and $A _ { \nu }$ , and a joint state transition function $T : S \times A _ { \alpha } \times A _ { \nu } \to \Delta ( S )$ where $\Delta \left( S \right)$ is a probability distribution on $S$ . The reward function $R _ { i } : S \times A _ { \alpha } \times A _ { \nu } \times S \to \mathbb { R }$ for player $i \in \{ \alpha , \nu \}$ depends on the current state, next state and both player’s actions. Each player wishes to maximize their (discounted) sum of rewards.
+
+The adversary is allowed unlimited black-box access to actions sampled from $\pi _ { v }$ , but is not given any white-box information such as weights or activations. We further assume the victim agent follows a fixed stochastic policy $\pi _ { v }$ , corresponding to the common case of a pre-trained model deployed with static weights. Note that in safety critical systems, where attacks like these would be most concerning, it is standard practice to validate a model and then freeze it, so as to ensure that the deployed model does not develop any new issues due to retraining. Therefore, a fixed victim is a realistic reflection of what we might see with RL-trained policies in real-world settings, such as with autonomous vehicles.
+
+Since the victim policy $\pi _ { \nu }$ is held fixed, the two-player Markov game $M$ reduces to a single-player MDP $M _ { \alpha } = ( S , A _ { \alpha } , T _ { \alpha } , R _ { \alpha } ^ { \prime } )$ that the attacker must solve. The state and action space of the adversary are the same as in $M$ , while the transition and reward function have the victim policy $\pi _ { \nu }$ embedded:
+
+$$
+T _ { \alpha } \left( s , a _ { \alpha } \right) = T \left( s , a _ { \alpha } , a _ { \nu } \right) \qquad \mathrm { a n d } \qquad R _ { \alpha } ^ { \prime } ( s , a _ { \alpha } , s ^ { \prime } ) = R _ { \alpha } ( s , a _ { \alpha } , a _ { \nu } , s ^ { \prime } ) ,
+$$
+
+where the victim’s action is sampled from the stochastic policy $a _ { \nu } \sim \pi _ { \nu } ( \cdot \mid s )$ . The goal of the attacker is to find an adversarial policy $\pi _ { \alpha }$ maximizing the sum of discounted rewards:
+
+$$
+\sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R _ { \alpha } \big ( s ^ { ( t ) } , a _ { \alpha } ^ { ( t ) } , s ^ { ( t + 1 ) } \big ) , \quad \mathrm { w h e r e ~ } s ^ { ( t + 1 ) } \sim T _ { \alpha } \big ( s ^ { ( t ) } , a _ { \alpha } ^ { ( t ) } \big ) \mathrm { ~ a n d ~ } a _ { \alpha } \sim \pi _ { \alpha } \big ( \cdot \big \vert s ^ { ( t ) } \big ) .
+$$
+
+Note the MDP’s dynamics $T _ { \alpha }$ will be unknown even if the Markov game’s dynamics $T$ are known since the victim policy $\pi _ { \nu }$ is a black-box. Consequently, the attacker must solve an RL problem.
+
+# 4 FINDING ADVERSARIAL POLICIES
+
+We demonstrate the existence of adversarial policies in zero-sum simulated robotics games. First, we describe how the victim policies were trained and the environments they operate in. Subsequently, we provide details of our attack method in these environments, and describe several baselines. Finally, we present a quantitative and qualitative evaluation of the adversarial policies and baseline opponents.
+
+![](images/63ad992ade8713c264a87ecf74028ea92cf983b9f0479c76b2508ed9933371d0.jpg)  
+Figure 2: Illustrations of the zero-sum simulated robotics games from Bansal et al. (2018a) we use for evaluation. Environments are further described in Section 4.1.
+
+# 4.1 ENVIRONMENTS AND VICTIM POLICIES
+
+We attack victim policies for the zero-sum simulated robotics games created by Bansal et al. (2018a), illustrated in Figure 2. The victims were trained in pairs via self-play against random old versions of their opponent, for between 680 and 1360 million timesteps. We use the pre-trained policy weights released in the “agent zoo” of Bansal et al. (2018b). In symmetric environments, the zoo agents are labeled ZooN where $N$ is a random seed. In asymmetric environments, they are labeled ZooVN and ZooON representing the Victim and Opponent agents.
+
+All environments are two-player games in the MuJoCo robotics simulator. Both agents observe the position, velocity and contact forces of joints in their body, and the position of their opponent’s joints. The episodes end when a win condition is triggered, or after a time limit, in which case the agents draw. We evaluate in all environments from Bansal et al. (2018a) except for Run to Goal, which we omit as the setup is identical to You Shall Not Pass except for the win condition. We describe the environments below, and specify the number of trained zoo policies and their type (MLP or LSTM):
+
+Kick and Defend (3, LSTM). A soccer penalty shootout between two Humanoid robots. The positions of the kicker, goalie and ball are randomly initialized. The kicker wins if the ball goes between the goalposts; otherwise, the goalie wins, provided it remains within 3 units of the goal.
+
+You Shall Not Pass (1, MLP). Two Humanoid agents are initialized facing each other. The runner wins if it reaches the finish line; the blocker wins if it does not.
+
+Sumo Humans (3, LSTM). Two Humanoid agents compete on a round arena. The players’ positions are randomly initialized. A player wins by remaining standing after their opponent has fallen.2
+
+Sumo Ants (4, LSTM). The same task as Sumo Humans, but with ‘Ant’ quadrupedal robot bodies.   
+We use this task in Section 5.2 to investigate the importance of dimensionality to this attack method.
+
+# 4.2 METHODS EVALUATED
+
+Following the RL formulation in Section 3, we train an adversarial policy to maximize Equation 1 using Proximal Policy Optimization (PPO; Schulman et al., 2017). We give a sparse reward at the end of the episode, positive when the adversary wins the game and negative when it loses or ties. Bansal et al. (2018a) trained the victim policies using a similar reward, with an additional dense component at the start of training. We train for 20 million timesteps using the PPO implementation from Stable Baselines (Hill et al., 2019). The hyperparameters were selected through a combination of manual tuning and a random search of 100 samples; see Section A in the appendix for details. We compare our methods to three baselines: a policy Rand taking random actions; a lifeless policy Zero that exerts zero control; and all pre-trained policies $\operatorname { Z O O } \star$ from Bansal et al. (2018a).
+
+# 4.3 RESULTS
+
+Quantitative Evaluation We find the adversarial policies reliably win against most victim policies, and outperform the pre-trained $\mathrm { { } } Z \circ \circ$ baseline for a majority of environments and victims. We report
+
+![](images/5fdb97b1cc131433ce97f556e45f0533c0970fc1aceba0d83f92e8aac651a348.jpg)  
+Figure 3: Win rates while training adversary Adv against the median victim in each environment (based on the difference between the win rate for Adv and Zoo). The adversary outperforms the $\mathrm { { } ^ { \it { Z } O O } }$ baseline against the median victim in Kick and Defend and You Shall Not Pass, and is competitive on Sumo Humans. For full results, see figure 4 below or figure C.1 in the supplementary material.
+
+Key: The solid line shows the median win rate for Adv across 5 random seeds, with the shaded region representing the minimum and maximum. The win rate is smoothed with a rolling average over 100 000 timesteps. Baselines are shown as horizontal dashed lines. Agents Rand and Zero take random and zero actions respectively. The $\mathrm { { } ^ { \it { Z } O O } }$ baseline is whichever ZooM (Sumo) or ZooOM (other environments) agent achieves the highest win rate. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the title above each figure.
+
+the win rate over time against the median victim in each environment in Figure 3, with full results in Figure C.1 in the supplementary material. Win rates against all victims are summarized in Figure 4.
+
+Qualitative Evaluation The adversarial policies beat the victim not by performing the intended task (e.g. blocking a goal), but rather by exploiting weaknesses in the victim’s policy. This effect is best seen by watching the videos at https://adversarialpolicies.github.io/. In Kick and Defend and You Shall Not Pass, the adversarial policy never stands up. The adversary instead wins by positioning their body to induce adversarial observations that cause the victim’s policy to take poor actions. A robust victim could easily win, a result we demonstrate in Section 5.1.
+
+This flavor of attacks is impossible in Sumo Humans, since the adversarial policy immediately loses if it falls over. Faced with this control constraint, the adversarial policy learns a more high-level strategy: it kneels in the center in a stable position. Surprisingly, this is very effective against victim 1, which in $8 8 \%$ of cases falls over attempting to tackle the adversary. However, it proves less effective against victims 2 and 3, achieving only a $62 \%$ and $45 \%$ win rate, below $\mathrm { { } } Z \circ \circ$ baselines. We further explore the importance of the number of dimensions the adversary can safely manipulate in Section 5.2.
+
+Distribution Shift One might wonder if the adversarial policies win because they are outside the training distribution of the victim. To test this, we evaluate victims against two simple off-distribution baselines: a random policy Rand (green) and a lifeless policy Zero (red). These baselines win as often as $30 \%$ to $50 \%$ in Kick and Defend, but less than $1 \%$ of the time in Sumo and You Shall Not Pass. This is well below the performance of our adversarial policies. We conclude that most victim policies are robust to off-distribution observations that are not adversarially optimized.
+
+# 5 UNDERSTANDING ADVERSARIAL POLICIES
+
+In the previous section we demonstrated adversarial policies exist for victims in a range of competitive simulated robotics environments. In this section, we focus on understanding why these policies exist. Specifically, we establish that adversarial policies manipulate the victim through their body position; that victims are more vulnerable to adversarial policies in high-dimensional environments; and that activations of the victim’s policy network differ substantially when playing an adversarial opponent.
+
+![](images/6ec52e417bcb3e7267f3fb29820029d441bddf2af7832124fe38958859ecbb0c.jpg)  
+Figure 4: Percentage of games won by opponent (out of 1000), the maximal cell in each row is in red. Key: Agents ZooYN are pre-trained policies from Bansal et al. (2018a), where $Y \in \{ { } ^ { \cdot } V ^ { \prime } , { } ^ { \cdot } O ^ { \prime } , { } ^ { \cdot \prime } \}$ denotes the agent plays as (V)ictim, (O)pponent or either side, and $N$ is a random seed. Opponents AdvN are the best adversarial policy of 5 seeds trained against the corresponding Zoo[V]N. Agents Rand and $\mathtt { Z e r o }$ are baseline agents taking random and zero actions respectively. Defended victims ZooXYN, where $X \in \{ \cdot S ^ { \prime } , \cdot \bar { D ^ { \prime } } , \cdot M ^ { \prime } \}$ , are derived from ZooYN by fine-tuning against a (S)ingle opponent AdvN, $\mathbf { ( D ) }$ ual opponents AdvN and Zoo[O]N, or by (M)asking the observations.
+
+(a) Gaussian Mixture Model (GMM): likelihood the activations of a victim’s policy network are “normal”. We collect activations for 20, 000 timesteps of victim Zoo[V]1 playing against each opponent. We fit a 20-component GMM to activations induced by Zoo[O]1. Error bars are a $9 5 \%$ confidence interval.
+
+![](images/ad1157c8cc95837c6beb757ea670845f8bdc73f610576c66c9a437afeabcc62f.jpg)  
+(b) t-SNE activations of Kick and Defend victim ZooV2 playing against different opponents. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. See Figures C.3 and C.4 in the supplementary results for visualizations of other environments and victims.
+
+Figure 5: Analysis of activations of the victim’s policy network. Both figures show the adversary Adv induces off-distribution activations. Key: legends specify opponent the victim played against. Adv is the best adversary trained against the victim, and Rand is a policy taking random actions. $\mathrm { ~ Z 0 O } \star \mathrm { N }$ corresponds to ZooN (Sumo) or ZooON (otherwise). ${ \mathrm { 2 0 0 } } \star { \mathrm { 1 T } }$ and ${ \mathsf { Z } } { \mathsf { O } } { \mathsf { O } } \star { \mathsf { 1 } } { \mathsf { V } }$ are the train and validation datasets, drawn from Zoo1 (Sumo) or ZooO1 (otherwise).
+
+# 5.1 MASKED POLICIES
+
+We have previously shown that adversarial policies are able to reliably win against victims. In this section, we demonstrate that they win by taking actions to induce natural observations that are adversarial to the victim, and not by physically interfering with the victim. To test this, we introduce a ‘masked‘ victim (labeled ZooMN or ZooMVN) that is the same as the normal victim ZooN or ZooVN, except the observation of the adversary’s position is set to a static value corresponding to a typical initial position. We use the same adversarial policy against the normal and masked victim.
+
+One would expect it to be beneficial to be able to see your opponent. Indeed, the masked victims do worse than a normal victim when playing normal opponents. For example, Figure 4b shows that in You Shall Not Pass the normal opponent ZooO1 wins $78 \%$ of the time against the masked victim ZooMV1 but only $47 \%$ of the time against the normal victim ZooV1. However, the relationship is reversed when playing an adversary. The normal victim $\mathrm { ~ Z o o V 1 }$ loses $86 \%$ of the time to adversary Adv1 whereas the masked victim ZooMV1 wins $9 9 \%$ of the time. This pattern is particularly clear in You Shall Not Pass, but the trend is similar in other environments.
+
+This result is surprising as it implies highly non-transitive relationships may exist between policies even in games that seem to be transitive. A game is said to be transitive if policies can be ranked such that higher-ranked policies beat lower-ranked policies. Prima facie, the games in this paper seem transitive: professional human soccer players and sumo wrestlers can reliably beat amateurs. Despite this, there is a non-transitive relationship between adversarial policies, victims and masked victims. Consequently, we urge caution when using methods such as self-play that assume transitivity, and would recommend more general methods where practical (Balduzzi et al., 2019; Brown et al., 2019).
+
+Our findings also suggest a trade-off in the size of the observation space. In benign environments, allowing more observation of the environment increases performance. However, this also makes the agent more vulnerable to adversaries. This is in contrast to an idealized Bayesian agent, where the value of information is always non-negative (Good, 1967). In the following section, we investigate further the connection between vulnerability to attack and the size of the observation space.
+
+# 5.2 DIMENSIONALITY
+
+A variety of work has concluded that classifiers are more vulnerable to adversarial examples on highdimensional inputs (Gilmer et al., 2018b; Khoury and Hadfield-Menell, 2018; Shafahi et al., 2019). We hypothesize a similar result for RL policies: the greater the dimensionality of the component $P$ of the observation space under control of the adversary, the more vulnerable the victim is to attack. We test this hypothesis in the Sumo environment, varying whether the agents are Ants or Humanoids. The results in Figures 4c and 4d support the hypothesis. The adversary has a much lower win-rate in the low-dimensional Sumo Ants (dim $1 P = 1 5$ ) environment than in the higher dimensional Sumo Humans (d $\operatorname* { l i m } P = 2 4$ ) environment, where $P$ is the position of the adversary’s joints.
+
+# 5.3 VICTIM ACTIVATIONS
+
+In Section 5.1 we showed that adversarial policies win by creating natural observations that are adversarial to the victim. In this section, we seek to better understand why these observations are adversarial. We record activations from each victim’s policy network playing a range of opponents, and analyze these using a Gaussian Mixture Model (GMM) and a t-SNE visualization. See Section B in the supplementary material for details of training and hyperparameters.
+
+We fit a GMM on activations ${ \mathrm { Z o o } } \star 1 { \mathrm { T } }$ collected playing against a normal opponent, Zoo1 or ZooV1, holding out ${ \mathrm { 2 0 0 + 1 0 } }$ for validation. Figure 5a shows that the adversarial policy Adv induces activations with the lowest log-likelihood, with random baseline Rand only slightly more probable. Normal opponents ${ \mathrm { Z o o } } \star 2$ and $\mathrm { Z o o } { \star 3 }$ induce activations with almost as high likelihood as the validation set ${ \mathrm { 2 0 0 \star 1 0 } }$ , except in Sumo Humans where they are as unlikely as Rand.
+
+We plot a t-SNE visualization of the activations of Kick and Defend victim ZooV2 in Figure 5b. As expected from the density model results, there is a clear separation between between Adv, Rand and the normal opponent ZooO2. Intriguingly, Adv induces activations more widely dispersed than the random policy Rand, which in turn are more widely dispersed than ZooO2. We report on the full set of victim policies in Figures C.3 and C.4 in the supplementary material.
+
+# 6 DEFENDING AGAINST ADVERSARIAL POLICIES
+
+The ease with which policies can be attacked highlights the need for effective defenses. A natural defense is to fine-tune the victim zoo policy against an adversary, which we term single training. We also investigate dual training, randomly picking either an adversary or a zoo policy at the start of each episode. The training procedure is otherwise the same as for adversaries, described in Section 4.2.
+
+We report on the win rates in You Shall Not Pass in Figure 4b. We find both the single ZooSV1 and dual ZooDV1 fine-tuned victims are robust to adversary Adv1, with the adversary win rate dropping from $87 \%$ to around $10 \%$ . However, ZooSV1 catastrophically forgets how to play against the normal opponent ZooO1. The dual fine-tuned victim ZooDV1 fares better, with opponent ZooO1 winning only $57 \%$ of the time. However, this is still an increase from ZooO1’s $48 \%$ win rate against the original victim ZooV1. This suggests ZooV1 may use features that are helpful against a normal opponent but which are easily manipulable (Ilyas et al., 2019).
+
+Although the fine-tuned victims are robust to the original adversarial policy Adv1, they are still vulnerable to our attack method. New adversaries AdvS1 and AdvD1 trained against ZooSV1 and ZooDV1 win at equal or greater rates than before, and transfer successfully to the original victim. However, the new adversaries AdvS1 and AdvD1 are qualitatively different, tripping the victim up by lying prone on the ground, whereas Adv1 causes ZooV1 to fall without ever touching it.
+
+# 7 DISCUSSION
+
+Contributions. Our paper makes three key contributions. First, we have proposed a novel threat model of natural adversarial observations produced by an adversarial policy taking actions in a shared environment. Second, we demonstrate that adversarial policies exist in a range of zero-sum simulated robotics games against state-of-the-art victims trained via self-play to be robust to adversaries. Third, we verify the adversarial policies win by confusing the victim, not by learning a generally strong policy. Specifically, we find the adversary induces highly off-distribution activations in the victim, and that victim performance increases when it is blind to the adversary’s position.
+
+Self-play. While it may at first appear unsurprising that a policy trained as an adversary against another RL policy would be able to exploit it, we believe that this observation is highly significant. The policies we have attacked were explicitly trained via self-play to be robust. Although it is known that self-play with deep RL may not converge, or converge only to a local rather than global Nash, self-play has been used with great success in a number of works focused on playing adversarial games directly against humans (Silver et al., 2018; OpenAI, 2018). Our work shows that even apparently strong self-play policies can harbor serious but hard to find failure modes, demonstrating these theoretical limitations are practically relevant and highlighting the need for careful testing.
+
+Our attack provides some amount of testing by constructively lower-bounding the exploitability of a victim policy – its performance against its worst-case opponent – by training an adversary. Since the victim’s win rate declines against our adversarial policy, we can confirm that the victim and its self-play opponent were not in a global Nash. Notably we expect our attack to succeed even for policies in a local Nash, as the adversary is trained starting from a random point that is likely outside the victim’s attractive basin.
+
+Defense. We implemented a simple defense: fine-tuning the victim against the adversary. We find our attack can be successfully reapplied to beat this defense, suggesting adversarial policies are difficult to eliminate. However, the defense does appear to protect against attacks that rely on confusing the victim: the new adversarial policy is forced to instead trip the victim up. We therefore believe that scaling up this defense is a promising direction for future work. In particular, we envisage a variant of population-based training where new agents are continually added to the pool to promote diversity, and agents train against a fixed opponent for a prolonged period of time to avoid local equilibria.
+
+Conclusion. Overall, we are excited about the implications the adversarial policy model has for the robustness, security and understanding of deep RL policies. Our results show the existence of a previously unrecognized problem in deep RL, and we hope this work encourages other researchers to investigate this area further. Videos and other supplementary material are available online at https://adversarialpolicies.github.io/ and our source code is available on GitHub at https://github.com/HumanCompatibleAI/adversarial-policies.
+
+# ACKNOWLEDGMENTS
+
+We thank Jakob Foerster, Matthew Rahtz, Dylan Hadfield-Menell, Catherine Olsson, Jan Leike, Rohin Shah, Victoria Krakovna, Daniel Filan, Steven Wang, Dawn Song, Sam Toyer and Dan Hendrycks for their suggestion and helpful feedback on earlier drafts of this paper. We thank Chris Northwood for assistance developing the website accompanying this paper. We are also grateful to our anonymous reviewers for valuable feedback and encouragement to explore defenses in this paper.
+
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+A TRAINING: HYPERPARAMETERS AND COMPUTATIONAL INFRASTRUCTURE   
+Table A.1: Hyperparameters for Proximal Policy Optimization.   
+
+<table><tr><td>Parameter</td><td>Value</td><td>Search Range</td><td>Search Distribution</td></tr><tr><td>Total Timesteps</td><td>20 ×106</td><td>[0,40 × 106]</td><td>Manual</td></tr><tr><td>Batch size</td><td>16 384</td><td>[2048, 65 536]</td><td>Log uniform</td></tr><tr><td>Number of environments</td><td>8</td><td>[1,16]</td><td>Manual</td></tr><tr><td>Mini-batches</td><td>4</td><td>[1,128]</td><td>Log uniform</td></tr><tr><td>Epochs per update</td><td>4</td><td>[1,11]</td><td>Uniform</td></tr><tr><td>Learning rate</td><td>3×10-4</td><td>[1 × i0-5,1 × 10-2]</td><td>Log uniform</td></tr><tr><td>Discount</td><td>0.99</td><td></td><td></td></tr><tr><td>Maximum Gradient Norm</td><td>0.5</td><td></td><td></td></tr><tr><td>Clip Range</td><td>0.2</td><td></td><td></td></tr><tr><td>Advantage Estimation Discount</td><td>0.95</td><td></td><td></td></tr><tr><td>Entropy coefficient</td><td>0.0</td><td></td><td></td></tr><tr><td>Value Function Loss Coefficient</td><td>0.5</td><td></td><td></td></tr></table>
+
+Table A.1 specifies the hyperparameters used for training. The number of environments was chosen for performance reasons after observing diminishing returns from using more than 8 parallel environments. The total timesteps was chosen by inspection after observing diminishing returns to additional training. The batch size, mini-batches, epochs per update, entropy coefficient and learning rate were tuned via a random search with 100 samples on two environments, Kick and Defend and Sumo Humans. All other hyperparameters are the defaults in the PPO2 implementation in Stable Baselines (Hill et al., 2019).
+
+We repeated the hyperparameter sweep for fine-tuning victim policies for the defense experiments, but obtained similar results. For simplicity, we therefore chose to use the same hyperparameters throughout.
+
+We used a mixture of in-house and cloud infrastructure to perform these experiments. It takes around 8 hours to train an adversary for a single victim using 4 cores of an Intel Xeon Platinum 8000 (Skylake) processor.
+
+# B ACTIVATION ANALYSIS: T-SNE AND GMM
+
+We collect activations from all feed forward layers of the victim’s policy network. This gives two 64-length vectors, which we concatenate into a single 128-dimension vector for analysis with a Gaussian Mixture Model and a t-SNE representation.
+
+# B.1 T-SNE HYPERPARAMETER SELECTION
+
+We fit models with perplexity 5, 10, 20, 50, 75, 100, 250 and 1000. We chose 250 since qualitatively it produced the clearest visualization of data with a moderate number of distinct clusters.
+
+B.2 GAUSSIAN MIXTURE MODEL HYPERPARAMETER SELECTION
+
+We fit models with 5, 10, 20, 40 and 80 components with a full (unrestricted) and diagonal covariance matrix. We used the Bayesian Information Criterion (BIC) and average log-likelihood on a heldout validation set as criteria for selecting hyperparameters. We found 20 components with a full covariance matrix achieved the lowest BIC and highest validation log-likelihood in the majority of environment-victim pairs, and was the runner-up in the remainder.
+
+# C FIGURES
+
+Supplementary figures are provided on the subsequent pages.
+
+![](images/5ce8d6572b4355a81fe77b7a828799cb26f91b82c957859a4434c50346bfc21d.jpg)  
+Figure C.1: Win rates while training adversary Adv. The adversary exceeds baseline win rates against most victims in Kick and Defend and You Shall Not Pass, is competitive on Sumo Humans, but performs poorly in the low-dimensional Sumo Ants environment. Key: The solid line shows the median win rate for Adv across 5 random seeds, with the shaded region representing the minimum and maximum. The win rate is smoothed with a rolling average over 100 000 timesteps. Baselines are shown as horizontal dashed lines. Agents Rand and Zero take random and zero actions respectively. The Zoo baseline is whichever ZooM (Sumo) or ZooOM (other environments) agent achieves the highest win rate. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the title above each figure.
+
+(a) Kick and Defend   
+
+<table><tr><td></td><td colspan="7">Opponent Win</td><td colspan="7">Victim Win</td><td colspan="7">Ties</td><td></td><td></td><td></td></tr><tr><td>ZooV1</td><td>79 53 50</td><td></td><td></td><td>32</td><td>四</td><td>42</td><td>30 30</td><td>21</td><td>4650</td><td></td><td>67</td><td>10</td><td>57</td><td>6461</td><td>0</td><td></td><td>0</td><td>0</td><td>2</td><td>1</td><td>1</td><td>6</td><td>9</td><td>100</td></tr><tr><td>ZooV2</td><td>54</td><td>因</td><td>63</td><td>9</td><td>80</td><td>63</td><td>50 46</td><td>46</td><td>6</td><td>37</td><td>90</td><td>20</td><td>36</td><td>45 44</td><td>0</td><td>0</td><td>0</td><td></td><td>0</td><td>1</td><td>0</td><td>5 10</td><td></td><td>80</td></tr><tr><td>ZooV3</td><td>31 31</td><td></td><td>4</td><td>50</td><td>66</td><td>64</td><td>2629</td><td></td><td>68 67</td><td>回</td><td>47</td><td>33</td><td>35</td><td>68 62</td><td>1</td><td>2</td><td></td><td>0</td><td>3</td><td>0</td><td>2</td><td>5 8</td><td></td><td>60</td></tr><tr><td>ZooMV1</td><td>1517</td><td></td><td>18</td><td>58</td><td>四</td><td>31</td><td>12 13</td><td>84</td><td>83</td><td>82</td><td>40</td><td>19</td><td>69</td><td>8478</td><td>0</td><td>0</td><td></td><td>0</td><td>2</td><td>2</td><td>0</td><td>4 9</td><td></td><td>40</td></tr><tr><td>ZooMV2</td><td>2728</td><td></td><td>27</td><td>36</td><td>因</td><td>42</td><td>2521</td><td></td><td>73 72</td><td>73</td><td>63</td><td>24</td><td>58</td><td>72</td><td>72</td><td>0</td><td>0</td><td>0</td><td>1</td><td>4</td><td>0</td><td>4</td><td>8</td><td></td></tr><tr><td rowspan="3">ZooMV3</td><td>262423</td><td></td><td></td><td></td><td>47 50</td><td>61</td><td>2119</td><td>74</td><td>75</td><td>77</td><td></td><td>51 48</td><td>36</td><td>74 72</td><td></td><td>0</td><td>1</td><td>0</td><td>2</td><td>1</td><td>3</td><td>5</td><td>10</td><td>20</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>1</td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td><td>0</td></tr><tr><td>[APA</td><td>£APA</td><td>1000Z 7000Z</td><td></td><td></td><td>£000Z</td><td>purd</td><td></td><td></td><td>3</td><td>1000Z</td><td>7000Z</td><td>£000Z</td><td>Purd </td><td>[APV</td><td>2</td><td>3</td><td>1000Z</td><td>7000Z</td><td>£000Z</td><td>Pury</td><td></td><td></td><td></td></tr></table>
+
+(b) You Shall Not Pass   
+
+<table><tr><td rowspan=1 colspan=1>ZooV1</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>81</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>48</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>97 99</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooSV1</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>99</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>91</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>88 82</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooDV1</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>91</td><td rowspan=1 colspan=1>88</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=2>8  6</td><td rowspan=1 colspan=1>89</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>43</td><td rowspan=1 colspan=1>9294</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooMV1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>83</td><td rowspan=1 colspan=1>70</td><td rowspan=1 colspan=1>78</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>99</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>9999</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=3 colspan=2>1</td><td rowspan=3 colspan=1>ISAPT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=1>A</td><td rowspan=2 colspan=1>0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=2>Pard  o.I9z</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Pard  o.az</td><td rowspan=1 colspan=1>APV</td><td rowspan=1 colspan=1>AAAPT</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td></td></tr></table>
+
+(c) Sumo Humans   
+
+<table><tr><td></td><td colspan="6">Opponent Win</td><td colspan="6">Victim Win</td><td colspan="6">Ties</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Z001</td><td>87</td><td>9</td><td>37</td><td>15</td><td>80</td><td>80</td><td>0</td><td>0</td><td>12</td><td>91</td><td>63</td><td>85</td><td>19 18</td><td></td><td>100100</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>2</td><td></td><td>0</td><td>0</td><td></td><td>100</td></tr><tr><td>Z002</td><td>10</td><td>63</td><td>14</td><td>34</td><td>54</td><td>71</td><td>0</td><td>0</td><td>76</td><td>19</td><td>64</td><td>62</td><td>35 回</td><td></td><td>100100</td><td>141</td><td>18</td><td>22</td><td>4</td><td>11</td><td>13</td><td></td><td>0</td><td>0</td><td></td><td>80</td></tr><tr><td>Z003</td><td>1718</td><td></td><td>44</td><td>10 31</td><td></td><td>57</td><td>0</td><td>1</td><td>81</td><td>79</td><td>2</td><td>88</td><td>61</td><td>34</td><td>100 99</td><td>2</td><td>4</td><td>33</td><td></td><td>1</td><td>8</td><td>9</td><td>0</td><td>0</td><td></td><td>60</td></tr><tr><td>ZooM1</td><td></td><td>8724</td><td>58</td><td>9391 92</td><td></td><td></td><td>0</td><td>1</td><td>12</td><td>76</td><td>42</td><td>□</td><td>9</td><td>8</td><td>100 99</td><td>0</td><td>0</td><td></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td><td>40</td></tr><tr><td>Z0oM2</td><td></td><td>3239</td><td>39</td><td>46</td><td></td><td>69</td><td>0</td><td>0</td><td>39</td><td>30</td><td>23</td><td>50</td><td>四</td><td>17</td><td>100100</td><td></td><td>29 3238</td><td></td><td></td><td>4</td><td>6</td><td>14</td><td>0</td><td>0</td><td></td><td>20</td></tr><tr><td>ZooM3</td><td></td><td>6268</td><td>75</td><td>66</td><td>82</td><td>四</td><td>1</td><td>1</td><td>30</td><td>20</td><td>10</td><td>31</td><td>13</td><td>□</td><td>99 99</td><td></td><td>9 1215</td><td></td><td></td><td>3</td><td>6</td><td>2</td><td>0</td><td>0</td><td></td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td>pury</td><td></td><td></td><td></td><td></td><td>[00Z</td><td>20</td><td>£00Z</td><td>puey</td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td><td></td><td></td><td>0</td></tr><tr><td></td><td>[APV</td><td>APA</td><td>1</td><td></td><td>2</td><td>2</td><td></td><td>0.I9z</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>00</td><td>[APV</td><td></td><td>3</td><td>[00Z</td><td>20</td><td>20</td><td>pued</td><td>20</td><td></td><td></td></tr></table>
+
+![](images/43e98aa0e379df5d4eb5e0e40961ab77cd0a094b151a07cfc26f1a9abd00e1cf.jpg)  
+Figure C.2: Percentage of episodes (out of 1000) won by the opponent, the victim or tied. The maximal opponent win rate in each row is in red. Victims are on the $y$ -axis and opponents on the $x$ -axis. Key: Agents ZooYN are pre-trained policies from Bansal et al. (2018a), where $Y \in \{ { } ^ { \cdot } V ^ { \prime } , { } ^ { \cdot } O ^ { \prime } , { } ^ { \cdot \prime } \}$ denotes the agent plays as (V)ictim, $\mathbf { \eta } ^ { ( \mathbf { 0 } ) }$ pponent or either side, and $N$ is a random seed. Opponents AdvN are the best adversarial policy of 5 seeds trained against the corresponding Zoo[V]N. Agents Rand and $\mathtt { Z e r o }$ are baseline agents taking random and zero actions respectively. Defended victims ZooXYN, where $X \in \{ \cdot S ^ { \prime } , \cdot \bar { D ^ { \prime } } , \cdot M ^ { \prime } \}$ , are derived from ZooYN by fine-tuning against a (S)ingle opponent AdvN, $\mathbf { ( D ) }$ ual opponents AdvN and Zoo[O]N, or by $\mathbf { \Psi } ^ { ( \mathbf { M } ) }$ asking the observations.
+
+![](images/305c21d93e648995cbf0f7f4c0ba99d2806afa39b254e780b4e6fcbbb26a5652.jpg)  
+Figure C.3: t-SNE activations of the victim when playing against different opponents. There is a clear separation between the activations induced by Adv and those of the normal opponent Zoo. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the figure caption. Opponent Adv is the best adversary trained against the victim. Opponent $\mathrm { { } ~ \ Z \ O O }$ corresponds to ZooN (Sumo) or ZooON (other environments). See Figure C.4 for activations for a single opponent at a time.
+
+![](images/bdce657631a365c6b4b9c7f5106a3f5b5288e8879e2f26cf31d40a00d4e27fb3.jpg)  
+Figure C.4: t-SNE activations of victim Zoo1 (Sumo) or ZooV1 (other environments). The results are the same as in Figure C.3 but decomposed into individual opponents for clarity. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. Opponent Adv is the best adversary trained against the victim. Opponent Zoo is Zoo1 (Sumo) or ZooO1 (other environments). See Figure C.3 for results for other victims (one plot per victim).

parse/train/HJgEMpVFwB/HJgEMpVFwB_content_list.json

@@ -0,0 +1,1413 @@
+[
+    {
+        "type": "text",
+        "text": "ADVERSARIAL POLICIES: ATTACKING DEEP REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            633,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adam Gleave1 Michael Dennis Cody Wild Neel Kant Sergey Levine Stuart Russell University of California, Berkeley ",
+        "bbox": [
+            176,
+            165,
+            816,
+            195
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "bbox": [
+            454,
+            207,
+            544,
+            222
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep reinforcement learning (RL) policies are known to be vulnerable to adversarial perturbations to their observations, similar to adversarial examples for classifiers. However, an attacker is not usually able to directly modify another agent’s observations. This might lead one to wonder: is it possible to attack an RL agent simply by choosing an adversarial policy acting in a multi-agent environment so as to create natural observations that are adversarial? We demonstrate the existence of adversarial policies in zero-sum games between simulated humanoid robots with proprioceptive observations, against state-of-the-art victims trained via self-play to be robust to opponents. The adversarial policies reliably win against the victims but generate seemingly random and uncoordinated behavior. We find that these policies are more successful in high-dimensional environments, and induce substantially different activations in the victim policy network than when the victim plays against a normal opponent. Fine-tuning protects a victim against a specific adversary, but the attack method can be successfully reapplied to find a new adversarial policy. Videos are available at https://adversarialpolicies.github.io/. ",
+        "bbox": [
+            232,
+            239,
+            766,
+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            476,
+            336,
+            491
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The discovery of adversarial examples for image classifiers prompted a new field of research into adversarial attacks and defenses (Szegedy et al., 2014). Recent work has shown that deep RL policies are also vulnerable to adversarial perturbations of image observations (Huang et al., 2017; Kos and Song, 2017). However, real-world RL agents inhabit natural environments populated by other agents, including humans, who can only modify another agent’s observations via their actions. We explore whether it’s possible to attack a victim policy by building an adversarial policy that takes actions in a shared environment, inducing natural observations which have adversarial effects on the victim. ",
+        "bbox": [
+            174,
+            507,
+            825,
+            604
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "RL has been applied in settings as varied as autonomous driving (Dosovitskiy et al., 2017), negotiation (Lewis et al., 2017) and automated trading (Noonan, 2017). In domains such as these, an attacker cannot usually directly modify the victim policy’s input. For example, in autonomous driving pedestrians and other drivers can take actions in the world that affect the camera image, but only in a physically realistic fashion. They cannot add noise to arbitrary pixels, or make a building disappear. Similarly, in financial trading an attacker can send orders to an exchange which will appear in the victim’s market data feed, but the attacker cannot modify observations of a third party’s orders. ",
+        "bbox": [
+            174,
+            612,
+            825,
+            709
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As a proof of concept, we show the existence of adversarial policies in zero-sum simulated robotics games with proprioceptive observations (Bansal et al., 2018a). The state-of-the-art victim policies were trained via self-play to be robust to opponents. We train each adversarial policy using model-free RL against a fixed black-box victim. We find the adversarial policies reliably beat their victim, despite training for less than $3 \\%$ of the timesteps initially used to train the victim policies. ",
+        "bbox": [
+            174,
+            717,
+            823,
+            786
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Critically, we find the adversaries win by creating natural observations that are adversarial, and not by becoming generally strong opponents. Qualitatively, the adversaries fall to the ground in contorted positions, as illustrated in Figure 1, rather than learning to run, kick or block like normal opponents. This strategy does not work when the victim is ‘masked’ and cannot see the adversary’s position, suggesting that the adversary succeeds by manipulating a victim’s observations through its actions. ",
+        "bbox": [
+            174,
+            792,
+            825,
+            863
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Having observed these results, we wanted to understand the sensitivity of the attack to the dimensionality of the victim’s observations. We find that victim policies in higher-dimensional Humanoid environments are substantially more vulnerable to adversarial policies than in lower-dimensional Ant environments. To gain insight into why adversarial policies succeed, we analyze the activations of the victim’s policy network using a Gaussian Mixture Model and t-SNE (Maaten and Hinton, 2008). We find adversarial policies induce significantly different activations than normal opponents, and that the adversarial activations are typically more widely dispersed between timesteps than normal activations. ",
+        "bbox": [
+            176,
+            869,
+            823,
+            898
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/bed7c6d84ef604ab225fc38240a0734aa665e13269680754a2d339a48b73f268.jpg",
+        "image_caption": [
+            "Figure 1: Illustrative snapshots of a victim (in blue) against normal and adversarial opponents (in red). The victim wins if it crosses the finish line; otherwise, the opponent wins. Despite never standing up, the adversarial opponent wins $86 \\%$ of episodes, far above the normal opponent’s $47 \\%$ win rate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            101,
+            813,
+            285
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            366,
+            825,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A natural defense is to fine-tune the victim against the adversary. We find this protects against that particular adversary, but that repeating the attack method finds a new adversary the fine-tuned victim is vulnerable to. However, this new adversary differs qualitatively by physically interfering with the victim. This suggests repeated fine-tuning might provide protection against a range of adversaries. ",
+        "bbox": [
+            174,
+            458,
+            825,
+            513
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our paper makes three contributions. First, we propose a novel, physically realistic threat model for adversarial examples in RL. Second, we demonstrate the existence of adversarial policies in this threat model for several simulated robotics games. Our adversarial policies reliably beat the victim, despite training with less than $3 \\%$ as many timesteps and generating seemingly random behavior. Third, we conduct a detailed analysis of why the adversarial policies work. We show they create natural observations that are adversarial to the victim and push the activations of the victim’s policy network off-distribution. Additionally, we find policies are easier to attack in high-dimensional environments. ",
+        "bbox": [
+            174,
+            520,
+            825,
+            618
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As deep RL is increasingly deployed in environments with potential adversaries, we believe it is important that practitioners are aware of this previously unrecognized threat model. Moreover, even in benign settings, we believe adversarial policies can be a useful tool for uncovering unexpected policy failure modes. Finally, we are excited by the potential of adversarial training using adversarial policies, which could improve robustness relative to conventional self-play by training against adversaries that exploit weaknesses undiscovered by the distribution of similar opponents present during self-play. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            709
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            729,
+            344,
+            746
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Most study of adversarial examples has focused on small $\\ell _ { p }$ norm perturbations to images, which Szegedy et al. (2014) discovered cause a variety of models to confidently misclassify the image, even though the changes are visually imperceptible to a human. Gilmer et al. (2018a) argued that attackers are not limited to small perturbations, and can instead construct new images or search for naturally misclassified images. Similarly, Uesato et al. (2018) argue that the near-ubiquitous $\\ell _ { p }$ model is merely a convenient local approximation for the true worst-case risk. We follow Goodfellow et al. (2017) in viewing adversarial examples more broadly, as “inputs to machine learning models that an attacker has intentionally designed to cause the model to make a mistake.” ",
+        "bbox": [
+            174,
+            762,
+            825,
+            875
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The little prior work studying adversarial examples in RL has assumed an $\\ell _ { p }$ -norm threat model. Huang et al. (2017) and Kos and Song (2017) showed that deep RL policies are vulnerable to small perturbations in image observations. Recent work by Lin et al. (2017) generates a sequence of perturbations guiding the victim to a target state. Our work differs from these previous approaches by using a physically realistic threat model that disallows direct modification of the victim’s observations. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Lanctot et al. (2017) showed agents may become tightly coupled to the agents they were trained with. Like adversarial policies, this results in seemingly strong polices failing against new opponents. However, the victims we attack win against a range of opponents, and so are not coupled in this way. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Adversarial training is a common defense to adversarial examples, achieving state-of-the-art robustness in image classification (Xie et al., 2019). Prior work has also applied adversarial training to improve the robustness of deep RL policies, where the adversary exerts a force vector on the victim or varies dynamics parameters such as friction (Pinto et al., 2017; Mandlekar et al., 2017; Pattanaik et al., 2018). Our defense of fine-tuning the victim against the adversary is inspired by this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work follows a rich tradition of worst-case analysis in RL. In robust MDPs, the transition function is chosen adversarially from an uncertainty set (Bagnell et al., 2001; Tamar et al., 2014). Doyle et al. (1996) solve the converse problem: finding the set of transition functions for which a policy is optimal. Methods also exist to verify controllers or find a counterexample to a specification. Bastani et al. (2018) verify decision trees distilled from RL policies, while Ghosh et al. (2018) test black-box closedloop simulations. Ravanbakhsh et al (2016) can even synthesize controllers robust to adversarial disturbances. Unfortunately, these techniques are only practical in simple environments with lowdimensional adversarial disturbances. By contrast, while our method lacks formal guarantees, it can test policies in complex multi-agent tasks and naturally scales with improvements in RL algorithms. ",
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+    {
+        "type": "text",
+        "text": "3 FRAMEWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We model the victim as playing against an opponent in a two-player Markov game (Shapley, 1953). Our threat model assumes the attacker can control the opponent, in which case we call the opponent an adversary. We denote the adversary and victim by subscript $\\alpha$ and $\\nu$ respectively. The game $M = ( S , ( \\bar { A } _ { \\alpha } , A _ { \\nu } ) , T , ( R _ { \\alpha } , R _ { \\nu } ) )$ consists of state set $S$ , action sets $A _ { \\alpha }$ and $A _ { \\nu }$ , and a joint state transition function $T : S \\times A _ { \\alpha } \\times A _ { \\nu } \\to \\Delta ( S )$ where $\\Delta \\left( S \\right)$ is a probability distribution on $S$ . The reward function $R _ { i } : S \\times A _ { \\alpha } \\times A _ { \\nu } \\times S \\to \\mathbb { R }$ for player $i \\in \\{ \\alpha , \\nu \\}$ depends on the current state, next state and both player’s actions. Each player wishes to maximize their (discounted) sum of rewards. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The adversary is allowed unlimited black-box access to actions sampled from $\\pi _ { v }$ , but is not given any white-box information such as weights or activations. We further assume the victim agent follows a fixed stochastic policy $\\pi _ { v }$ , corresponding to the common case of a pre-trained model deployed with static weights. Note that in safety critical systems, where attacks like these would be most concerning, it is standard practice to validate a model and then freeze it, so as to ensure that the deployed model does not develop any new issues due to retraining. Therefore, a fixed victim is a realistic reflection of what we might see with RL-trained policies in real-world settings, such as with autonomous vehicles. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the victim policy $\\pi _ { \\nu }$ is held fixed, the two-player Markov game $M$ reduces to a single-player MDP $M _ { \\alpha } = ( S , A _ { \\alpha } , T _ { \\alpha } , R _ { \\alpha } ^ { \\prime } )$ that the attacker must solve. The state and action space of the adversary are the same as in $M$ , while the transition and reward function have the victim policy $\\pi _ { \\nu }$ embedded: ",
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+    {
+        "type": "equation",
+        "img_path": "images/56bc8b8d1ccbb03d81b6031c2b5128aeb8efa33d44c3496beeea3630fe691ff8.jpg",
+        "text": "$$\nT _ { \\alpha } \\left( s , a _ { \\alpha } \\right) = T \\left( s , a _ { \\alpha } , a _ { \\nu } \\right) \\qquad \\mathrm { a n d } \\qquad R _ { \\alpha } ^ { \\prime } ( s , a _ { \\alpha } , s ^ { \\prime } ) = R _ { \\alpha } ( s , a _ { \\alpha } , a _ { \\nu } , s ^ { \\prime } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where the victim’s action is sampled from the stochastic policy $a _ { \\nu } \\sim \\pi _ { \\nu } ( \\cdot \\mid s )$ . The goal of the attacker is to find an adversarial policy $\\pi _ { \\alpha }$ maximizing the sum of discounted rewards: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ad21be910b629356f0d71badbef7892f53b1838688ea7cf4eebeb074d95b9c2d.jpg",
+        "text": "$$\n\\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R _ { \\alpha } \\big ( s ^ { ( t ) } , a _ { \\alpha } ^ { ( t ) } , s ^ { ( t + 1 ) } \\big ) , \\quad \\mathrm { w h e r e ~ } s ^ { ( t + 1 ) } \\sim T _ { \\alpha } \\big ( s ^ { ( t ) } , a _ { \\alpha } ^ { ( t ) } \\big ) \\mathrm { ~ a n d ~ } a _ { \\alpha } \\sim \\pi _ { \\alpha } \\big ( \\cdot \\big \\vert s ^ { ( t ) } \\big ) .\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Note the MDP’s dynamics $T _ { \\alpha }$ will be unknown even if the Markov game’s dynamics $T$ are known since the victim policy $\\pi _ { \\nu }$ is a black-box. Consequently, the attacker must solve an RL problem. ",
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+    {
+        "type": "text",
+        "text": "4 FINDING ADVERSARIAL POLICIES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We demonstrate the existence of adversarial policies in zero-sum simulated robotics games. First, we describe how the victim policies were trained and the environments they operate in. Subsequently, we provide details of our attack method in these environments, and describe several baselines. Finally, we present a quantitative and qualitative evaluation of the adversarial policies and baseline opponents. ",
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+        "type": "image",
+        "img_path": "images/63ad992ade8713c264a87ecf74028ea92cf983b9f0479c76b2508ed9933371d0.jpg",
+        "image_caption": [
+            "Figure 2: Illustrations of the zero-sum simulated robotics games from Bansal et al. (2018a) we use for evaluation. Environments are further described in Section 4.1. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 ENVIRONMENTS AND VICTIM POLICIES ",
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+        "text": "We attack victim policies for the zero-sum simulated robotics games created by Bansal et al. (2018a), illustrated in Figure 2. The victims were trained in pairs via self-play against random old versions of their opponent, for between 680 and 1360 million timesteps. We use the pre-trained policy weights released in the “agent zoo” of Bansal et al. (2018b). In symmetric environments, the zoo agents are labeled ZooN where $N$ is a random seed. In asymmetric environments, they are labeled ZooVN and ZooON representing the Victim and Opponent agents. ",
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+        "type": "text",
+        "text": "All environments are two-player games in the MuJoCo robotics simulator. Both agents observe the position, velocity and contact forces of joints in their body, and the position of their opponent’s joints. The episodes end when a win condition is triggered, or after a time limit, in which case the agents draw. We evaluate in all environments from Bansal et al. (2018a) except for Run to Goal, which we omit as the setup is identical to You Shall Not Pass except for the win condition. We describe the environments below, and specify the number of trained zoo policies and their type (MLP or LSTM): ",
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+    {
+        "type": "text",
+        "text": "Kick and Defend (3, LSTM). A soccer penalty shootout between two Humanoid robots. The positions of the kicker, goalie and ball are randomly initialized. The kicker wins if the ball goes between the goalposts; otherwise, the goalie wins, provided it remains within 3 units of the goal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "You Shall Not Pass (1, MLP). Two Humanoid agents are initialized facing each other. The runner wins if it reaches the finish line; the blocker wins if it does not. ",
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+    {
+        "type": "text",
+        "text": "Sumo Humans (3, LSTM). Two Humanoid agents compete on a round arena. The players’ positions are randomly initialized. A player wins by remaining standing after their opponent has fallen.2 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sumo Ants (4, LSTM). The same task as Sumo Humans, but with ‘Ant’ quadrupedal robot bodies.   \nWe use this task in Section 5.2 to investigate the importance of dimensionality to this attack method. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.2 METHODS EVALUATED ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following the RL formulation in Section 3, we train an adversarial policy to maximize Equation 1 using Proximal Policy Optimization (PPO; Schulman et al., 2017). We give a sparse reward at the end of the episode, positive when the adversary wins the game and negative when it loses or ties. Bansal et al. (2018a) trained the victim policies using a similar reward, with an additional dense component at the start of training. We train for 20 million timesteps using the PPO implementation from Stable Baselines (Hill et al., 2019). The hyperparameters were selected through a combination of manual tuning and a random search of 100 samples; see Section A in the appendix for details. We compare our methods to three baselines: a policy Rand taking random actions; a lifeless policy Zero that exerts zero control; and all pre-trained policies $\\operatorname { Z O O } \\star$ from Bansal et al. (2018a). ",
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+    {
+        "type": "text",
+        "text": "4.3 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Quantitative Evaluation We find the adversarial policies reliably win against most victim policies, and outperform the pre-trained $\\mathrm { { } } Z \\circ \\circ$ baseline for a majority of environments and victims. We report ",
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+    {
+        "type": "image",
+        "img_path": "images/5fdb97b1cc131433ce97f556e45f0533c0970fc1aceba0d83f92e8aac651a348.jpg",
+        "image_caption": [
+            "Figure 3: Win rates while training adversary Adv against the median victim in each environment (based on the difference between the win rate for Adv and Zoo). The adversary outperforms the $\\mathrm { { } ^ { \\it { Z } O O } }$ baseline against the median victim in Kick and Defend and You Shall Not Pass, and is competitive on Sumo Humans. For full results, see figure 4 below or figure C.1 in the supplementary material. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Key: The solid line shows the median win rate for Adv across 5 random seeds, with the shaded region representing the minimum and maximum. The win rate is smoothed with a rolling average over 100 000 timesteps. Baselines are shown as horizontal dashed lines. Agents Rand and Zero take random and zero actions respectively. The $\\mathrm { { } ^ { \\it { Z } O O } }$ baseline is whichever ZooM (Sumo) or ZooOM (other environments) agent achieves the highest win rate. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the title above each figure. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the win rate over time against the median victim in each environment in Figure 3, with full results in Figure C.1 in the supplementary material. Win rates against all victims are summarized in Figure 4. ",
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+    {
+        "type": "text",
+        "text": "Qualitative Evaluation The adversarial policies beat the victim not by performing the intended task (e.g. blocking a goal), but rather by exploiting weaknesses in the victim’s policy. This effect is best seen by watching the videos at https://adversarialpolicies.github.io/. In Kick and Defend and You Shall Not Pass, the adversarial policy never stands up. The adversary instead wins by positioning their body to induce adversarial observations that cause the victim’s policy to take poor actions. A robust victim could easily win, a result we demonstrate in Section 5.1. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "This flavor of attacks is impossible in Sumo Humans, since the adversarial policy immediately loses if it falls over. Faced with this control constraint, the adversarial policy learns a more high-level strategy: it kneels in the center in a stable position. Surprisingly, this is very effective against victim 1, which in $8 8 \\%$ of cases falls over attempting to tackle the adversary. However, it proves less effective against victims 2 and 3, achieving only a $62 \\%$ and $45 \\%$ win rate, below $\\mathrm { { } } Z \\circ \\circ$ baselines. We further explore the importance of the number of dimensions the adversary can safely manipulate in Section 5.2. ",
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+    {
+        "type": "text",
+        "text": "Distribution Shift One might wonder if the adversarial policies win because they are outside the training distribution of the victim. To test this, we evaluate victims against two simple off-distribution baselines: a random policy Rand (green) and a lifeless policy Zero (red). These baselines win as often as $30 \\%$ to $50 \\%$ in Kick and Defend, but less than $1 \\%$ of the time in Sumo and You Shall Not Pass. This is well below the performance of our adversarial policies. We conclude that most victim policies are robust to off-distribution observations that are not adversarially optimized. ",
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+    {
+        "type": "text",
+        "text": "5 UNDERSTANDING ADVERSARIAL POLICIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the previous section we demonstrated adversarial policies exist for victims in a range of competitive simulated robotics environments. In this section, we focus on understanding why these policies exist. Specifically, we establish that adversarial policies manipulate the victim through their body position; that victims are more vulnerable to adversarial policies in high-dimensional environments; and that activations of the victim’s policy network differ substantially when playing an adversarial opponent. ",
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+    {
+        "type": "image",
+        "img_path": "images/6ec52e417bcb3e7267f3fb29820029d441bddf2af7832124fe38958859ecbb0c.jpg",
+        "image_caption": [
+            "Figure 4: Percentage of games won by opponent (out of 1000), the maximal cell in each row is in red. Key: Agents ZooYN are pre-trained policies from Bansal et al. (2018a), where $Y \\in \\{ { } ^ { \\cdot } V ^ { \\prime } , { } ^ { \\cdot } O ^ { \\prime } , { } ^ { \\cdot \\prime } \\}$ denotes the agent plays as (V)ictim, (O)pponent or either side, and $N$ is a random seed. Opponents AdvN are the best adversarial policy of 5 seeds trained against the corresponding Zoo[V]N. Agents Rand and $\\mathtt { Z e r o }$ are baseline agents taking random and zero actions respectively. Defended victims ZooXYN, where $X \\in \\{ \\cdot S ^ { \\prime } , \\cdot \\bar { D ^ { \\prime } } , \\cdot M ^ { \\prime } \\}$ , are derived from ZooYN by fine-tuning against a (S)ingle opponent AdvN, $\\mathbf { ( D ) }$ ual opponents AdvN and Zoo[O]N, or by (M)asking the observations. "
+        ],
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+    {
+        "type": "text",
+        "text": "(a) Gaussian Mixture Model (GMM): likelihood the activations of a victim’s policy network are “normal”. We collect activations for 20, 000 timesteps of victim Zoo[V]1 playing against each opponent. We fit a 20-component GMM to activations induced by Zoo[O]1. Error bars are a $9 5 \\%$ confidence interval. ",
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+    {
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+        "img_path": "images/ad1157c8cc95837c6beb757ea670845f8bdc73f610576c66c9a437afeabcc62f.jpg",
+        "image_caption": [
+            "(b) t-SNE activations of Kick and Defend victim ZooV2 playing against different opponents. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. See Figures C.3 and C.4 in the supplementary results for visualizations of other environments and victims. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 5: Analysis of activations of the victim’s policy network. Both figures show the adversary Adv induces off-distribution activations. Key: legends specify opponent the victim played against. Adv is the best adversary trained against the victim, and Rand is a policy taking random actions. $\\mathrm { ~ Z 0 O } \\star \\mathrm { N }$ corresponds to ZooN (Sumo) or ZooON (otherwise). ${ \\mathrm { 2 0 0 } } \\star { \\mathrm { 1 T } }$ and ${ \\mathsf { Z } } { \\mathsf { O } } { \\mathsf { O } } \\star { \\mathsf { 1 } } { \\mathsf { V } }$ are the train and validation datasets, drawn from Zoo1 (Sumo) or ZooO1 (otherwise). ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 MASKED POLICIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We have previously shown that adversarial policies are able to reliably win against victims. In this section, we demonstrate that they win by taking actions to induce natural observations that are adversarial to the victim, and not by physically interfering with the victim. To test this, we introduce a ‘masked‘ victim (labeled ZooMN or ZooMVN) that is the same as the normal victim ZooN or ZooVN, except the observation of the adversary’s position is set to a static value corresponding to a typical initial position. We use the same adversarial policy against the normal and masked victim. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One would expect it to be beneficial to be able to see your opponent. Indeed, the masked victims do worse than a normal victim when playing normal opponents. For example, Figure 4b shows that in You Shall Not Pass the normal opponent ZooO1 wins $78 \\%$ of the time against the masked victim ZooMV1 but only $47 \\%$ of the time against the normal victim ZooV1. However, the relationship is reversed when playing an adversary. The normal victim $\\mathrm { ~ Z o o V 1 }$ loses $86 \\%$ of the time to adversary Adv1 whereas the masked victim ZooMV1 wins $9 9 \\%$ of the time. This pattern is particularly clear in You Shall Not Pass, but the trend is similar in other environments. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This result is surprising as it implies highly non-transitive relationships may exist between policies even in games that seem to be transitive. A game is said to be transitive if policies can be ranked such that higher-ranked policies beat lower-ranked policies. Prima facie, the games in this paper seem transitive: professional human soccer players and sumo wrestlers can reliably beat amateurs. Despite this, there is a non-transitive relationship between adversarial policies, victims and masked victims. Consequently, we urge caution when using methods such as self-play that assume transitivity, and would recommend more general methods where practical (Balduzzi et al., 2019; Brown et al., 2019). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our findings also suggest a trade-off in the size of the observation space. In benign environments, allowing more observation of the environment increases performance. However, this also makes the agent more vulnerable to adversaries. This is in contrast to an idealized Bayesian agent, where the value of information is always non-negative (Good, 1967). In the following section, we investigate further the connection between vulnerability to attack and the size of the observation space. ",
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+    {
+        "type": "text",
+        "text": "5.2 DIMENSIONALITY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A variety of work has concluded that classifiers are more vulnerable to adversarial examples on highdimensional inputs (Gilmer et al., 2018b; Khoury and Hadfield-Menell, 2018; Shafahi et al., 2019). We hypothesize a similar result for RL policies: the greater the dimensionality of the component $P$ of the observation space under control of the adversary, the more vulnerable the victim is to attack. We test this hypothesis in the Sumo environment, varying whether the agents are Ants or Humanoids. The results in Figures 4c and 4d support the hypothesis. The adversary has a much lower win-rate in the low-dimensional Sumo Ants (dim $1 P = 1 5$ ) environment than in the higher dimensional Sumo Humans (d $\\operatorname* { l i m } P = 2 4$ ) environment, where $P$ is the position of the adversary’s joints. ",
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+    {
+        "type": "text",
+        "text": "5.3 VICTIM ACTIVATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Section 5.1 we showed that adversarial policies win by creating natural observations that are adversarial to the victim. In this section, we seek to better understand why these observations are adversarial. We record activations from each victim’s policy network playing a range of opponents, and analyze these using a Gaussian Mixture Model (GMM) and a t-SNE visualization. See Section B in the supplementary material for details of training and hyperparameters. ",
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+        "type": "text",
+        "text": "We fit a GMM on activations ${ \\mathrm { Z o o } } \\star 1 { \\mathrm { T } }$ collected playing against a normal opponent, Zoo1 or ZooV1, holding out ${ \\mathrm { 2 0 0 + 1 0 } }$ for validation. Figure 5a shows that the adversarial policy Adv induces activations with the lowest log-likelihood, with random baseline Rand only slightly more probable. Normal opponents ${ \\mathrm { Z o o } } \\star 2$ and $\\mathrm { Z o o } { \\star 3 }$ induce activations with almost as high likelihood as the validation set ${ \\mathrm { 2 0 0 \\star 1 0 } }$ , except in Sumo Humans where they are as unlikely as Rand. ",
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+    {
+        "type": "text",
+        "text": "We plot a t-SNE visualization of the activations of Kick and Defend victim ZooV2 in Figure 5b. As expected from the density model results, there is a clear separation between between Adv, Rand and the normal opponent ZooO2. Intriguingly, Adv induces activations more widely dispersed than the random policy Rand, which in turn are more widely dispersed than ZooO2. We report on the full set of victim policies in Figures C.3 and C.4 in the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "6 DEFENDING AGAINST ADVERSARIAL POLICIES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The ease with which policies can be attacked highlights the need for effective defenses. A natural defense is to fine-tune the victim zoo policy against an adversary, which we term single training. We also investigate dual training, randomly picking either an adversary or a zoo policy at the start of each episode. The training procedure is otherwise the same as for adversaries, described in Section 4.2. ",
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+    {
+        "type": "text",
+        "text": "We report on the win rates in You Shall Not Pass in Figure 4b. We find both the single ZooSV1 and dual ZooDV1 fine-tuned victims are robust to adversary Adv1, with the adversary win rate dropping from $87 \\%$ to around $10 \\%$ . However, ZooSV1 catastrophically forgets how to play against the normal opponent ZooO1. The dual fine-tuned victim ZooDV1 fares better, with opponent ZooO1 winning only $57 \\%$ of the time. However, this is still an increase from ZooO1’s $48 \\%$ win rate against the original victim ZooV1. This suggests ZooV1 may use features that are helpful against a normal opponent but which are easily manipulable (Ilyas et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Although the fine-tuned victims are robust to the original adversarial policy Adv1, they are still vulnerable to our attack method. New adversaries AdvS1 and AdvD1 trained against ZooSV1 and ZooDV1 win at equal or greater rates than before, and transfer successfully to the original victim. However, the new adversaries AdvS1 and AdvD1 are qualitatively different, tripping the victim up by lying prone on the ground, whereas Adv1 causes ZooV1 to fall without ever touching it. ",
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Contributions. Our paper makes three key contributions. First, we have proposed a novel threat model of natural adversarial observations produced by an adversarial policy taking actions in a shared environment. Second, we demonstrate that adversarial policies exist in a range of zero-sum simulated robotics games against state-of-the-art victims trained via self-play to be robust to adversaries. Third, we verify the adversarial policies win by confusing the victim, not by learning a generally strong policy. Specifically, we find the adversary induces highly off-distribution activations in the victim, and that victim performance increases when it is blind to the adversary’s position. ",
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+    {
+        "type": "text",
+        "text": "Self-play. While it may at first appear unsurprising that a policy trained as an adversary against another RL policy would be able to exploit it, we believe that this observation is highly significant. The policies we have attacked were explicitly trained via self-play to be robust. Although it is known that self-play with deep RL may not converge, or converge only to a local rather than global Nash, self-play has been used with great success in a number of works focused on playing adversarial games directly against humans (Silver et al., 2018; OpenAI, 2018). Our work shows that even apparently strong self-play policies can harbor serious but hard to find failure modes, demonstrating these theoretical limitations are practically relevant and highlighting the need for careful testing. ",
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+    {
+        "type": "text",
+        "text": "Our attack provides some amount of testing by constructively lower-bounding the exploitability of a victim policy – its performance against its worst-case opponent – by training an adversary. Since the victim’s win rate declines against our adversarial policy, we can confirm that the victim and its self-play opponent were not in a global Nash. Notably we expect our attack to succeed even for policies in a local Nash, as the adversary is trained starting from a random point that is likely outside the victim’s attractive basin. ",
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+    {
+        "type": "text",
+        "text": "Defense. We implemented a simple defense: fine-tuning the victim against the adversary. We find our attack can be successfully reapplied to beat this defense, suggesting adversarial policies are difficult to eliminate. However, the defense does appear to protect against attacks that rely on confusing the victim: the new adversarial policy is forced to instead trip the victim up. We therefore believe that scaling up this defense is a promising direction for future work. In particular, we envisage a variant of population-based training where new agents are continually added to the pool to promote diversity, and agents train against a fixed opponent for a prolonged period of time to avoid local equilibria. ",
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+    {
+        "type": "text",
+        "text": "Conclusion. Overall, we are excited about the implications the adversarial policy model has for the robustness, security and understanding of deep RL policies. Our results show the existence of a previously unrecognized problem in deep RL, and we hope this work encourages other researchers to investigate this area further. Videos and other supplementary material are available online at https://adversarialpolicies.github.io/ and our source code is available on GitHub at https://github.com/HumanCompatibleAI/adversarial-policies. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank Jakob Foerster, Matthew Rahtz, Dylan Hadfield-Menell, Catherine Olsson, Jan Leike, Rohin Shah, Victoria Krakovna, Daniel Filan, Steven Wang, Dawn Song, Sam Toyer and Dan Hendrycks for their suggestion and helpful feedback on earlier drafts of this paper. We thank Chris Northwood for assistance developing the website accompanying this paper. We are also grateful to our anonymous reviewers for valuable feedback and encouragement to explore defenses in this paper. ",
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+    {
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+    {
+        "type": "table",
+        "img_path": "images/99d96ecdc4823ee83c61884b27fb4eeac9e230e5980172f5ce90dbc65cc40b0e.jpg",
+        "table_caption": [
+            "A TRAINING: HYPERPARAMETERS AND COMPUTATIONAL INFRASTRUCTURE ",
+            "Table A.1: Hyperparameters for Proximal Policy Optimization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameter</td><td>Value</td><td>Search Range</td><td>Search Distribution</td></tr><tr><td>Total Timesteps</td><td>20 ×106</td><td>[0,40 × 106]</td><td>Manual</td></tr><tr><td>Batch size</td><td>16 384</td><td>[2048, 65 536]</td><td>Log uniform</td></tr><tr><td>Number of environments</td><td>8</td><td>[1,16]</td><td>Manual</td></tr><tr><td>Mini-batches</td><td>4</td><td>[1,128]</td><td>Log uniform</td></tr><tr><td>Epochs per update</td><td>4</td><td>[1,11]</td><td>Uniform</td></tr><tr><td>Learning rate</td><td>3×10-4</td><td>[1 × i0-5,1 × 10-2]</td><td>Log uniform</td></tr><tr><td>Discount</td><td>0.99</td><td></td><td></td></tr><tr><td>Maximum Gradient Norm</td><td>0.5</td><td></td><td></td></tr><tr><td>Clip Range</td><td>0.2</td><td></td><td></td></tr><tr><td>Advantage Estimation Discount</td><td>0.95</td><td></td><td></td></tr><tr><td>Entropy coefficient</td><td>0.0</td><td></td><td></td></tr><tr><td>Value Function Loss Coefficient</td><td>0.5</td><td></td><td></td></tr></table>",
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table A.1 specifies the hyperparameters used for training. The number of environments was chosen for performance reasons after observing diminishing returns from using more than 8 parallel environments. The total timesteps was chosen by inspection after observing diminishing returns to additional training. The batch size, mini-batches, epochs per update, entropy coefficient and learning rate were tuned via a random search with 100 samples on two environments, Kick and Defend and Sumo Humans. All other hyperparameters are the defaults in the PPO2 implementation in Stable Baselines (Hill et al., 2019). ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We repeated the hyperparameter sweep for fine-tuning victim policies for the defense experiments, but obtained similar results. For simplicity, we therefore chose to use the same hyperparameters throughout. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We used a mixture of in-house and cloud infrastructure to perform these experiments. It takes around 8 hours to train an adversary for a single victim using 4 cores of an Intel Xeon Platinum 8000 (Skylake) processor. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ACTIVATION ANALYSIS: T-SNE AND GMM ",
+        "text_level": 1,
+        "bbox": [
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+            609
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We collect activations from all feed forward layers of the victim’s policy network. This gives two 64-length vectors, which we concatenate into a single 128-dimension vector for analysis with a Gaussian Mixture Model and a t-SNE representation. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 T-SNE HYPERPARAMETER SELECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            684,
+            482,
+            699
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We fit models with perplexity 5, 10, 20, 50, 75, 100, 250 and 1000. We chose 250 since qualitatively it produced the clearest visualization of data with a moderate number of distinct clusters. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            738
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2 GAUSSIAN MIXTURE MODEL HYPERPARAMETER SELECTION ",
+        "bbox": [
+            174,
+            756,
+            638,
+            770
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We fit models with 5, 10, 20, 40 and 80 components with a full (unrestricted) and diagonal covariance matrix. We used the Bayesian Information Criterion (BIC) and average log-likelihood on a heldout validation set as criteria for selecting hyperparameters. We found 20 components with a full covariance matrix achieved the lowest BIC and highest validation log-likelihood in the majority of environment-victim pairs, and was the runner-up in the remainder. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            852
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C FIGURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            871,
+            282,
+            887
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Supplementary figures are provided on the subsequent pages. ",
+        "bbox": [
+            174,
+            904,
+            575,
+            917
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/5ce8d6572b4355a81fe77b7a828799cb26f91b82c957859a4434c50346bfc21d.jpg",
+        "image_caption": [
+            "Figure C.1: Win rates while training adversary Adv. The adversary exceeds baseline win rates against most victims in Kick and Defend and You Shall Not Pass, is competitive on Sumo Humans, but performs poorly in the low-dimensional Sumo Ants environment. Key: The solid line shows the median win rate for Adv across 5 random seeds, with the shaded region representing the minimum and maximum. The win rate is smoothed with a rolling average over 100 000 timesteps. Baselines are shown as horizontal dashed lines. Agents Rand and Zero take random and zero actions respectively. The Zoo baseline is whichever ZooM (Sumo) or ZooOM (other environments) agent achieves the highest win rate. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the title above each figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            98,
+            812,
+            791
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d1bd2987326cae10e72f8090ba7d66f7f1b8976519dc5076cf74aa9cbbe307f2.jpg",
+        "table_caption": [
+            "(a) Kick and Defend "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"7\">Opponent Win</td><td colspan=\"7\">Victim Win</td><td colspan=\"7\">Ties</td><td></td><td></td><td></td></tr><tr><td>ZooV1</td><td>79 53 50</td><td></td><td></td><td>32</td><td>四</td><td>42</td><td>30 30</td><td>21</td><td>4650</td><td></td><td>67</td><td>10</td><td>57</td><td>6461</td><td>0</td><td></td><td>0</td><td>0</td><td>2</td><td>1</td><td>1</td><td>6</td><td>9</td><td>100</td></tr><tr><td>ZooV2</td><td>54</td><td>因</td><td>63</td><td>9</td><td>80</td><td>63</td><td>50 46</td><td>46</td><td>6</td><td>37</td><td>90</td><td>20</td><td>36</td><td>45 44</td><td>0</td><td>0</td><td>0</td><td></td><td>0</td><td>1</td><td>0</td><td>5 10</td><td></td><td>80</td></tr><tr><td>ZooV3</td><td>31 31</td><td></td><td>4</td><td>50</td><td>66</td><td>64</td><td>2629</td><td></td><td>68 67</td><td>回</td><td>47</td><td>33</td><td>35</td><td>68 62</td><td>1</td><td>2</td><td></td><td>0</td><td>3</td><td>0</td><td>2</td><td>5 8</td><td></td><td>60</td></tr><tr><td>ZooMV1</td><td>1517</td><td></td><td>18</td><td>58</td><td>四</td><td>31</td><td>12 13</td><td>84</td><td>83</td><td>82</td><td>40</td><td>19</td><td>69</td><td>8478</td><td>0</td><td>0</td><td></td><td>0</td><td>2</td><td>2</td><td>0</td><td>4 9</td><td></td><td>40</td></tr><tr><td>ZooMV2</td><td>2728</td><td></td><td>27</td><td>36</td><td>因</td><td>42</td><td>2521</td><td></td><td>73 72</td><td>73</td><td>63</td><td>24</td><td>58</td><td>72</td><td>72</td><td>0</td><td>0</td><td>0</td><td>1</td><td>4</td><td>0</td><td>4</td><td>8</td><td></td></tr><tr><td rowspan=\"3\">ZooMV3</td><td>262423</td><td></td><td></td><td></td><td>47 50</td><td>61</td><td>2119</td><td>74</td><td>75</td><td>77</td><td></td><td>51 48</td><td>36</td><td>74 72</td><td></td><td>0</td><td>1</td><td>0</td><td>2</td><td>1</td><td>3</td><td>5</td><td>10</td><td>20</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>1</td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td><td>0</td></tr><tr><td>[APA</td><td>£APA</td><td>1000Z 7000Z</td><td></td><td></td><td>£000Z</td><td>purd</td><td></td><td></td><td>3</td><td>1000Z</td><td>7000Z</td><td>£000Z</td><td>Purd </td><td>[APV</td><td>2</td><td>3</td><td>1000Z</td><td>7000Z</td><td>£000Z</td><td>Pury</td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            181,
+            106,
+            821,
+            248
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/56c77d842352608a0c57df2736f22b4a8ec31a94f488d6db62e1a8298145cdb2.jpg",
+        "table_caption": [
+            "(b) You Shall Not Pass "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>ZooV1</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>81</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>48</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>97 99</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooSV1</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>99</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>91</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>88 82</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooDV1</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>91</td><td rowspan=1 colspan=1>88</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=2>8  6</td><td rowspan=1 colspan=1>89</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>43</td><td rowspan=1 colspan=1>9294</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=1 colspan=1>ZooMV1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>83</td><td rowspan=1 colspan=1>70</td><td rowspan=1 colspan=1>78</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>99</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>9999</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0  0</td></tr><tr><td rowspan=3 colspan=2>1</td><td rowspan=3 colspan=1>ISAPT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=1>A</td><td rowspan=2 colspan=1>0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=2>Pard  o.I9z</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Pard  o.az</td><td rowspan=1 colspan=1>APV</td><td rowspan=1 colspan=1>AAAPT</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td></td></tr></table>",
+        "bbox": [
+            184,
+            276,
+            823,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/62d704ca84eef08c8313448310a9133f8b1e00fa57dc36fb2b59b85b5b38caac.jpg",
+        "table_caption": [
+            "(c) Sumo Humans "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"6\">Opponent Win</td><td colspan=\"6\">Victim Win</td><td colspan=\"6\">Ties</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Z001</td><td>87</td><td>9</td><td>37</td><td>15</td><td>80</td><td>80</td><td>0</td><td>0</td><td>12</td><td>91</td><td>63</td><td>85</td><td>19 18</td><td></td><td>100100</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>2</td><td></td><td>0</td><td>0</td><td></td><td>100</td></tr><tr><td>Z002</td><td>10</td><td>63</td><td>14</td><td>34</td><td>54</td><td>71</td><td>0</td><td>0</td><td>76</td><td>19</td><td>64</td><td>62</td><td>35 回</td><td></td><td>100100</td><td>141</td><td>18</td><td>22</td><td>4</td><td>11</td><td>13</td><td></td><td>0</td><td>0</td><td></td><td>80</td></tr><tr><td>Z003</td><td>1718</td><td></td><td>44</td><td>10 31</td><td></td><td>57</td><td>0</td><td>1</td><td>81</td><td>79</td><td>2</td><td>88</td><td>61</td><td>34</td><td>100 99</td><td>2</td><td>4</td><td>33</td><td></td><td>1</td><td>8</td><td>9</td><td>0</td><td>0</td><td></td><td>60</td></tr><tr><td>ZooM1</td><td></td><td>8724</td><td>58</td><td>9391 92</td><td></td><td></td><td>0</td><td>1</td><td>12</td><td>76</td><td>42</td><td>□</td><td>9</td><td>8</td><td>100 99</td><td>0</td><td>0</td><td></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td><td>40</td></tr><tr><td>Z0oM2</td><td></td><td>3239</td><td>39</td><td>46</td><td></td><td>69</td><td>0</td><td>0</td><td>39</td><td>30</td><td>23</td><td>50</td><td>四</td><td>17</td><td>100100</td><td></td><td>29 3238</td><td></td><td></td><td>4</td><td>6</td><td>14</td><td>0</td><td>0</td><td></td><td>20</td></tr><tr><td>ZooM3</td><td></td><td>6268</td><td>75</td><td>66</td><td>82</td><td>四</td><td>1</td><td>1</td><td>30</td><td>20</td><td>10</td><td>31</td><td>13</td><td>□</td><td>99 99</td><td></td><td>9 1215</td><td></td><td></td><td>3</td><td>6</td><td>2</td><td>0</td><td>0</td><td></td><td></td></tr><tr><td colspan=\"2\"></td><td></td><td></td><td></td><td></td><td></td><td>pury</td><td></td><td></td><td></td><td></td><td>[00Z</td><td>20</td><td>£00Z</td><td>puey</td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td><td></td><td></td><td>0</td></tr><tr><td></td><td>[APV</td><td>APA</td><td>1</td><td></td><td>2</td><td>2</td><td></td><td>0.I9z</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>00</td><td>[APV</td><td></td><td>3</td><td>[00Z</td><td>20</td><td>20</td><td>pued</td><td>20</td><td></td><td></td></tr></table>",
+        "bbox": [
+            189,
+            433,
+            821,
+            569
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/43e98aa0e379df5d4eb5e0e40961ab77cd0a094b151a07cfc26f1a9abd00e1cf.jpg",
+        "image_caption": [
+            "Figure C.2: Percentage of episodes (out of 1000) won by the opponent, the victim or tied. The maximal opponent win rate in each row is in red. Victims are on the $y$ -axis and opponents on the $x$ -axis. Key: Agents ZooYN are pre-trained policies from Bansal et al. (2018a), where $Y \\in \\{ { } ^ { \\cdot } V ^ { \\prime } , { } ^ { \\cdot } O ^ { \\prime } , { } ^ { \\cdot \\prime } \\}$ denotes the agent plays as (V)ictim, $\\mathbf { \\eta } ^ { ( \\mathbf { 0 } ) }$ pponent or either side, and $N$ is a random seed. Opponents AdvN are the best adversarial policy of 5 seeds trained against the corresponding Zoo[V]N. Agents Rand and $\\mathtt { Z e r o }$ are baseline agents taking random and zero actions respectively. Defended victims ZooXYN, where $X \\in \\{ \\cdot S ^ { \\prime } , \\cdot \\bar { D ^ { \\prime } } , \\cdot M ^ { \\prime } \\}$ , are derived from ZooYN by fine-tuning against a (S)ingle opponent AdvN, $\\mathbf { ( D ) }$ ual opponents AdvN and Zoo[O]N, or by $\\mathbf { \\Psi } ^ { ( \\mathbf { M } ) }$ asking the observations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            603,
+            820,
+            797
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/305c21d93e648995cbf0f7f4c0ba99d2806afa39b254e780b4e6fcbbb26a5652.jpg",
+        "image_caption": [
+            "Figure C.3: t-SNE activations of the victim when playing against different opponents. There is a clear separation between the activations induced by Adv and those of the normal opponent Zoo. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. The victim is ZooN (Sumo) or ZooVN (other environments), where $N$ is given in the figure caption. Opponent Adv is the best adversary trained against the victim. Opponent $\\mathrm { { } ~ \\ Z \\ O O }$ corresponds to ZooN (Sumo) or ZooON (other environments). See Figure C.4 for activations for a single opponent at a time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            95,
+            820,
+            821
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/bdce657631a365c6b4b9c7f5106a3f5b5288e8879e2f26cf31d40a00d4e27fb3.jpg",
+        "image_caption": [
+            "Figure C.4: t-SNE activations of victim Zoo1 (Sumo) or ZooV1 (other environments). The results are the same as in Figure C.3 but decomposed into individual opponents for clarity. Model fitted with a perplexity of 250 to activations from 5000 timesteps against each opponent. Opponent Adv is the best adversary trained against the victim. Opponent Zoo is Zoo1 (Sumo) or ZooO1 (other environments). See Figure C.3 for results for other victims (one plot per victim). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            84,
+            823,
+            830
+        ],
+        "page_idx": 15
+    }
+]

parse/train/HJlnC1rKPB/HJlnC1rKPB.md

@@ -0,0 +1,433 @@
+# ON THE RELATIONSHIP BETWEEN SELF-ATTENTION AND CONVOLUTIONAL LAYERS
+
+Jean-Baptiste Cordonnier, Andreas Loukas & Martin Jaggi Ecole Polytechnique F ´ ed´ erale de Lausanne (EPFL) ´ {first.last}@epfl.ch
+
+# ABSTRACT
+
+Recent trends of incorporating attention mechanisms in vision have led researchers to reconsider the supremacy of convolutional layers as a primary building block. Beyond helping CNNs to handle long-range dependencies, Ramachandran et al. (2019) showed that attention can completely replace convolution and achieve state-of-the-art performance on vision tasks. This raises the question: do learned attention layers operate similarly to convolutional layers? This work provides evidence that attention layers can perform convolution and, indeed, they often learn to do so in practice. Specifically, we prove that a multi-head self-attention layer with sufficient number of heads is at least as expressive as any convolutional layer. Our numerical experiments then show that self-attention layers attend to pixel-grid patterns similarly to CNN layers, corroborating our analysis. Our code is publicly available1.
+
+# 1 INTRODUCTION
+
+Recent advances in Natural Language Processing (NLP) are largely attributed to the rise of the transformer (Vaswani et al., 2017). Pre-trained to solve an unsupervised task on large corpora of text, transformer-based architectures, such as GPT-2 (Radford et al., 2018), BERT (Devlin et al., 2018) and Transformer-XL (Dai et al., 2019), seem to possess the capacity to learn the underlying structure of text and, as a consequence, to learn representations that generalize across tasks. The key difference between transformers and previous methods, such as recurrent neural networks (Hochreiter & Schmidhuber, 1997) and convolutional neural networks (CNN), is that the former can simultaneously attend to every word of their input sequence. This is made possible thanks to the attention mechanism—originally introduced in Neural Machine Translation to better handle long-range dependencies (Bahdanau et al., 2015). With self-attention in particular, the similarity of two words in a sequence is captured by an attention score measuring the distance of their representations. The representation of each word is then updated based on those words whose attention score is highest.
+
+Inspired by its capacity to learn meaningful inter-dependencies between words, researchers have recently considered utilizing self-attention in vision tasks. Self-attention was first added to CNN by either using channel-based attention (Hu et al., 2018) or non-local relationships across the image (Wang et al., 2018). More recently, Bello et al. (2019) augmented CNNs by replacing some convolutional layers with self-attention layers, leading to improvements on image classification and object detection tasks. Interestingly, Ramachandran et al. (2019) noticed that, even though state-of-the art results are reached when attention and convolutional features are combined, under same computation and model size constraints, self-attention-only architectures also reach competitive image classification accuracy.
+
+These findings raise the question, do self-attention layers process images in a similar manner to convolutional layers? From a theoretical perspective, one could argue that transfomers have the capacity to simulate any function—including a CNN. Indeed, Perez et al. (2019) showed that a multi- ´ layer attention-based architecture with additive positional encodings is Turing complete under some strong theoretical assumptions, such as unbounded precision arithmetic. Unfortunately, universality results do not reveal how a machine solves a task, only that it has the capacity to do so. Thus, the question of how self-attention layers actually process images remains open.
+
+Contributions. In this work, we put forth theoretical and empirical evidence that self-attention layers can (and do) learn to behave similar to convolutional layers:
+
+I. From a theoretical perspective, we provide a constructive proof showing that self-attention layers can express any convolutional layers.
+
+Specifically, we show that a single multi-head self-attention layer using relative positional encoding can be re-parametrized to express any convolutional layer.
+
+II. Our experiments show that the first few layers of attention-only architectures (Ramachandran et al., 2019) do learn to attend on grid-like pattern around each query pixel, similar to our theoretical construction.
+
+Strikingly, this behavior is confirmed both for our quadratic encoding, but also for relative encoding that is learned. Our results seem to suggest that localized convolution is the right inductive bias for the first few layers of an image classifying network. We provide an interactive website2 to explore how self-attention exploits localized position-based attention in lower layers and contentbased attention in deeper layers. For reproducibility purposes, our code is publicly available.
+
+# 2 BACKGROUND ON ATTENTION MECHANISMS FOR VISION
+
+We here recall the mathematical formulation of self-attention layers and emphasize the role of positional encodings.
+
+# 2.1 THE MULTI-HEAD SELF-ATTENTION LAYER
+
+Let $\pmb { X } \in \mathbb { R } ^ { T \times D _ { i n } }$ be an input matrix consisting of $T$ tokens in of $D _ { i n }$ dimensions each. While in NLP each token corresponds to a word in a sentence, the same formalism can be applied to any sequence of $T$ discrete objects, e.g. pixels. A self-attention layer maps any query token $t \in [ \bar { T } ]$ from $D _ { i n }$ to $D _ { o u t }$ dimensions as follows:
+
+$$
+{ \mathrm { S e l f - A t t e n t i o n } } ( X ) _ { t , : } : = { \mathrm { s o f t m a x } } \left( A _ { t , : } \right) X W _ { v a l } ,
+$$
+
+where we refer to the elements of the $T \times T$ matrix
+
+$$
+A : = X W _ { q r y } W _ { k e y } ^ { \top } X ^ { \top }
+$$
+
+as attention scores and the softmax output3 as attention probabilities. The layer is parametrized by a query matrix $W _ { q r y } \in \mathbb { R } ^ { D _ { i n } \times D _ { k } }$ , a key matrix $W _ { k e y } ^ { \mathrm { ~ ~ } } \in \mathbb { R } ^ { D _ { i n } \times D _ { k } }$ and a value matrix $W _ { \nu a l } ~ \in$ $\mathbb { R } ^ { D _ { i n } \times D _ { o u t } }$ .For simplicity, we exclude any residual connections, batch normalization and constant factors.
+
+A key property of the self-attention model described above is that it is equivariant to reordering, that is, it gives the same output independently of how the $T$ input tokens are shuffled. This is problematic for cases we expect the order of things to matter. To alleviate the limitation, a positional encoding is learned for each token in the sequence (or pixel in an image), and added to the representation of the token itself before applying self-attention
+
+$$
+\pmb { A } : = ( \pmb { X } + \pmb { P } ) \pmb { W } _ { q r y } \pmb { W } _ { k e y } ^ { \top } ( \pmb { X } + \pmb { P } ) ^ { \top } ,
+$$
+
+where $P \in \mathbb { R } ^ { T \times D _ { i n } }$ contains the embedding vectors for each position. More generally, $_ { r }$ may be substituted by any function that returns a vector representation of the position.
+
+It has been found beneficial in practice to replicate this self-attention mechanism into multiple heads, each being able to focus on different parts of the input by using different query, key and value matrices. In multi-head self-attention, the output of the $N _ { h }$ heads of output dimension $D _ { h }$ are concatenated and projected to dimension $D _ { o u t }$ as follows:
+
+$$
+\operatorname { M H S A } ( X ) : = \operatorname { c o n c a t } _ { h \in [ N _ { h } ] } \left[ \operatorname { S e l f - A t t e n t i o n } _ { h } ( X ) \right] W _ { o u t } + b _ { o u t }
+$$
+
+and two new parameters are introduced: the projection matrix $W _ { o u t } \in \mathbb { R } ^ { N _ { h } D _ { h } \times D _ { o u t } }$ and a bias term bout ∈ RDout .
+
+# 2.2 ATTENTION FOR IMAGES
+
+Convolutional layers are the de facto choice for building neural networks that operate on images. We recall that, given an image tensor $\pmb { \mathsf { X } } \in \mathbb { R } ^ { W \times H \times D _ { i n } }$ of width $W$ , height $H$ and $D _ { i n }$ channels, the output of a convolutional layer for pixel $( i , j )$ is given by
+
+$$
+\mathrm { C o n v } ( \boldsymbol { X } ) _ { i , j , : } : = \sum _ { ( \delta _ { 1 } , \delta _ { 2 } ) \in \Delta _ { K } } \mathsf { X } _ { i + \delta _ { 1 } , j + \delta _ { 2 } , : } \mathsf { W } _ { \delta _ { 1 } , \delta _ { 2 } , : , : } + \boldsymbol { b } ,
+$$
+
+where $\boldsymbol { \mathsf { W } }$ is the $K \times K \times D _ { i n } \times D _ { o u t }$ weight tensor 4, $\pmb { b } \in \mathbb { R } ^ { D _ { o u t } }$ is the bias vector and the set
+
+$$
+\mathbb { \Delta } _ { K } : = \left[ - \left\lfloor { \frac { K } { 2 } } \right\rfloor , \cdots , \left\lfloor { \frac { K } { 2 } } \right\rfloor \right] \times \left[ - \left\lfloor { \frac { K } { 2 } } \right\rfloor , \cdots , \left\lfloor { \frac { K } { 2 } } \right\rfloor \right]
+$$
+
+contains all possible shifts appearing when convolving the image with a $K \times K$ kernel.
+
+In the following, we review how self-attention can be adapted from 1D sequences to images.
+
+With images, rather than tokens, we have query and key pixels $q , k \in [ W ] \times [ H ]$ . Accordingly, the input is a tensor $\pmb { \chi }$ of dimension $W \times H \times D _ { i n }$ and each attention score associates a query and a key pixel.
+
+To keep the formulas consistent with the 1D case, we abuse notation and slice tensors by using a 2D index vector: if $\mathbf { \boldsymbol { p } } = ( i , j )$ , we write $\mathsf { \pmb X } _ { p } ,$ : and $\mathsf { \pmb { A } } _ { p } ,$ : to mean $\mathsf { X } _ { i , j } ,$ : and $\mathsf { \pmb { A } } _ { i , j , : , : }$ , respectively. With this notation in place, the multi-head self attention layer output at pixel $\pmb q$ can be expressed as follows:
+
+$$
+\mathrm { S e l f - A t t e n t i o n } ( X ) _ { q , : } = \sum _ { k } \mathrm { s o f t m a x } \left( \mathbb { A } _ { q , : } \right) _ { k } \mathbb { X } _ { k , : } W _ { \nu a l }
+$$
+
+and accordingly for the multi-head case.
+
+# 2.3 POSITIONAL ENCODING FOR IMAGES
+
+There are two types of positional encoding that has been used in transformer-based architectures: the absolute and relative encoding (see also Table 3 in the Appendix).
+
+With absolute encodings, a (fixed or learned) vector $\mathsf { P } _ { p } ,$ ,: is assigned to each pixel $\pmb { p }$ . The computation of the attention scores we saw in eq. (2) can then be decomposed as follows:
+
+$$
+\begin{array} { r l } & { \pmb { \mathrm { A } } _ { q , k } ^ { \mathrm { a b s } } = ( \pmb { \mathrm { X } } _ { q , : } + \pmb { \mathrm { P } } _ { q , : } ) W _ { q r y } W _ { k e y } ^ { \top } ( \pmb { \mathrm { X } } _ { k , : } + \pmb { \mathrm { P } } _ { k , : } ) ^ { \top } } \\ & { \qquad = \pmb { \mathrm { X } } _ { q , : } W _ { q r y } W _ { k e y } ^ { \top } \pmb { \mathrm { X } } _ { k , : } ^ { \top } + \pmb { \mathrm { X } } _ { q , : } W _ { q r y } W _ { k e y } ^ { \top } \pmb { \mathrm { P } } _ { k , : } ^ { \top } + \pmb { \mathrm { P } } _ { q , : } W _ { q r y } W _ { k e y } ^ { \top } \pmb { \mathrm { X } } _ { k , : } + \pmb { \mathrm { P } } _ { q , : } W _ { q r y } W _ { k e y } ^ { \top } \pmb { \mathrm { P } } _ { k , : } } \end{array}
+$$
+
+where $\pmb q$ and $\boldsymbol { k }$ correspond to the query and key pixels, respectively.
+
+The relative positional encoding was introduced by Dai et al. (2019). The main idea is to only consider the position difference between the query pixel (pixel we compute the representation of) and the key pixel (pixel we attend) instead of the absolute position of the key pixel:
+
+$$
+\begin{array} { r } { \pmb { \mathsf { A } } _ { q , k } ^ { \mathrm { r e l } } : = \pmb { \mathsf { X } } _ { q , : } ^ { \top } W _ { q r y } ^ { \top } W _ { k e y } \pmb { \mathsf { X } } _ { k , : } + \pmb { \mathsf { X } } _ { q , : } ^ { \top } W _ { q r y } ^ { \top } \widehat { W } _ { k e y } \pmb { r } _ { \delta } + \boldsymbol { u } ^ { \top } W _ { k e y } \pmb { \mathsf { X } } _ { k , : } + \boldsymbol { v } ^ { \top } \widehat { W } _ { k e y } \pmb { r } _ { \delta } } \end{array}
+$$
+
+In this manner, the attention scores only depend on the shift $\delta : = k - q$ . Above, the learnable vectors $\textbf { \em u }$ and $\pmb { v }$ are unique for each head, whereas for every shift $\delta$ the relative positional encoding $\boldsymbol { r } _ { \delta } \in \mathbb { R } ^ { D _ { p } }$ is shared by all layers and heads. Moreover, now the key weights are split into two types: $W _ { k e y }$ pertain to the input and $\widehat { W } _ { k e y }$ to the relative position of pixels.
+
+# 3 SELF-ATTENTION AS A CONVOLUTIONAL LAYER
+
+This section derives sufficient conditions such that a multi-head self-attention layer can simulate a convolutional layer. Our main result is the following:
+
+Theorem 1. A multi-head self-attention layer with $N _ { h }$ heads of dimension $D _ { h }$ , output dimension $D _ { o u t }$ and a relative positional encoding of dimension $D _ { p } \geq 3$ can express any convolutional layer of kernel size $\sqrt { N _ { h } } \times \sqrt { N _ { h } }$ and $\operatorname* { m i n } ( D _ { h } , D _ { o u t } )$ output channels.
+
+4To simplify notation, we index the first two dimensions of the tensor from $- \lfloor K / 2 \rfloor$ to $\lfloor K / 2 \rfloor$
+
+The theorem is proven constructively by selecting the parameters of the multi-head self-attention layer so that the latter acts like a convolutional layer. In the proposed construction, the attention scores of each self-attention head should attend to a different relative shift within the set $\Delta _ { K } =$ $\{ - \lfloor K / 2 \rfloor , \ldots , \lfloor K / 2 \rfloor \} ^ { 2 }$ of all pixel shifts in a $K \times K$ kernel. The exact condition can be found in the statement of Lemma 1.
+
+Then, Lemma 2 shows that the aforementioned condition is satisfied for the relative positional encoding that we refer to as the quadratic encoding:
+
+$$
+\pmb { v } ^ { ( h ) } : = - \alpha ^ { ( h ) } \left( 1 , - 2 \pmb { \Delta } _ { 1 } ^ { ( h ) } , - 2 \pmb { \Delta } _ { 2 } ^ { ( h ) } \right) \quad r _ { \delta } : = \left( \| \delta \| ^ { 2 } , \delta _ { 1 } , \delta _ { 2 } \right) \quad W _ { q r \gamma } = W _ { k e \gamma } : = \mathbf { 0 } \quad \widehat { W _ { k e \gamma } } : = I
+$$
+
+The learned parameters $\Delta ^ { ( h ) } = ( \Delta _ { 1 } ^ { ( h ) } , \Delta _ { 2 } ^ { ( h ) } )$ and $\alpha ^ { ( h ) }$ determine the center and width of attention of each head, respectively. On the other hand, $\pmb { \delta } = ( \delta _ { 1 } , \delta _ { 2 } )$ is fixed and expresses the relative shift between query and key pixels.
+
+It is important to stress that the above encoding is not the only one for which the conditions of Lemma 1 are satisfied. In fact, in our experiments, the relative encoding learned by the neural network also matched the conditions of the lemma (despite being different from the quadratic encoding). Nevertheless, the encoding defined above is very efficient in terms of size, as only $D _ { p } = 3$ dimensions suffice to encode the relative position of pixels, while also reaching similar or better empirical performance (than the learned one).
+
+The theorem covers the general convolution operator as defined in eq. (17). However, machine learning practitioners using differential programming frameworks (Paszke et al., 2017; Abadi et al., 2015) might question if the theorem holds for all hyper-parameters of 2D convolutional layers:
+
+• Padding: a multi-head self-attention layer uses by default the "SAME" padding while a convolutional layer would decrease the image size by $K - 1$ pixels. The correct way to alleviate these boundary effects is to pad the input image with $\lfloor K / 2 \rfloor$ zeros on each side. In this case, the cropped output of a MHSA and a convolutional layer are the same. • Stride: a strided convolution can be seen as a convolution followed by a fixed pooling operation—with computational optimizations. Theorem 1 is defined for stride 1, but a fixed pooling layer could be appended to the Self-Attention layer to simulate any stride. • Dilation: a multi-head self-attention layer can express any dilated convolution as each head can attend a value at any pixel shift and form a (dilated) grid pattern.
+
+Remark for the 1D case. Convolutional layers acting on sequences are commonly used in the literature for text (Kim, 2014), as well as audio (van den Oord et al., 2016) and time series (Franceschi et al., 2019). Theorem 1 can be straightforwardly extended to show that multi-head self-attention with $N _ { h }$ heads can also simulate a 1D convolutional layer with a kernel of size $K \ : = \ : N _ { h }$ with $\operatorname* { m i n } ( D _ { h } , D _ { o u t } )$ output channels using a positional encoding of dimension $D _ { p } \geq 2$ . Since we have not tested empirically if the preceding construction matches the behavior of 1D self-attention in practice, we cannot claim that it actually learns to convolve an input sequence—only that it has the capacity to do so.
+
+# PROOF OF MAIN THEOREM
+
+The proof follows directly from Lemmas 1 and 2 stated below:
+
+Lemma 1. Consider a multi-head self-attention layer consisting of $N _ { h } = K ^ { 2 }$ heads, $D _ { h } \geq D _ { o u t }$ and let $f ~ : ~ [ N _ { h } ] ~  ~ \mathbb { \Delta } _ { K }$ be a bijective mapping of heads onto shifts. Further, suppose that for every head the following holds:
+
+$$
+\mathrm { s o f t m a x } ( A _ { q , : } ^ { ( h ) } ) _ { k } = \left\{ \begin{array} { l l } { { 1 } } & { { \ i f f ( h ) = q - k } } \\ { { 0 } } & { { \ o t h e r w i s e . } } \end{array} \right.
+$$
+
+Then, for any convolutional layer with a $K \times K$ kernel and $D _ { o u t }$ output channels, there exists $\{ W _ { \nu a l } ^ { ( h ) } \} _ { h \in [ N _ { h } ] }$ such that $\mathrm { M H S A } ( X ) = \operatorname { C o n v } ( X )$ for every $\pmb { X } \in \mathbb { R } ^ { W \times H \times D _ { i n } }$ .
+
+![](images/5138108bfade7d47b518c0f0d260b27648a33c5c76f853d34ffceb5c5e6bdec9.jpg)  
+Figure 1: Illustration of a Multi-Head Self-Attention layer applied to a tensor image $\pmb { \chi }$ . Each head $h$ attends pixel values around shift ∆(h) and learn a filter matrix W (h)val . We show attention maps computed for a query pixel at position $\pmb q$ .
+
+Proof. Our first step will be to rework the expression of the Multi-Head Self-Attention operator from equation (1) and equation (4) such that the effect of the multiple heads becomes more transparent:
+
+$$
+\mathrm { M H S A } ( { \pmb X } ) = b _ { o u t } + \sum _ { h \in [ N _ { h } ] } \mathrm { s o f t m a x } ( { \pmb A } ^ { ( h ) } ) { \pmb X } \underbrace { W _ { v a l } ^ { ( h ) } W _ { o u t } [ ( h - 1 ) D _ { h } + 1 : h D _ { h } + 1 ] } _ { { \pmb W } ^ { ( h ) } }
+$$
+
+Note that each head’s value matrix W (h)val $W _ { \nu a l } ^ { ( h ) } \in \mathbb { R } ^ { D _ { i n } \times D _ { h } }$ and each block of the projection matrix $W _ { o u t }$ of dimension $D _ { h } \times D _ { o u t }$ are learned. Assuming that $D _ { h } \ \ge \ D _ { o u t }$ , we can replace each pair of matrices by a learned matrix $W ^ { ( h ) }$ for each head. We consider one output pixel of the multi-head self-attention:
+
+$$
+\mathrm { M H S A } ( \boldsymbol { X } ) _ { q , : } = \sum _ { h \in [ N _ { h } ] } \left( \sum _ { k } \mathrm { s o f t m a x } ( \pmb { \mathscr { A } } _ { q , : } ^ { ( h ) } ) _ { k } \pmb { \mathscr { X } } _ { k , : } \right) \boldsymbol { W } ^ { ( h ) } + b _ { o u t }
+$$
+
+Due to the conditions of the Lemma, for the $h$ -th attention head the attention probability is one when $\pmb { k } = \pmb { q } - \pmb { f } ( h )$ and zero otherwise. The layer’s output at pixel $\pmb q$ is thus equal to
+
+$$
+\mathrm { M H S A } ( \mathbf { X } ) _ { q } = \sum _ { h \in [ N _ { h } ] } \mathbf { X } _ { q - f ( h ) , : } \mathbf { {  { W } } } ^ { ( h ) } + \pmb { b } _ { o u t }
+$$
+
+For $K = \sqrt { N _ { h } }$ , the above can be seen to be equivalent to a convolutional layer expressed in eq. 17: there is a one to one mapping (implied by map $f$ ) between the matrices $W ^ { ( \bar { h } ) }$ for $h = [ N _ { h } ]$ and the matrices $\mathsf { W } _ { k _ { 1 } , k _ { 2 } , : , : }$ for all $( k _ { 1 } , k _ { 2 } ) \in [ K ] ^ { 2 }$ . □
+
+Remark about $D _ { h }$ and $D _ { o u t }$ . It is frequent in transformer-based architectures to set $D _ { h } ~ = ~ D _ { o u t } / N _ { h }$ , hence $D _ { h } < D _ { o u t }$ . In that case, $W ^ { ( h ) }$ can be seen to be of rank $D _ { o u t } - D _ { h }$ , which does not suffice to express every convolutional layer with $D _ { o u t }$ channels. Nevertheless, it can be seen that any $D _ { h }$ out of $D _ { o u t }$ outputs of $\operatorname { M H S A } ( X )$ can express the output of any convolutional layer with $D _ { h }$ output channels. To cover both cases, in the statement of the main theorem we assert that the output channels of the convolutional layer should be $\operatorname* { m i n } ( D _ { h } , D _ { o u t } )$ . In practice, we advise to concatenate heads of dimension $D _ { h } = D _ { o u t }$ instead of splitting the $D _ { o u t }$ dimensions among heads to have exact re-parametrization and no “unused” channels.
+
+Lemma 2. There exists a relative encoding scheme $\{ r _ { \delta } \in \mathbb { R } ^ { D _ { p } } \} _ { \pmb { \delta } \in \mathbb { Z } ^ { 2 } }$ with $D _ { p } \geq 3$ and parameters $W _ { q r y }$ , $W _ { k e y }$ , $\widehat { W } _ { k e y } , u$ with $D _ { p } \leq D _ { k }$ such that, for every $\pmb { \Delta } \in \mathbb { \Delta } _ { K }$ there exists some vector $\textbf {  { v } }$ (conditioned on $\pmb { \Delta }$ ) yielding softm $\arg ( \mathbf { A } _ { q , : } ) _ { k } = 1 i f k - q = \Delta$ and zero, otherwise.
+
+Proof. We show by construction the existence of a $D _ { p } = 3$ dimensional relative encoding scheme yielding the required attention probabilities.
+
+As the attention probabilities are independent of the input tensor $\pmb { \chi }$ , we set $\boldsymbol { W _ { k e y } } = \boldsymbol { W _ { q r y } } = \mathbf { 0 }$ which leaves only the last term of eq. (8). Setting $\widehat { \pmb { W } } _ { k e y } \in \mathbb { R } ^ { D _ { k } \times D _ { p } }$ to the identity matrix (with appropriate row padding), yields $\pmb { \mathsf { A } } _ { q , \pmb { k } } = \pmb { v } ^ { \top } \pmb { r } _ { \delta }$ where $\delta : = k - q$ . Above, we have assumed that $D _ { p } \leq D _ { k }$ such that no information from $\mathbf { \Delta } _ { \pmb { r } \delta }$ is lost.
+
+Now, suppose that we could write:
+
+$$
+\mathsf { \pmb { A } } _ { q , k } = - \alpha ( \| \pmb { \delta } - \pmb { \Delta } \| ^ { 2 } + c )
+$$
+
+for some constant $c$ . In the above expression, the maximum attention score over $\mathsf { \pmb { A } } _ { \pmb { q } , }$ ,: is $- \alpha c$ and it is reached for $\pmb { \Delta } _ { q , k }$ with $\delta = \Delta$ . On the other hand, the $\alpha$ coefficient can be used to scale arbitrarily the difference between $\mathsf { \pmb { A } } _ { q , \Delta }$ and the other attention scores.
+
+In this way, for $\delta = \Delta$ , we have
+
+$$
+\begin{array} { l } { { \displaystyle \operatorname* { l i m } _ { \imath \to \infty } \mathrm { s o f t m a x } ( \mathbb { A } _ { q , : } ) _ { k } = \operatorname* { l i m } _ { \alpha \to \infty } \frac { e ^ { - \alpha ( \| \delta - \Delta \| ^ { 2 } + c ) } } { \sum _ { k ^ { \prime } } e ^ { - \alpha ( \| ( k - q ^ { \prime } ) - \Delta \| ^ { 2 } + c ) } } } \ ~ } \\ { { \displaystyle = \operatorname* { l i m } _ { \alpha \to \infty } \frac { e ^ { - \alpha \| \delta - \Delta \| ^ { 2 } } } { \sum _ { k ^ { \prime } } e ^ { - \alpha \| ( k - q ^ { \prime } ) - \Delta \| ^ { 2 } } } = \frac { 1 } { 1 + \operatorname* { l i m } _ { \alpha \to \infty } \sum _ { k ^ { \prime } \neq k } e ^ { - \alpha \| ( k - q ^ { \prime } ) - \Delta \| ^ { 2 } } } = 1 } } \end{array}
+$$
+
+and for $\delta \neq \Delta$ , the equation becomes $\begin{array} { r } { \operatorname* { l i m } _ { \alpha  \infty } \mathrm { s o f t m a x } ( \pmb { \mathsf { A } } _ { q , : } ) _ { k } = 0 } \end{array}$ , exactly as needed to satisfy the lemma statement.
+
+What remains is to prove that there exist $\textbf {  { v } }$ and $\{ r _ { \delta } \} _ { \delta \in \mathbb { Z } ^ { 2 } }$ for which eq. (14) holds. Expanding the RHS of the equation, we have $- \alpha ( \| \pmb { \delta } - \pmb { \Delta } \| ^ { 2 } + \underline { { { c } } } ) = - \alpha ( \| \pmb { \delta } \| ^ { 2 } + \| \pmb { \Delta } \| ^ { 2 } - 2 \langle \pmb { \delta } , \pmb { \Delta } \rangle + c )$ . Now if we set $v = - \alpha \left( 1 , - 2 \Delta _ { 1 } , - 2 \Delta _ { 2 } \right)$ and $r _ { \delta } = ( \| \delta \| ^ { 2 } , \delta _ { 1 } , \delta _ { 2 } )$ , then
+
+$$
+\lambda _ { q , k } = v ^ { \top } r _ { \delta } = - \alpha \big ( \| \delta \| ^ { 2 } - 2 \Delta _ { 1 } \delta _ { 1 } - 2 \Delta _ { 2 } \delta _ { 2 } \big ) = - \alpha \big ( \| \delta \| ^ { 2 } - 2 \langle \delta , \Delta \rangle \big ) = - \alpha \big ( \| \delta - \Delta \| ^ { 2 } - \| \Delta \| ^ { 2 } \big ) ,
+$$
+
+which matches eq. (14) with $c = - \| \Delta \| ^ { 2 }$ and the proof is concluded.
+
+Remark on the magnitude of $\alpha$ . The exact representation of one pixel requires $\alpha$ (or the matrices $W _ { q r y }$ and $W _ { k e y , }$ ) to be arbitrary large, despite the fact that the attention probabilities of all other pixels converge exponentially to 0 as $\alpha$ grows. Nevertheless, practical implementations always rely on finite precision arithmetic for which a constant $\alpha$ suffices to satisfy our construction. For instance, since the smallest positive float32 scalar is approximately $1 0 ^ { - 4 \bar { 5 } }$ , setting $\alpha = 4 6$ would suffice to obtain hard attention.
+
+# 4 EXPERIMENTS
+
+The aim of this section is to validate the applicability of our theoretical results—which state that self-attention can perform convolution—and to examine whether self-attention layers in practice do actually learn to operate like convolutional layers when trained on standard image classification tasks. In particular, we study the relationship between self-attention and convolution with quadratic and learned relative positional encodings. We find that, for both cases, the attention probabilities learned tend to respect the conditions of Lemma 1, supporting our hypothesis.
+
+# 4.1 IMPLEMENTATION DETAILS
+
+We study a fully attentional model consisting of six multi-head self-attention layers. As it has already been shown by Bello et al. (2019) that combining attention features with convolutional features improves performance on Cifar-100 and ImageNet, we do not focus on attaining state-of-the-art performance. Nevertheless, to validate that our model learns a meaningful classifier, we compare it to the standard ResNet18 (He et al., 2015) on the CIFAR-10 dataset (Krizhevsky et al.). In all experiments, we use a $2 \times 2$ invertible down-sampling (Jacobsen et al., 2018) on the input to reduce the size of the image. As the size of the attention coefficient tensors (stored during forward) scales quadratically with the size of the input image, full attention cannot be applied to bigger images. The fixed size representation of the input image is computed as the average pooling of the last layer representations and given to a linear classifier.
+
+![](images/c25ba2d97b90d00bd1dec6470894d68608b4f937513a6998c4ba69932b425272.jpg)  
+Figure 2: Test accuracy on CIFAR-10.
+
+Table 1: Test accuracy on CIFAR-10 and model sizes. SA stands for Self-Attention.   
+
+<table><tr><td>Models</td><td>accuracy</td><td># of params</td><td># of FLOPS</td></tr><tr><td>ResNet18</td><td>0.938</td><td>11.2M</td><td>1.1B</td></tr><tr><td>SA quadratic emb.</td><td>0.938</td><td>12.1M</td><td>6.2B</td></tr><tr><td>SA learned emb.</td><td>0.918</td><td>12.3M</td><td>6.2B</td></tr><tr><td>SA learned emb.+ content</td><td>0.871</td><td>29.5M</td><td>15B</td></tr></table>
+
+![](images/4f98864f99089743c52d62975bc2c83b4b4e30f9048971de668f1822da395adb.jpg)
+
+![](images/d382e8252424a5d4675c9ab87eab3b192e5a233eb9d5c84c96d935476db8a570.jpg)  
+Figure 3: Centers of attention of each attention head (different colors) at layer 4 during the training with quadratic relative positional encoding. The central black square is the query pixel, whereas solid and dotted circles represent the $50 \%$ and $90 \%$ percentiles of each Gaussian, respectively.
+
+We used the PyTorch library (Paszke et al., 2017) and based our implementation on PyTorch Transformers5. We release our code on Github6 and hyper-parameters are listed in Table 2 (Appendix).
+
+Remark on accuracy. To verify that our self-attention models perform reasonably well, we display in Figure 6 the evolution of the test accuracy on CIFAR-10 over the 300 epochs of training for our self-attention models against a small ResNet (Table 1). The ResNet is faster to converge, but we cannot ascertain whether this corresponds to an inherent property of the architecture or an artifact of the adopted optimization procedures. Our implementation could be optimized to exploit the locality of Gaussian attention probabilities and reduce significantly the number of FLOPS. We observed that learned embeddings with content-based attention were harder to train probably due to their increased number of parameters. We believe that the performance gap can be bridged to match the ResNet performance, but this is not the focus of this work.
+
+# 4.2 QUADRATIC ENCODING
+
+As a first step, we aim to verify that, with the relative position encoding introduced in equation (9), attention layers learn to behave like convolutional layers. We train nine attention heads at each layer to be on par with the $3 \times 3$ kernels used predominantly by the ResNet architecture. The center of attention of each head $h$ is initialized to $\bar { \Delta ^ { ( h ) } } \sim \mathcal { N } ( \mathbf { 0 } , \bar { 2 } I _ { 2 } )$ .
+
+Figure 3 shows how the initial positions of the heads (different colors) at layer 4 changed during training. We can see that after optimization, the heads attend on specific pixel of the image forming a grid around the query pixel. Our intuition that Self-Attention applied to images learns convolutional filters around the queried pixel is confirmed.
+
+Figure 4 displays all attention head at each layer of the model at the end of the training. It can be seen that in the first few layers the heads tend to focus on local patterns (layers 1 and 2), while deeper layers (layers 3-6) also attend to larger patterns by positioning the center of attention further from the queried pixel position. We also include in the Appendix a plot of the attention positions for a higher number of heads $N _ { h } = 1 6$ ). Figure 14 displays both local patterns similar to CNN and long range dependencies. Interestingly, attention heads do not overlap and seem to take an arrangement maximizing the coverage of the input space.
+
+![](images/fc807102ee254a57bd94d2aa351c72dff80d58e101c4766a723397c8b3f59435.jpg)  
+Figure 4: Centers of attention of each attention head (different colors) for the 6 self-attention layers using quadratic positional encoding. The central black square is the query pixel, whereas solid and dotted circles represent the $50 \%$ and $90 \%$ percentiles of each Gaussian, respectively.
+
+# 4.3 LEARNED RELATIVE POSITIONAL ENCODING
+
+We move on to study the positional encoding used in practice by fully-attentional models on images.
+
+We implemented the 2D relative positional encoding scheme used by (Ramachandran et al., 2019; Bello et al., 2019): we learn a $\lfloor D _ { p } / 2 \rfloor$ position encoding vector for each row and each column pixel shift. Hence, the relative positional encoding of a key pixel at position $\boldsymbol { k }$ with a query pixel at position $\pmb q$ is the concatenation of the row shift embedding $\delta _ { 1 }$ and the column shift embedding $\delta _ { 2 }$ (where $\pmb { \delta } = \pmb { k } - \pmb { q } )$ . We chose $D _ { p } = D _ { o u t } = 4 0 0$ in the experiment. We differ from their (unpublished) implementation in the following points: $( i )$ we do not use convolution stem and ResNet bottlenecks for downsampling, but only a $2 \times 2$ invertible downsampling layer (Jacobsen et al., 2018) at input, $( i i )$ we use $D _ { h } = D _ { o u t }$ instead of $D _ { h } = D _ { o u t } / N _ { h }$ backed by our theory that the effective number of learned filters is $\operatorname* { m i n } ( D _ { h } , D _ { o u t } )$ .
+
+At first, we discard the input data and compute the attention scores solely as the last term of eq. (8). The attention probabilities of each head at each layer are displayed on Figure 5. The figure confirms our hypothesis for the first two layers and partially for the third: even when left to learn the positional encoding scheme from randomly initialized vectors, certain self-attention heads (depicted on the left) learn to attend to individual pixels, closely matching the condition of Lemma 1 and thus Theorem 1. At the same time, other heads pay attention to horizontally-symmetric but non-localized patterns, as well as to long-range pixel inter-dependencies.
+
+We move on to a more realistic setting where the attention scores are computed using both positional and content-based attention (i.e., $q ^ { \top } \bar { k } + q ^ { \top } r$ in (Ramachandran et al., 2019)) which corresponds to a full-blown standalone self-attention model.
+
+The attention probabilities of each head at each layer are displayed in Figure 6. We average the attention probabilities over a batch of 100 test images to outline the focus of each head and remove the dependency on the input image. Our hypothesis is confirmed for some heads of layer 2 and 3: even when left to learn the encoding from the data, certain self-attention heads only exploit positionbased attention to attend to distinct pixels at a fixed shift from the query pixel reproducing the receptive field of a convolutional kernel. Other heads use more content-based attention (see Figures 8 to 10 in Appendix for non-averaged probabilities) leveraging the advantage of Self-Attention over CNN which does not contradict our theory. In practice, it was shown by Bello et al. (2019) that combining CNN and self-attention features outperforms each taken separately. Our experiments shows that such combination is learned when optimizing an unconstrained fully-attentional model.
+
+The similarity between convolution and multi-head self-attention is striking when the query pixel is slid over the image: the localized attention patterns visible in Figure 6 follow the query pixel. This characteristic behavior materializes when comparing Figure 6 with the attention probabilities at a different query pixel (see Figure 7 in Appendix). Attention patterns in layers 2 and 3 are not only localized but stand at a constant shift from the query pixel, similarly to convolving the receptive field of a convolutional kernel over an image. This phenomenon is made evident on our interactive website7. This tool is designed to explore different components of attention for diverse images with or without content-based attention. We believe that it is a useful instrument to further understand how MHSA learns to process images.
+
+![](images/52b64f91759547b07bafc71621616367c44d407c8960280d6fdf956828dba220.jpg)  
+Figure 5: Attention probabilities of each head (column) at each layer $( r o w )$ using learned relative positional encoding without content-based attention. The central black square is the query pixel. We reordered the heads for visualization and zoomed on the $7 \mathbf { x } 7$ pixels around the query pixel.
+
+![](images/76c6943801b5627dd99707874d7d218c80f37bf4302118758a5d9af1692dc347.jpg)  
+Figure 6: Attention probabilities for a model with 6 layers (rows) and 9 heads (columns) using learned relative positional encoding and content-content based attention. Attention maps are averaged over 100 test images to display head behavior and remove the dependence on the input content. The black square is the query pixel. More examples are presented in Appendix A.
+
+# 5 RELATED WORK
+
+In this section, we review the known differences and similarities between CNNs and transformers.
+
+The use of CNN networks for text—at word level (Gehring et al., 2017) or character level (Kim, 2014)—is more seldom than transformers (or RNN). Transformers and convolutional models have been extensively compared empirically on tasks of Natural Language Processing and Neural Machine Translation. It was observed that transformers have a competitive advantage over convolutional model applied to text (Vaswani et al., 2017). It is only recently that Bello et al. (2019); Ramachandran et al. (2019) used transformers on images and showed that they achieve similar accuracy as ResNets. However, their comparison only covers performance and number of parameters and FLOPS but not expressive power.
+
+Beyond performance and computational-cost comparisons of transformers and CNN, the study of expressiveness of these architectures has focused on their ability to capture long-term dependencies (Dai et al., 2019). Another interesting line of research has demonstrated that transformers are Turingcomplete (Dehghani et al., 2018; Perez et al., 2019), which is an important theoretical result but is ´ not informative for practitioners. To the best of our knowledge, we are the first to show that the class of functions expressed by a layer of self-attention encloses all convolutional filters.
+
+The closest work in bridging the gap between attention and convolution is due to Andreoli (2019). They cast attention and convolution into a unified framework leveraging tensor outerproduct. In this framework, the receptive field of a convolution is represented by a “basis” tensor $\mathbf { \lambda } \mathbf { A } ~ \in ~ \mathbb { R } ^ { K \times K \times H \times W \times H \times W }$ . For instance, the receptive field of a classical $K \times K$ convolutional kernel would be encoded by $\mathsf { A } _ { \Delta , q , k } = \mathbb { 1 } \{ k - q \overset { - } { = } \Delta \}$ for $\pmb { \Delta } \in \mathbb { \Delta } _ { K }$ . The author distinguishes this index-based convolution with content-based convolution where $\pmb { \mathsf { A } }$ is computed from the value of the input, e.g., using a key/query dot-product attention. Our work moves further and presents sufficient conditions for relative positional encoding injected into the input content (as done in practice) to allow content-based convolution to express any index-based convolution. We further show experimentally that such behavior is learned in practice.
+
+# 6 CONCLUSION
+
+We showed that self-attention layers applied to images can express any convolutional layer (given sufficiently many heads) and that fully-attentional models learn to combine local behavior (similar to convolution) and global attention based on input content. More generally, fully-attentional models seem to learn a generalization of CNNs where the kernel pattern is learned at the same time as the filters—similar to deformable convolutions (Dai et al., 2017; Zampieri, 2019). Interesting directions for future work include translating existing insights from the rich CNNs literature back to transformers on various data modalities, including images, text and time series.
+
+# ACKNOWLEDGMENTS
+
+Jean-Baptiste Cordonnier is thankful to the Swiss Data Science Center (SDSC) for funding this work. Andreas Loukas was supported by the Swiss National Science Foundation (project “Deep Learning for Graph Structured Data”, grant number PZ00P2 179981).
+
+# REFERENCES
+
+Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mane, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, ´ Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, Martin Watten- ´ berg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.
+
+Jean-Marc Andreoli. Convolution, attention and structure embedding. NeurIPS 2019 workshop on Graph Representation Learning, Dec 13, 2019, Vancouver, BC, Canada, 2019.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.
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+Irwan Bello, Barret Zoph, Ashish Vaswani, Jonathon Shlens, and Quoc V. Le. Attention Augmented Convolutional Networks. arXiv:1904.09925 [cs], April 2019.
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+Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong Zhang, Han Hu, and Yichen Wei. Deformable convolutional networks. CoRR, abs/1703.06211, 2017.
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+Zihang Dai, Zhilin Yang, Yiming Yang, Jaime G. Carbonell, Quoc V. Le, and Ruslan Salakhutdinov. Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context. CoRR, abs/1901.02860, 2019.
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+Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Lukasz Kaiser. Universal transformers. CoRR, abs/1807.03819, 2018.
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+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: pre-training of deep bidirectional transformers for language understanding. CoRR, abs/1810.04805, 2018.
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+Jean-Yves Franceschi, Aymeric Dieuleveut, and Martin Jaggi. Unsupervised scalable representation learning for multivariate time series. In NeurIPS 2019, 2019.
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+Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N. Dauphin. Convolutional sequence to sequence learning. CoRR, abs/1705.03122, 2017.
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+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015.
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+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural Computation, 9(8): 1735–1780, 1997.
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+Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation networks. In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018, pp. 7132–7141, 2018.
+
+Jorn-Henrik Jacobsen, Arnold W.M. Smeulders, and Edouard Oyallon. i-revnet: Deep invertible ¨ networks. In International Conference on Learning Representations, 2018.
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+Yoon Kim. Convolutional neural networks for sentence classification. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 1746– 1751, Doha, Qatar, October 2014. Association for Computational Linguistics. doi: 10.3115/v1/ D14-1181.
+
+Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced research).
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+Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In NIPS Autodiff Workshop, 2017.
+
+Jorge Perez, Javier Marinkovic, and Pablo Barcel ´ o. On the turing completeness of modern neural ´ network architectures. CoRR, abs/1901.03429, 2019.   
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. 2018.   
+Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan Bello, Anselm Levskaya, and Jonathon Shlens. Stand-alone self-attention in vision models. CoRR, abs/1906.05909, 2019.   
+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alexander ¨ Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.   
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. CoRR, abs/1706.03762, 2017.   
+Xiaolong Wang, Ross B. Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018, pp. 7794–7803, 2018.   
+Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G. Carbonell, Ruslan Salakhutdinov, and Quoc V. Le. Xlnet: Generalized autoregressive pretraining for language understanding. CoRR, abs/1906.08237, 2019.   
+Luca Zampieri. Geometric deep learning for volumetric computational fluid dynamics. pp. 67, 2019.
+
+# APPENDIX
+
+# A MORE EXAMPLES WITH CONTENT-BASED ATTENTION
+
+We present more examples of attention probabilities computed by self-attention model. Figure 7 shows average attention at a different query pixel than Figure 6. Figures 8 to 10 display attention for single images.
+
+![](images/1f9da53a96695486ecb2a8dfcfc5f346f28b7e97999df113830159b383a6512b.jpg)  
+Figure 7: Attention probabilities for a model with 6 layers (rows) and 9 heads (columns) using learned relative positional encoding and content-content attention. We present the average of 100 test images. The black square is the query pixel.
+
+![](images/54c07ce4dbf8afdfc3feb029a764183a00b9ab96f97a7cab5466b938d7ea8b6d.jpg)  
+Figure 8: Attention probabilities for a model with 6 layers (rows) and 9 heads (columns) using learned relative positional encoding and content-content based attention. The query pixel (black square) is on the frog head.
+
+![](images/df740f9a54c6afe9b6f34f9e34922741ca33e50591ad0962c61134903c094529.jpg)  
+Figure 9: Attention probabilities for a model with 6 layers (rows) and 9 heads (columns) using learned relative positional encoding and content-content based attention. The query pixel (black square) is on the horse head.
+
+![](images/ae348ab4c5f0409f19d197dd4ff26f9fa5350f7cd5166e7a59676e5f5749ceb3.jpg)  
+Figure 10: Attention probabilities for a model with 6 layers (rows) and 9 heads (columns) using learned relative positional encoding and content-content based attention. The query pixel (black square) is on the building in the background.
+
+B HYPER-PARAMETERS USED IN OUR EXPERIMENTS   
+Table 2: Self-attention network parameters   
+
+<table><tr><td colspan="2">Hyper-parameters</td></tr><tr><td>number of layers number of heads hidden dimension intermediate dimension invertible pooling width dropout probability</td><td>6 9 400 512 2 0.1</td></tr><tr><td>layer normalization epsilon number of epochs batch size learning rate weight decay</td><td>10-12 300 100 0.1 0.0001 0.9</td></tr></table>
+
+C POSITIONAL ENCODING REFERENCES   
+Table 3: Types of positional encoding used by transformers models applied to text (top) and images (bottom). When multiple encoding types have been tried, we report the one advised by the authors.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">type of positional encoding</td><td rowspan="2">relative</td></tr><tr><td>sinusoids</td><td>learned</td><td>quadratic</td></tr><tr><td>Vaswani et al. (2017)</td><td>√</td><td></td><td></td><td></td></tr><tr><td>Radford et al. (2018)</td><td></td><td></td><td></td><td></td></tr><tr><td>Devlin et al. (2018)</td><td></td><td>√</td><td></td><td></td></tr><tr><td>Dai et al. (2019)</td><td>4</td><td></td><td></td><td>√</td></tr><tr><td>Yang et al. (2019)</td><td></td><td></td><td></td><td>√</td></tr><tr><td>Bello et al. (2019)</td><td></td><td>√</td><td></td><td>√</td></tr><tr><td>Ramachandran et al. (2019)</td><td></td><td></td><td></td><td>√</td></tr><tr><td>Our work</td><td></td><td>&lt;</td><td>√</td><td>√</td></tr></table>
+
+# D GENERALIZED LEMMA 1
+
+We present a generalization of Lemma 1 that replaces the necessity of hard attention (to single pixels) by a milder assumption: the attention probabilities should span the grid receptive field. The conditions of this Lemma are still satisfied by Lemma 2, hence Theorem 1 follows.
+
+Lemma 3. Consider a multi-head self-attention layer consisting of $N _ { h } \ge K ^ { 2 }$ heads, $D _ { h } \geq D _ { o u t }$ and let $\omega : [ H ] \times [ W ] \to [ H W ]$ be a pixel indexing. Then, for any convolutional layer with a $K \times$ $K$ kernel and $D _ { o u t }$ output channels, there exists $\{ W _ { \nu a l } ^ { ( h ) } \} _ { h \in [ N _ { h } ] }$ and $W _ { o u t }$ such that $\mathrm { M H S A } ( { \pmb X } ) =$ $\mathrm { C o n v } ( { \pmb X } )$ for every $\pmb { \chi } \in \mathbb { R } ^ { W \times H \times D _ { i n } }$ if and only $i f ,$ for all $\pmb q \in [ H ] \times [ W ]$ , 8
+
+$$
+\mathrm { s p a n } ( \{ e _ { \omega ( \mathbf { q } + \Delta ) } \in \mathbb { R } ^ { H W } : \Delta \in \mathbb { \Delta } _ { K } \} ) \subseteq \mathrm { s p a n } ( \{ \mathrm { v e c t } ( \mathrm { s o f t m a x } ( \pmb { \hat { a } } _ { \mathbf { q } ; \cdot } ^ { ( h ) } ) ) : h \in [ N _ { h } ] \} ) .
+$$
+
+![](images/4d9558b8b43de028204e9089f3e90f033b175ba5533fa457b9da869eddd00161.jpg)  
+Figure 11: Factorization of the vectorized weight matrices $V _ { \pmb { q } } ^ { \mathrm { c o n v } }$ and $V _ { q } ^ { \mathrm { S A } }$ used to compute the output at position $\pmb q$ for an input image of dimension $H \times W$ . On the left: a convolution of kernel $2 \times 2$ , on the right: a self-attention with $N _ { h } = 5$ heads. $D _ { i n } = 2$ , $D _ { o u t } = 3$ in both cases.
+
+Proof. Our first step will be to rework the expression of the Multi-Head Self-Attention operator from equation (1) and equation (4) such that the effect of the multiple heads becomes more transparent:
+
+$$
+\mathrm { M H S A } ( \mathbf { X } ) = b _ { o u t } + \sum _ { h \in [ N _ { h } ] } \mathrm { s o f t m a x } ( \mathbf { A } ^ { ( h ) } ) \mathbf { X } \underbrace { W _ { v a l } ^ { ( h ) } W _ { o u t } [ ( h - 1 ) D _ { h } + 1 : h D _ { h } + 1 ] } _ { W ^ { ( h ) } }
+$$
+
+Note that each head’s value matrix $W _ { \nu a l } ^ { ( h ) } \in \mathbb { R } ^ { D _ { i n } \times D _ { h } }$ and each block of the projection matrix $W _ { o u t }$ of dimension $D _ { h } \times D _ { o u t }$ are learned. Assuming that $D _ { h } \ \ge \ D _ { o u t }$ , we can replace each pair of matrices by a learned matrix $W ^ { ( h ) }$ for each head. We consider one output pixel of the multi-head self-attention and drop the bias term for simplicity:
+
+$$
+\mathrm { M H S A } ( \pmb { \mathsf { X } } ) _ { q ; \mathit { = } } \sum _ { h \in [ N _ { h } ] } \Big ( \sum _ { k } a _ { q , k } ^ { ( h ) } \pmb { \mathsf { X } } _ { k , : } \Big ) \pmb { W } ^ { ( h ) } = \sum _ { k } \pmb { \mathsf { X } } _ { k , : } \underbrace { \Big ( \sum _ { h \in [ N _ { h } ] } a _ { q , k } ^ { ( h ) } \pmb { W } ^ { ( h ) } \Big ) } _ { W _ { q , k } ^ { \mathrm { S A } } \in \mathbb { R } ^ { D _ { i n } \times D _ { o u t } } } ,
+$$
+
+with a(h)q,k $a _ { \pmb { q } , \pmb { k } } ^ { ( h ) } = \mathrm { s o f t m a x } ( \pmb { \mathsf { A } } _ { \pmb { q } , \colon } ^ { ( h ) } ) _ { \pmb { k } }$ . We rewrite the output of a convolution at pixel $\pmb q$ in the same manner:
+
+$$
+\operatorname { C o n v } ( \mathbf { X } ) _ { q , : } = \sum _ { \Delta \in \Delta _ { K } } \mathbf { X } _ { q + \Delta , : } \mathsf { W } _ { \Delta , : , : } = \sum _ { k \in [ H ] \times [ W ] } \mathbf { X } _ { k , : } \underbrace { \mathbb { 1 } _ { \{ k - q \in \Delta _ { K } \} } \mathbf { W } _ { k - q , : , : } } _ { W _ { q , k } ^ { \mathrm { c o n v } } \in \mathbb { R } ^ { D _ { i n } \times D _ { o u t } } } .
+$$
+
+Equality between equations (16) and (17) holds for any input $\pmb { \chi }$ if and only if the linear transformations for each pair of key/query pixels aweight matrices into matrices of dimension $\dot { W } _ { q , k } ^ { \mathrm { c o n v } } = W _ { q , k } ^ { \mathrm { S A } } \dot { \forall q } , k$ $D _ { i n } D _ { o u t } \times H W$ $V _ { \pmb { q } } ^ { \mathrm { c o n v } } : = [ \mathrm { v e c t } ( { W } _ { \pmb { q } , \pmb { k } } ^ { \mathrm { c o n v } } ) ] _ { { \pmb { k } } \in [ H ] \times [ W ] }$ and $V _ { \pmb { q } } ^ { \mathrm { S A } } : = [ \mathrm { v e c t } ( { W } _ { \pmb { q } , \pmb { k } } ^ { \mathrm { S A } } ) ] _ { \pmb { k } \in [ H ] \times [ W ] }$ . Hence, to show that $\mathrm { C o n v } ( \pmb { \mathsf { X } } ) = \mathrm { M H S A } ( \pmb { \mathsf { X } } )$ for all $\pmb { \chi }$ , we must show that $V _ { q } ^ { \mathrm { c o n v } } = V _ { q } ^ { \mathrm { S A } }$ for all $\pmb q$ .
+
+q in the receptive field of pixel The matrix has a restricted support: only the columns associated with a pixel shift $\pmb q$ can be non-zero. This leads to the factorization $V _ { q } ^ { \mathrm { c o n v } } = W ^ { \mathrm { c o n v } } E _ { q }$ $\pmb { \Delta } \in \mathbb { \Delta } _ { K }$ displayed in Figure 11 where $W ^ { \mathrm { c o n v } } \in \mathbb R ^ { D _ { i n } D _ { o u t } \times K ^ { 2 } }$ and $E _ { q } \in \mathbb { R } ^ { K ^ { 2 } \times H W }$ . Given an ordering of the shifts $\pmb { \Delta } \in \mathbb { \Delta } _ { K }$ indexed by $j$ , set $( W ^ { \mathrm { c o n v } } ) _ { : , j } = \mathrm { v e c t } ( \mathbf { W } _ { \Delta , : , : } )$ and $( E _ { q } ) _ { j , : } = e _ { \omega ( q + \Delta ) } .$ . On the other hand, we decompose $V _ { q } ^ { \mathrm { S A } } = W ^ { \mathrm { S A } } A _ { q }$ with $( W ^ { \mathrm { S A } } ) _ { : , h } = \mathrm { v e c t } ( W ^ { ( h ) } )$ and $( \boldsymbol { A } _ { q } ) _ { h , i } = \boldsymbol { a } _ { \boldsymbol { q } , \omega ( i ) } ^ { ( h ) }$ .
+
+The proof is concluded by showing that $\operatorname { r o w } ( E _ { q } ) \subseteq \operatorname { r o w } ( A _ { q } )$ is a necessary and sufficient condition for the existence of a $W ^ { \mathrm { S A } }$ such that any $V _ { q } ^ { \mathrm { c o n v } } = W ^ { \mathrm { c o n v } } E _ { q }$ can be written as $W ^ { \mathrm { S A } } A _ { q }$ .
+
+Sufficient. Given that $\operatorname { r o w } ( E _ { q } ) \subseteq \operatorname { r o w } ( A _ { q } )$ , there exists $\Phi \in \mathbb { R } ^ { K ^ { 2 } \times N _ { h } }$ such that $E _ { q } = \Phi A _ { q }$ and a valid decomposition is $W ^ { \mathrm { S A } } = W ^ { \mathrm { c o n v } } \Phi$ which gives $W ^ { \mathrm { S A } } A _ { q } = V _ { q } ^ { \mathrm { c o n v } }$ .
+
+Necessary. Assume there exists $\boldsymbol { x } \in \mathbb { R } ^ { H W }$ such that $\pmb { x } \in \operatorname { r o w } ( E _ { q } )$ and $\pmb { x } \not \in \operatorname { r o w } ( A _ { q } )$ and set $\pmb { x } ^ { \top }$ to be a row of $V _ { q } ^ { \mathrm { c o n v } }$ . Then, $W ^ { \mathrm { S A } } A _ { q } \ne V _ { q } ^ { \mathrm { c o n v } }$ for any $W ^ { \mathrm { S A } }$ and there is no possible decomposition.
+
+# E GENERALIZED QUADRATIC POSITIONAL ENCODING
+
+We noticed the similarity of the attention probabilities in the quadratic positional encoding (Section 3) to isotropic bivariate Gaussian distributions with bounded support:
+
+$$
+\mathrm { s o f t m a x } ( \pmb { \mathsf { A } } _ { q , : } ) _ { \pmb { k } } = \frac { e ^ { - \alpha \parallel ( \pmb { k } - \pmb { q } ) - \pmb { \Delta } \parallel ^ { 2 } } } { \sum _ { \pmb { k } ^ { \prime } \in [ W ] \times [ H ] } e ^ { - \alpha \parallel ( \pmb { k } ^ { \prime } - \pmb { q } ) - \pmb { \Delta } \parallel ^ { 2 } } } .
+$$
+
+Building on this observation, we further extended our attention mechanism to non-isotropic Gaussian distribution over pixel positions. Each head is parametrized by a center of attention $\pmb { \Delta }$ and a covariance matrix $\pmb { \Sigma }$ to obtain the following attention scores,
+
+$$
+{ \pmb A } _ { q , k } = - \frac { 1 } { 2 } ( { \pmb \delta } - { \pmb \Delta } ) ^ { \top } { \pmb \Sigma } ^ { - 1 } ( { \pmb \delta } - { \pmb \Delta } ) = - \frac { 1 } { 2 } { \pmb \delta } ^ { \top } { \pmb \Sigma } ^ { - 1 } { \pmb \delta } + { \pmb \delta } ^ { \top } { \pmb \Sigma } ^ { - 1 } { \pmb \Delta } - \frac { 1 } { 2 } { \pmb \Delta } ^ { \top } { \pmb \Sigma } ^ { - 1 } { \pmb \Delta } ,
+$$
+
+where, once more, $\delta = k - q$ . The last term can be discarded because the softmax is shift invariant and we rewrite the attention coefficient as a dot product between the head target vector $\textbf {  { v } }$ and the relative position encoding $\mathbf { \Delta } _ { \pmb { r } \delta }$ (consisting of the first and second order combinations of the shift in pixels $\pmb { \delta }$ ):
+
+$$
+\boldsymbol { v } = \frac { 1 } { 2 } ( 2 ( \boldsymbol { \Sigma } ^ { - 1 } \boldsymbol { \Delta } ) _ { 1 } , 2 ( \boldsymbol { \Sigma } ^ { - 1 } \boldsymbol { \Delta } ) _ { 2 } , - \boldsymbol { \Sigma } _ { 1 , 1 } ^ { - 1 } , - \boldsymbol { \Sigma } _ { 2 , 2 } ^ { - 1 } , - 2 \cdot \boldsymbol { \Sigma } _ { 1 , 2 } ^ { - 1 } ) ^ { \top } \mathrm { ~ a n d ~ } \boldsymbol { r } _ { \delta } = ( \delta _ { 1 } , \delta _ { 2 } , \delta _ { 1 } ^ { 2 } , \delta _ { 2 } ^ { 2 } , \delta _ { 1 } \delta _ { 2 } ) ^ { \top } .
+$$
+
+Evaluation. We trained our model using this generalized quadratic relative position encoding. We were curious to see if, using the above encoding the self-attention model would learn to attend to non-isotropic groups of pixels—thus forming unseen patterns in CNNs. Each head was parametrized by $\pmb { \Delta } \in \mathbb { R } ^ { 2 }$ and $\bar { \mathbf { \boldsymbol { \Sigma } } } ^ { - 1 / 2 } \in \mathbb { R } ^ { 2 \times 2 }$ to ensure that the covariance matrix remained positive semi-definite. We initialized the center of attention to $\Delta ^ { ( h ) } \sim \mathcal { N } ( \mathbf { 0 } , 2 I _ { 2 } )$ and $\pmb { \Sigma } ^ { - 1 / 2 } = \pmb { I } _ { 2 } + \pmb { \mathcal { N } } ( \mathbf { 0 } , 0 . 0 1 \pmb { I } _ { 2 } )$ so that initial attention probabilities were close to an isotropic Gaussian. Figure 12 shows that the network did learn non-isotropic attention probability patterns, especially in high layers. Nevertheless, the fact that we do not obtain any performance improvement seems to suggest that attention non-isotropy is not particularly helpful in practice—the quadratic positional encoding suffices.
+
+![](images/85fddc4966d733e6ae88cf4ef57aa879deed51a40c66fe9ce49a20a237f242fb.jpg)  
+Figure 12: Centers of attention of each attention head (different colors) for the 6 self-attention layers using non-isotropic Gaussian parametrization. The central black square is the query pixel, whereas solid and dotted circles represent the $50 \%$ and $90 \%$ percentiles of each Gaussian, respectively.
+
+Pruning degenerated heads. Some non-isotropic attention heads attend on “non-intuitive” patches of pixels: either attending a very thin stripe of pixels, when $\Sigma ^ { - 1 }$ was almost singular, or attending all pixels uniformly, when $\pmb { \Sigma } ^ { - 1 }$ was close to 0 (i.e. constant attention scores). We asked ourselves, are such attention patterns indeed useful for the model or are these heads degenerated and unused? To find out, we pruned all heads having largest eigen-values smaller than $1 0 ^ { - 5 }$ or condition number (ratio of the biggest and smallest eigen-values) greater than $1 0 ^ { 5 }$ . Specifically in our model with 6-layer and 9-heads each, we pruned $[ 2 , 4 , 1 , 2 , 6 , 0 ]$ heads from the first to the last layer. This means that these layers cannot express a $3 \times 3$ kernel anymore. As shown in yellow on fig. 2, this ablation initially hurts a bit the performance, probably due to off biases, but after a few epochs of continued training with a smaller learning rate (divided by 10) the accuracy recovers its unpruned value. Hence, without sacrificing performance, we reduce the size of the parameters and the number of FLOPS by a fourth.
+
+# F INCREASING THE NUMBER OF HEADS
+
+For completeness, we also tested increasing the number of heads of our architecture from 9 to 16.
+
+![](images/42abecd8bb4a4b3709a5f575d5388cb696db4e4503e6c3d9ae68ca1cbcdd2ffe.jpg)  
+Figure 13: Evolution of test accuracy on CIFAR10. Pruned model (yellow) is continued training of the non-isotropic model (orange).
+
+<table><tr><td>Models</td><td>accuracy</td><td># of params</td><td># of FLOPS</td></tr><tr><td>ResNet18</td><td>0.938</td><td>11.2M</td><td>1.1B</td></tr><tr><td>SA quadratic emb.</td><td>0.938</td><td>12.1M</td><td>6.2B</td></tr><tr><td>SA quadratic emb. gen.</td><td>0.934</td><td>12.1M</td><td>6.2B</td></tr><tr><td>SA quadratic emb. gen. pruned</td><td>0.934</td><td>9.7M</td><td>4.9B</td></tr><tr><td>SA learned emb.</td><td>0.918</td><td>12.3M</td><td>6.2B</td></tr><tr><td>SA learned emb. + content</td><td>0.871</td><td>29.5M</td><td>15B</td></tr></table>
+
+![](images/48e1e5eadeb35cca9236528ed4d9cf72d3e173c6d73a6ecd261fa5d4863eabe5.jpg)  
+Table 4: Number of parameters and accuracy on CIFAR-10 per model. SA stands for SelfAttention.   
+Figure 14: Centers of attention for 16 attention heads (different colors) for the 6 self-attention layers using quadratic positional encoding. The central black square is the query pixel, whereas solid and dotted circles represent the $50 \%$ and $90 \%$ percentiles of each Gaussian, respectively.
+
+Similar to Figure 4, we see that the network distinguishes two main types of attention patterns. Localized heads (i.e., those that attend to nearly individual pixels) appear more frequently in the first few layers. The self-attention layer uses these heads to act in a manner similar to how convolutional layers do. Heads with less-localized attention become more common at higher layers.

parse/train/HJx38iC5KX/HJx38iC5KX.md

@@ -0,0 +1,303 @@
+# DOMAIN GENERALIZATION VIA INVARIANT REPRESENTATION UNDER DOMAIN-CLASS DEPENDENCY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning domain-invariant representation is a dominant approach for domain generalization, where we need to build a classifier that is robust toward domain shifts induced by change of users, acoustic or lighting conditions, etc. However, prior domain-invariance-based methods overlooked the underlying dependency of classes (target variable) on source domains during optimization, which causes the trade-off between classification accuracy and domain-invariance, and often interferes with the domain generalization performance. This study first provides the notion of domain generalization under domain-class dependency and elaborates on the importance of considering the dependency by expanding the analysis of Xie et al. (2017). We then propose a method, invariant feature learning under optimal classifier constrains (IFLOC), which explicitly considers the dependency and maintains accuracy while improving domain-invariance. Specifically, the proposed method regularizes the representation so that it has as much domain information as the class labels, unlike prior methods that remove all domain information. Empirical validations show the superior performance of IFLOC to baseline methods, supporting the importance of the domain-class dependency in domain generalization and the efficacy of the proposed method for overcoming the issue.
+
+# 1 INTRODUCTION
+
+In supervised learning problems we typically assume that samples are obtained from the same distribution in training and testing; however, such an assumption does not hold in many practical situations, depressing the classification accuracy for the test data (Torralba & Efros (2011)). One typical situation is domain generalization (e.g., Blanchard et al. (2011)): we have labeled data from several source domains and collectively exploit them so that the trained system generalizes to other, unseen but somewhat similar, target domains. Such challenges arise in many applications, e.g., hand-writing recognition (Shankar et al. (2018)), robust speech recognition (Sriram et al. (2018)), and sensor data interpretation (Erfani et al. (2016)).
+
+To address domain generalization, many methods take advantage of invariant feature learning (Muandet et al. (2013); Erfani et al. (2016); Ghifary et al. (2017); Xie et al. (2017)). Such methods assume that learning the representation $( h )$ that is invariant to domains $( d )$ from input data $( x )$ prevents $h$ to overfit to source domains and leads to higher classification accuracy for unseen domains. To obtain such $h$ , we used various methods to measure the invariance of $h$ to $d$ and imposed some regularization on the measurement. For example, domain adversarial networks (DAN) (Ganin et al. (2016); Xie et al. (2017)) measure the invariance using a domain classifier (also called a discriminator) parameterized by deep neural networks and impose regularization by deceiving it.
+
+Most prior works, however, overlooked the underlying dependency of classes on source domains, which we refer to as domain-class dependency. More specifically, we define domain-class dependency as the situation where domain and class labels are statistically dependent due to some common latent factor $( z )$ of $y$ and $d$ (Figure 1-right). Under the domain-class dependency, merely forcing the optimal domain-invariance harms the classification accuracy, as shown in Figure 1-(c). Intuitively speaking, since $y$ contains information about $d$ under domain-class dependency, $h$ must keep at least as much domain information as $y$ to achieve the optimal classification accuracy; however, invariant feature learning attempts to remove all domain information from $h$ , which causes the trade-off. It might be similar to the situation where $p ( y | x )$ and $p ( x )$ change across domains due to the causal structure $y  x$ (Zhang et al. (2013); Gong et al. (2016) in domain adaptation and Li et al. (2018c) in domain generalization), which we call conditional probability shift. However, the shift does not cause the trade-off as long as $y$ and $\cdot$ are independent (Figure 1-left), so it is necessary to focus on the relationship between $\cdot$ and $\cdot$ .
+
+![](images/a1c8f47a0c3956668c94bc462a79e08ab22c86612b5a33a6144b2afac2070354.jpg)  
+Figure 1: The illustration of the domain-class dependency problem. While Li et al. (2018c) focused on the causal relationship between $x$ and $y$ , we focus on the relationship between $\cdot$ and $\cdot$ because it causes the following trade-off problem. (a) When domain and class are independent, domain invariance and classification accuracy can be optimized at the same time. (b,c) In domain-class dependency, there is a trade-off between these two: (b) optimal invariance cannot be achieved when optimal classification accuracy is achieved, and (c) vice versa. We propose a method to lead explicitly to (b) rather than (c), because the primary purpose for domain generalization is classification, not domain-invariance itself.
+
+Unfortunately, domain-class dependency is common in real-world datasets as shown in Zhang et al. (2013). The dependency can be caused by both the characteristics of data and errors in collecting data. For example, the WISDM Activity Prediction dataset (Kwapisz et al. (2011)), where classes and domains correspond to activities and users, exhibits the dependency because (1) some activities (jogging and climbing stairs) are strenuous (data characteristics) and (2) other activities (sitting and standing) and some users were added only after the study began (data-collection errors).
+
+In this paper, we address domain generalization under domain-class dependency. We first expand the analysis about DAN by Xie et al. (2017), show that domain-class dependency causes the tradeoff problem, and then derive a way to evade the trade-off. Specifically, we investigate the condition where the domain-invariance is maximized under the constraint that it does not interfere with classification accuracy (Figure 1 (b)), because the primary purpose of domain generalization is classification rather than domain-invariance itself. We then propose a novel method invariant feature learning under optimal classifier constraint (IFLOC), modifying DAN’s regularization term to make the learned representation have as much domain information as the class labels, i.e., $H ( d | h ) = H ( d | y )$ holds (here $H$ denotes entropy). Like DAN, IFLOC has an encoder, classifier, and domain discriminator, and also takes over the good properties of DAN: it does not depend on pre-defined metrics (e.g., maximum mean discrepancy (Tzeng et al. (2014))), and it can be trained in an end-to-end manner. Empirical validations show the superior performance of IFLOC to baseline methods, supporting the importance of considering domain-class dependency in domain generalization tasks and the efficacy of the proposed approach for overcoming the issue.
+
+The main contributions of this paper can be summarized as follows. Firstly, we elaborate on the trade-off problem under domain-class dependency, both theoretically and experimentally, for the first time in domain generalization context. Secondly, to address the issue we provide theoretical analysis, which shows to what extent latent representations can become invariant to domains without interfering with classification accuracy. Finally, based on the analysis we propose a novel method IFLOC, and validated its efficacy by the experiments on both synthetic and real world datasets.
+
+# 2 RELATED WORKS
+
+Invariant feature learning is a general-purpose method applicable to domain generalization as well as to domain adaptation (e.g., Tzeng et al. (2014); Ganin et al. (2016)), style transfer (e.g., Lample et al. (2017); Chou et al. (2018)), and fairness-aware classification (e.g., Zemel et al. (2013);
+
+Louizos et al. (2016); Madras et al. (2018)). However, it is likely that adjusting it to each specific task can improve performance. For example, in the fairness-aware classification task Madras et al. (2018) proposed to optimize the fairness criterion directly instead of applying invariance to sensitive variables. By analogy, we adapted invariant feature learning for domain generalization so as to address the domain-class dependency problem.
+
+Domain generalization has been attracting considerable attention in recent years (Blanchard et al. (2011); Muandet et al. (2013); Shankar et al. (2018)). Note that it is different from domain adaptation in that we cannot obtain input and label data from target domain(s). Although the efficacy of domain-invariance-based methods had been known, Li et al. (2017) showed that non end-to-end methods such as DICA (Muandet et al. (2013)) and MTAE (Ghifary et al. (2015)) do not tend to outperform even vanilla CNN. Thus, end-to-end methods are desirable and can be divided into two categories: adversarial-learning-based methods such as DAN (Ganin et al. (2016); Xie et al. (2017)) and pre-defined-metric-based methods (e.g., Ghifary et al. (2017); Li et al. (2018b)).
+
+In particular, IFLOC closely relates to DAN. Although DAN was originally invented for domain adaptation, Xie et al. (2017) showed its efficacy in domain generalization. Also, Xie et al. (2017) provided the intuitive explanation of the trade-off between classification accuracy and domain invariance. However, they did not provide any way to deal with the problem because its focus is invariant feature learning itself. Louppe et al. (2017) provided the similar analysis with Xie et al. (2017), but differs in that they focused on the relation between nuisance parameters (domains) and output distribution of a domain classifier. IFLOC also relates to domain confusion loss (Tzeng et al. (2015)) in that their encoders attempt to directly minimize Kullback-Leibler divergence (KLD) between output distribution of the discriminators and some domain distribution $( p ( d | y )$ in IFLOC and uniform distribution in domain confusion loss), rather than deceive the discriminator as DAN.
+
+There are several studies that address domain generalization without utilizing invariant feature learning. For example, Motiian et al. (2017); Li et al. (2018c) proposed to make use of semantic alignment, which attempts to make latent representation given class label $( p ( h | y ) )$ identical within source domains. This approach was originally proposed in Gong et al. (2016) in domain adaptation context, but its efficacy for domain-class dependency is not obvious because it focuses on conditional probability shift. CrossGrad (Shankar et al. (2018)) is one of the recent state-of-the-art domain generalization methods, which utilizes data augmentation with adversarial examples. However, since the method relies on the assumption that $y$ and $d$ are independent, it might not be directly applicable to our setting. MLDG (Li et al. (2018a)), also one of the state-of-the-art methods, utilizes metalearning. Since it makes no assumption about the relation between $y$ and $d$ , it could be combined with our proposed method, though we have not experimentally confirmed it.
+
+There are several kinds of distributional shifts other than domain-class dependency, such as conditional probability shift. Although the distinction between that shift and domain-class dependency is important, it has been received less attention. For example, Li et al. (2018c) claimed that conditional probability shift might harm the performance of domain-invariance-based methods, but our analysis in Sec.4.1.1 suggests that the root cause of the performance degradation is not it but domain-class dependency. They also proposed to correct the shift of $\cdot$ across source domains by aligning sampling frequency of each class across domains. However, this approach is not applicable to when some classes are rarely or never appear in some domains (e.g., as in WISDM dataset). In domain adaptation, Zhang et al. (2013); Gong et al. (2016) address the situation where $p ( y )$ changes across source and target domains by estimating $p ( y )$ change using unlabeled target data. However, this approach is not applicable (or necessary) to domain generalization because our problem setting is different from theirs in that we are agnostic on target domain and aim to care about $\cdot$ change within source domains instead.
+
+# 3 PRELIMINARIES
+
+# 3.1 PROBLEM STATEMENT OF DOMAIN GENERALIZATION
+
+Denote $\mathcal { X } , \mathcal { y }$ , and $\mathcal { D }$ as the input feature, label, and domain spaces, respectively. With random variables $x \in { \mathcal { X } } , \ y \in { \mathcal { Y } }$ , and $d \in \mathcal { D }$ ，we can define the probability distribution for each domain $d$ as $p ( x , y | d )$ . Here, we assume that $y$ and $d$ are discrete variables for simplicity. In domain generalization, we are given a training dataset consisting of $D _ { s } = \{ x _ { i } ^ { s } , y _ { i } ^ { s } \} _ { i = 1 } ^ { n ^ { s } }$ for all $s \in \{ 1 , 2 , . . . , m \}$ . Here, each $D _ { s }$ corresponds to samples drawn from the source domain $p ( x , y | d = s )$ . Using the training dataset, we train a classifier $f : \mathcal { X }  \mathcal { Y }$ , and use the classifier to predict labels of samples drawn from the unknown target domain $p ( x , y | d = t )$ .
+
+# 3.2 DOMAIN ADVERSARIAL NETWORKS FOR DOMAIN GENERALIZATION
+
+In this section, we give a brief overview of DAN (Ganin et al. (2016)) given that our proposed method is an extension of it. DAN trains a domain discriminator that attempts to predict domains from latent representations encoded by an encoder, while simultaneously trains the encoder to remove domain information by deceiving the discriminator. This procedure ensures that there is no or little domain information in the representations, so a label classifier attached to the encoder can make robust predictions regarding unseen target domains.
+
+Formally, we denote $f _ { E } ( x ) , q _ { M } ( y | h )$ , and $q _ { D } ( d | h ) \ ( E , M$ , and $D$ are the parameters) as deterministic encoder, probabilistic model of label classifier, and that of domain discriminator, respectively. Then, the objective function of DAN is described as follows:
+
+$$
+\operatorname* { m i n } _ { E , M } \operatorname* { m a x } _ { D } J ( E , M , D ) = \mathbb { E } _ { x , d , y \sim p ( x , d , y ) } [ \gamma \log q _ { D } ( d | h = f _ { E } ( x ) ) - \log q _ { M } ( y | h = f _ { E } ( x ) ) ] .
+$$
+
+Here, the second term in Eq.1 simply maximizes the log likelihood of $q _ { M }$ as well as in standard classification problems. On the other hand, the first term corresponds to a minimax game between the encoder and discriminator, where the decoder $q _ { D } ( d | h )$ tries to predict $d$ from $h$ and the encoder $f _ { E } ( x )$ tries to fool $q _ { D } ( d | h )$ .
+
+As Xie et al. (2017) originally showed, the minmax game ensures that the learned representation has no or little domain information, i.e., the representation becomes domain-invariant. Such invariance makes a prediction from $h$ to $y$ independent from $d$ , and therefore hopefully helps to build a classifier that correctly handle samples drawn from unknown domains. Below is a brief explanation.
+
+Since $h$ is a deterministic mapping of $x$ , the joint probability distribution of $h , d$ and $y$ can be defined as follows:
+
+$$
+\begin{array} { l } { \displaystyle \tilde { p } _ { E } ( h , d , y ) = \int _ { x } \tilde { p } _ { E } ( x , d , h , y ) d x } \\ { \displaystyle = \int _ { x } p ( x , d , y ) \delta ( f _ { E } ( x ) = h ) d x } \end{array}
+$$
+
+Here, we use the notation of $\tilde { p } _ { E }$ for the true probability distribution that depends on the encoder’s parameter $E$ . Using Eq.2, Eq.1 can be replaced as follows:
+
+$$
+\operatorname* { m i n } _ { E , M } \operatorname* { m a x } _ { D } J ( E , M , D ) = \mathbb { E } _ { h , d , y \sim \tilde { p } _ { E } ( h , d , y ) } [ \gamma \log q _ { D } ( d | h ) - \log q _ { M } ( y | h ) ]
+$$
+
+Assuming $E$ is fixed, the solutions $M ^ { * }$ and $D ^ { * }$ to Eq.3 obviously satisfy $q _ { M ^ { * } } ( y | h ) = \tilde { p } _ { E } ( y | h )$ and $q _ { D ^ { * } } ( d | h ) = \tilde { p } _ { E } ( d | h )$ . Then, substituting $q _ { M ^ { * } }$ and $q _ { D ^ { * } }$ into Eq.3, we can obtain the following optimization problem depending only on $E$ :
+
+$$
+\displaystyle \operatorname* { m i n } _ { E } J ( E ) = - \gamma H _ { \tilde { p } _ { E } } ( d | h ) + H _ { \tilde { p } _ { E } } ( y | h )
+$$
+
+Solving Eq.4, we can obtain the solutions $M ^ { * } , D ^ { * }$ , and $E ^ { * }$ , which are in Nash equilibrium. Here, $H _ { \tilde { p } _ { E } } ( d | h )$ means conditional entropy with joint probability distribution $\tilde { p } _ { E } ( d , h )$ . Thus, minimizing the second term in Eq.4 intuitively means learning (the mapping function $f _ { E }$ to) the latent representation $h$ which contains as much information about $y$ as possible. On the other hand, the first term can be regarded as a regularizer that attempts to learn $h$ which is invariant to $d$ .
+
+# 4 OUR APPROACH
+
+# 4.1 ANALYSIS OF DOMAIN-CLASS DEPENDENCY
+
+We address domain generalization under domain-class dependency, i.e., the situation where $p ( y | d ) \neq p ( y )$ holds. Although the issue had been overlooked, it is common in real-world datasets given that they can have the dependency in nature, e.g., nocturnal annimals (class) do not tend to appear in daylight (domain), and the dependency in such datasets is often not corrected unlike in standard benchmark datasets. To address the problem, we expand the analysis of Xie et al. (2017) to theoretically show that domain-class dependency causes the trade-off between accuracy and invariance, and to consider to what extent the latent representation should become invariant.
+
+# 4.1.1 TRADE-OFF CAUSED BY DOMAIN-CLASS DEPENDENCY
+
+We first show that the performance of DAN explained in the previous section suffers from the existence of domain-class dependency. The following analysis also suggests that all of the methods that utilize domain-invariant representation suffer from the dependency. Concretely, we show that the domain-class dependency causes the trade-off between classification accuracy and domain invariance: when $d$ and $y$ are not statistically independent, any $E$ cannot optimize the first and second term in Eq.4 at the same time. In this analysis, for simplicity, we assume that we can obtain any $\tilde { p } _ { E } ( y | h ) , \tilde { p } _ { E } ( d | h )$ , i.e., the models have enough capacity and there are no optimization difficulties.
+
+To begin with, we consider only the first term in Eq.4 and address the optimization problem:
+
+$$
+\operatorname* { m i n } _ { E } J _ { 1 } ( E ) = - \gamma H _ { \tilde { p } _ { E } } ( d | h )
+$$
+
+Using the property of entropy, $H _ { \tilde { p } _ { E } } ( d | h )$ is bounded as follows:
+
+$$
+H _ { \tilde { p } _ { E } } ( d | h ) \leq H ( d )
+$$
+
+Here, $H _ { \tilde { p } _ { E } } ( d | h ) = H ( d )$ holds only if $h$ and $d$ are independent. Thus, Eq.5 has the solution $E ^ { 1 * }$ , which satisfies the following condition:
+
+$$
+H _ { \tilde { p } _ { E ^ { 1 * } } } ( d | h ) = H ( d )
+$$
+
+Eq.7 suggests that the regularizer in DAN is intended to remove all information about domains from latent variables, thereby making domains and latent variables independent.
+
+Next, we analogically consider only the second term in Eq.4, thereby addressing the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { E } J _ { 2 } ( E ) = H _ { \tilde { p } _ { E } } ( y | h )
+$$
+
+Since conditional entropy $H ( a | b )$ has a minimum value when $b$ contains all information about $a$ , Eq.8 has the solution $E ^ { 2 * }$ , which satisfies the following equation:
+
+$$
+H _ { \tilde { p } _ { E ^ { 2 * } } } ( d | h ) = H _ { \tilde { p } _ { E ^ { 2 * } } } ( d | h , y )
+$$
+
+Using Eq.9 and the property of entropy: $H ( a | b , c ) \leq H ( a | b )$ , we can obtain the following condition:
+
+$$
+H _ { \tilde { p } _ { E ^ { 2 * } } } ( d | h ) = H _ { \tilde { p } _ { E ^ { 2 * } } } ( d | h , y ) \leq H ( d | y )
+$$
+
+Eq.10 implies that $h$ has at least as much information about $d$ as $y$ does. Now, we assume that $y$ and $d$ are not independent, i.e., domain-class dependency exists, and obtain the following condition:
+
+$$
+H _ { \tilde { p } _ { E ^ { 2 * } } } ( d | h ) \leq H ( d | y ) < H ( d )
+$$
+
+Considering Eq.7 and Eq.11, $E ^ { 1 * } \neq E ^ { 2 * }$ holds. This means that when $y$ and $d$ are not independent, there is no solution $E$ that optimizes Eq.5 and Eq.8 at the same time, i.e., there is a trade-off between classification accuracy and domain invariance.
+
+It is worth noting that although Li et al. (2018c) claimed that conditional probability shift (the causal structure $\cdot$ ) could harm the domain generalization performance of invariance-based methods, this analysis suggests that it does not harm DAN as long as domain and class are independent. It can be confirmed by considering Eq.7 and Eq.10; even when the shift occurs, i.e., $H ( y | x , d ) < H ( y | x )$ holds and then $H _ { \tilde { p } _ { E } } ( y | h , d ) \le H _ { \tilde { p } _ { E } } ( y | h )$ holds, it does not conflict with $H _ { \tilde { p } _ { E ^ { * } } } ( d | h ) = H ( d | y ) =$ $\cdot$ as long as $H ( d | y ) = H ( d )$ holds. In other words, we only need to infer latent variable $\cdot$ that satisfies the causal structure $y  h  x$ to avoid the trade-off. Although Gong et al. (2016) showed the similar result in domain adaptation context, it has been overlooked in domain generalization.
+
+# 4.1.2 OPTIMAL DOMAIN-INVARIANCE UNDER DOMAIN-CLASS DEPENDENCY
+
+If we cannot avoid the trade-off, the next question is how to deal with it, i.e., to what extent the representation should become domain-invariant for domain generalization tasks. We propose to maximize domain-invariance within a range that does not interfere with classification accuracy, rather than merely enforcing domain-invariance without any constraint. The reason for the constraint is that the primary purpose of domain generalization is classification for unseen domains rather than domain-invariance itself, and the improvement of the invariance could harm the classification performance for them. For example, in WISDM, if we know the target activity (class) was performed by not an old but yound man (domain), we can predict it was jogging with higher probability, so we should avoid removing such domain information that is useful in the classification task. As another example, if the target domain has the similar characteristics as a certain source domain (or as an extreme case, $p ( x , \bar { y } | d = s ) = p ( x , y | d = t )$ holds), giving priority to domain-invariance obviously interferes with the domain generalization performance.
+
+Given that Eq.10 is the necessary condition where we can build an optimal classifier, we can write the optimization problem of maximizing domain-invariance within a range that does not interfere with classification accuracy as follows:
+
+$$
+\begin{array} { r l } & { \underset { E } { \operatorname* { m i n } } J ( E ) = - \gamma H _ { \tilde { p } _ { E } } ( d | h ) } \\ & { \qquad s u b j e c t t o \ : H _ { \tilde { p } _ { E } } ( d | h ) \leq H ( d | y ) } \end{array}
+$$
+
+Continuing, we can obtain the solution $E ^ { * }$ , which obviously satisfies $H _ { \tilde { p } _ { E ^ { * } } } ( d | h ) = H ( d | y )$ . More specifically, when we want to maximize domain-invariance (Eq.12) within the range that does not interfere with accuracy (Eq.13), the solution satisfies $H _ { \tilde { p } _ { E ^ { * } } } ( d | h ) \ : = \ : H ( d | y )$ . So without interfering with classification accuracy we can remove domain information from $h$ to the extent that $\dot { H } _ { \tilde { p } _ { E } } ( d | h ) = H ( \dot { d } | y )$ holds, i.e., $h$ has as much information about $d$ as $y$ does.
+
+# 4.2 PROPOSED METHOD
+
+Based on the above analysis, the remaining challenge is how to impose such regularization that makes $H _ { \tilde { p } _ { E } } ( d | h ) = H ( d | y )$ hold. Although DAN might be able to achive that condition by carefully tuning the regularizer ( $\langle \gamma$ in Eq.1), such tuning is time-consuming and impracticable as suggested in our experiments. Alternatively, we propose a novel method called IFLOC, modifying DAN’s regularization term: while the encoder of DAN attempts to fool a discriminator, that of IFLOC attempts to directly minimize KLD between $p ( d | y )$ and $q _ { D } ( d | h )$ . Formally, IFLOC solves the following joint optimization problem by alternating gradient descent.
+
+$$
+\begin{array} { r l } & { \underset { E , M } { \mathop { \operatorname* { m i n } } } J ( E , M ) = \mathbb { E } _ { x , d , y \sim p ( x , d , y ) } [ \gamma D _ { K L } [ p ( d | y ) | q _ { D } ( d | h - f _ { E } ( x ) ) ] - \log q _ { M } ( y | h - f _ { E } ( x ) ) ] } \\ & { } \\ & { \underset { D } { \mathop { \operatorname* { m i n } } } J ( E , D ) = \mathbb { E } _ { x , d \sim p ( x , d ) } [ - \log q _ { D } ( d | h = f _ { E } ( x ) ) ] } \end{array}
+$$
+
+The second term in Eq.14 and Eq.15 respectively means maximization of log-likelihood of $q _ { M }$ and $q _ { D }$ as well as DAN. However, the first term in Eq.14 differs from DAN in that it is intended to satisfy $\bar { q } _ { D } ( d | h ) = p ( d | y )$ for almost every $( y , h )$ pair.
+
+Next we show that the regularization of IFLOC is intended to achieve $H _ { \tilde { p } _ { E } } ( d | h ) = H ( d | y )$ . Similarly to Section 3.2, $D ^ { * }$ and $M ^ { * }$ , which are the solutions to Eq.14 and Eq.15 with fixed $E$ , obviously satisfy $q _ { D } ^ { * } = \tilde { p } _ { E } ( d | h ) , q _ { M } ^ { * } = \tilde { p } _ { E } ( y | h )$ . Thus Eq.14 can be written as follows:
+
+$$
+\displaystyle \operatorname* { m i n } _ { E } J ( E ) = \mathbb { E } _ { h , y \sim \tilde { p } _ { E } ( h , y ) } [ \gamma D _ { K L } [ p ( d | y ) | \tilde { p } _ { E } ( d | h ) ] ] + H _ { \tilde { p } _ { E } } ( y | h )
+$$
+
+Since the minimization of the KLD term does not interfere with the second term optimization, $E ^ { * }$ , which is the solution to Eq.16 and in Nash equilibrium, satisfies $\mathbb { E } _ { h , y \sim \tilde { p } _ { E ^ { * } } ( h , y ) } [ D _ { K L } [ p ( d | y ) | \tilde { p } _ { E ^ { * } } ( d | h ) ] ] = 0$ . Then, $\tilde { H _ { \tilde { p } _ { E ^ { * } } } } ( d | h ) = H ( d | y )$ obviously holds.
+
+Note that we cannot obtain true $p ( d | y )$ , but we can use a maximum likelihood or maximum a posteriori estimator for it. Also, we could use some divergences other than $D _ { K L } [ p ( d | y ) | q _ { D } ( d | h ) ]$ in Eq.14, e.g., $D _ { K L } [ q _ { D } ( d | h ) | p ( d | y ) ]$ , but in doing so, we could not observe performance gain, so we discontinued testing them.
+
+Table 1: Sample sizes for each domain-class pair in BMNISTR. Those for the classes $0 { \sim } 4$ are variable across domains, whereas the classes $5 { \sim } 9$ have identical sample sizes across domains.   
+
+<table><tr><td>Dataset</td><td>Class</td><td>M0</td><td>M15</td><td>M30</td><td>M45</td><td>M60</td><td>M75</td></tr><tr><td>BMNISTR-1</td><td>0~4 5~9</td><td>100 100</td><td>85 100</td><td>70 100</td><td>55 100</td><td>40 100</td><td>25 100</td></tr><tr><td>BMNISTR-2</td><td>0~4 5~9</td><td>100 100</td><td>80 100</td><td>60 100</td><td>40 100</td><td>20 100</td><td>0 100</td></tr><tr><td>BMNISTR-3</td><td>0~4 5~9</td><td>100 100</td><td>90 100</td><td>80 100</td><td>70 100</td><td>60 100</td><td>50 100</td></tr><tr><td>BMNISTR-4</td><td>0~4 5~9</td><td>100 100</td><td>25 100</td><td>100 100</td><td>25 100</td><td>100 100</td><td>25 100</td></tr></table>
+
+# 5 EXPERIMENTS
+
+# 5.1 DATASETS
+
+BMNISTR We created the Biased Rotated MNIST dataset (BMNISTR) by modifying the sample size of MNISTR (Ghifary et al. (2015)) so that class distribution differs among the domains. Specifically, we created four variants of MNISTR that have different types of domain-class dependency, referred to as BMNISTR-1 through BMNISTR-4. As shown in Table 1, BMNISTR-1, -2, and -3 have similar trends but different degrees of dependency; BMNISTR-1 and BMNISTR-4 differ in trends. In MNISTR, each class is represented by 10 digits. Each domain was created by rotating images by 15 degree increments: 0, 15, 30, 45, 60, and 75 (referred to as M0, ..., M75). Each image is cropped to $1 6 \mathrm { ~ x ~ } 1 6$ in accordance with Ghifary et al. (2015). In training, we employed one-domainleave-out setting: trained on five of the six domains and then tested using the remaining one. We used two convolution layers and two fully-connected (FC) layers (with nonlinear activations) as the encoder, three FC layers as the classifier, and two FC layers as the discriminator.
+
+PACS The PACS dataset (Li et al. (2017)) has 9991 images across 7 categories (dog, elephant, giraffe, guitar, house, horse, and person) and 4 domains comprising different stylistic depictions (Photo, Art painting, Cartoon, and Sketch). It has domain-class dependency probably because samples in some $<$ <domain, class $>$ pairs are difficult to obtain. For example, $-$ ) is much higher than $-$ , which indicates that photos of person are easier to obtain than those of animals, but sketches of persons are more difficult to obtain than those of animals in the wild. The concrete sample sizes for each category and style is shown in Table 4 in appendix. In training, we employed one-domain-leave-out setting as well as in BMNISTR, and used the ImageNet pre-trained AlexNet CNN (Krizhevsky et al. (2012)) as the base network, following previous studies (Li et al. (2017; 2018a)). The two-FC-layer discriminator was connected to the last FC layer, following Ganin et al. (2016).
+
+WISDM The WISDM Activity Prediction dataset contains sensor data of accelerometers for six human activities (walking, jogging, upstairs, downstairs, sitting, and standing) performed by 36 users (domains). WISDM suffers from the dependency due to the reason noted in Sec.1. The concrete sample sizes for each user and activity is shown in Table 5 in appendix. Referring to Andrey (2017), we use the sliding-window procedure with 60 frames $\left( = 3 \right.$ seconds) and 20-frame overlap. The total number of samples was 54455. In training, we used randomly chosen ${ < } 1 0 / 2 6 >$ , ${ < } 1 6 / 2 0 >$ , and ${ < } 2 6 / 1 0 >$ users as $<$ source / target $>$ domains. We parameterized the encoder using three convolution layers followed by one FC layer and the classifier by logistic regression, following previous studies (Yang et al. (2015); Iwasawa et al. (2017)). The two-FC-layer discriminator was connected to the output of the encoder.
+
+# 5.2 BASELINES
+
+To demonstrate the efficacy of the proposed method IFLOC, we compared it with the following methods. (1) CNN is a vanilla convolutional networks trained on the aggregation of data from all source domains. Although CNN has no special treatments for domain generalization, Li et al. (2017) reports that it outperforms many traditional domain generalization methods. (2) DAN (Xie et al.
+
+Table 2: Mean F scores for the classes $0 { \sim } 4$ and classes $5 { \sim } 9$ with the target domain M0.   
+
+<table><tr><td>Dataset</td><td>Class</td><td>CNN</td><td>DAN</td><td>IFLOC-Abl</td><td>IFLOC</td><td>Relative Improvement of IFLOC to IFLOC-Abl</td></tr><tr><td>BMNISTR-1</td><td>0~4</td><td>83.86</td><td>84.54</td><td>87.46</td><td>90.62</td><td>3.6%</td></tr><tr><td></td><td>5~9</td><td>83.90</td><td>85.24</td><td>86.46</td><td>88.10</td><td>1.9%</td></tr><tr><td>BMNISTR-2</td><td>0~4</td><td>84.76</td><td>86.20</td><td>86.42</td><td>89.58</td><td>3.7%</td></tr><tr><td></td><td>5~9</td><td>83.36</td><td>85.22</td><td>85.62</td><td>86.86</td><td>1.4%</td></tr><tr><td>BMNISTR-3</td><td>0~4</td><td>82.54</td><td>85.30</td><td>88.60</td><td>89.64</td><td>1.2%</td></tr><tr><td></td><td>5~9</td><td>82.18</td><td>85.80</td><td>87.60</td><td>89.04</td><td>1.6%</td></tr><tr><td>BMNISTR-4</td><td>0~4</td><td>71.26</td><td>79.22</td><td>76.56</td><td>80.02</td><td>4.5%</td></tr><tr><td></td><td>5~9</td><td>78.62</td><td>83.14</td><td>82.94</td><td>82.80</td><td>-0.2%</td></tr></table>
+
+(2017)) is expected to generalize across domains via invariant feature learning, but it has the tradeoff between domain invariance and classification accuracy as explained in Section 4.1.1. We trained DAN with a gradient reverse layer following Ganin et al. (2016); Xie et al. (2017). Also, we used (3) IFLOC-Abl, which is a version of IFLOC modified for ablation studies. IFLOC-Abl replaces $D _ { K L } [ p ( d | y ) | \tilde { p } _ { E } ( d | h ) ]$ in Eq.14 of $D _ { K L } [ p ( d ) | \tilde { p } _ { E } ( d | h ) ]$ , so it attempts to learn the representation that is completely invariant to domains or make $\begin{array} { r } { H ( d | h ) = H ( d ) } \end{array}$ hold as well as DAN. Comparing IFLOC and IFLOC-Abl, we measured the genuine effect of taking domain-class dependency into account. In training IFLOC and IFLOC-Abl, we cannot obtain true $p ( d | y )$ and $p ( \bar { d } )$ , so we used maximum likelihood estimators of them for calculating the KLD terms.
+
+# 5.3 EXPERIMENTAL SETTINGS
+
+For all the datasets and methods, we used RMSprop for optimization. And we set the learning rate, batch size, and the number of iterations as 5e-4, 128, and 10k for BMNISTR; 5e-5, 64, and $1 0 \mathrm { k }$ for PACS; 1e-4, 128, and $3 0 \mathrm { k }$ for WISDM, respectively. For DAN, IFLOC-Abl, and IFLOC we optimized the weighting parameter $\gamma$ from $\{ 0 . 0 \bar { 0 0 } 1 , 0 . 0 \bar { 0 1 } , 0 . 0 1 , 0 . 1 , 1 , 1 0 \}$ , and used the $\gamma$ annealing following Ganin et al. (2016). In all the experiments, we split source data into $80 \%$ of training data and $20 \%$ of validation data, assuming that target data are not absolutely available in the training phase. We conducted experiments multiple times with different seeds. Specifically, we trained on 10 and 25 seeds in BMNISTR and WISDM, chose the best hyperparameter that achieved the highest validation accuracies measured in each epoch, and reported the mean scores (accuracies and f-values) for the hyperparameter. In PACS, since it requires a long time to train on, we chose the best $\gamma$ from $\{ 0 . 0 0 \dot { 0 } \dot { 1 } , \dot { 0 } . 0 0 1 , 0 . 0 1 , 0 . 1 \}$ with three experiment, and reported the mean scores in experiments with 20 seeds in total. Also, we empirically measured the level of domain-invariance by training a post-hoc classifier that is intended to predict $d$ over learned representation, following previous studies (Xie et al. (2017); Iwasawa et al. (2017); Moyer et al. (2018)). Specifically, we trained the classifier with 400 hidden units on $1 0 \mathrm { k }$ iterations (by RMSprop optimizer with a 0.001 learning rate and 128 batch size) with the data that is used for training the models. We then evaluated the domain classification accuracy (referred to as D-Acc) 10 times at equal intervals during training, and reported D-Acc in the nearest time when the validation accuracy is maximized.
+
+# 5.4 RESULTS
+
+We first investigated how domain-class dependency affects the performance of domain-invariancebased methods. In Table 2, we compared mean f-scores for the classes 0 through 4 and classes 5 through 9 in BMNISTR with the target domain M0. Recall that sample sizes for the classes $0 { \sim } 4$ are variable across domains, whereas the classes $5 { \sim } 9$ has identical sample sizes across domains (Table 1). The f-scores show that IFLOC outperformed DAN and IFLOC-Abl in most datasetclass pairs, which supports that domain-class dependency depresses the performance of domaininvariance-based methods and that IFLOC can mitigate the problem. Futher, relative improvement of IFLOC to IFLOC-Abl is more significant for the classes $0 { \sim } 4$ than $5 { \sim } 9$ in BMNISTR-1, BMNISTR2, and BMNISTR-4, suggesting that IFLOC tends to increase performance more significantly for classes where the domain-class dependency occurs. Also, the improvement is more significant in
+
+![](images/a6a9970777ab61ce57842bcd699cce01520f999dd90594796c444b68de6cd6fc.jpg)  
+Figure 2: Class accuracy (Y-Acc) and domain accuracy (D-Acc) with various $\gamma$ in BMNISTR-1. Each caption shows the metric name (Y-Acc or D-Acc) and target domain.
+
+![](images/6748a095442cbef294cb517e35607deabed7150afa87ba2160db7597f3f864e5.jpg)  
+Figure 3: Class accuracy (Y-Acc) and domain accuracy (D-Acc) with various $\gamma$ in WISDM. Each caption shows the metric name (Y-Acc or D-Acc) and target domain.
+
+BMNISTR-1 than in BMNISTR-3, suggesting that the stronger the domain-class dependency is, the lower the performance of domain-invariance-based methods becomes. Finally, although the dependencies of BMNISTR-1 and BMNISTR-4 have different trends as described in Table 1, IFLOC improved f-scores in both datasets.
+
+Next we investigated the relationship between the strength of regularization and performance. Figures 2 and 3 show the hyperparameter sensitivity of class accuracies (Y-Acc) and domain accuracies (D-Acc) for DAN, IFLOC-Abl, and IFLOC. Note that the gray line in Figures 2-(c) and 3-(c) shows the trivial baseline predicting the majority label. From these figures, we can make the following observations. (1) All the methods including IFLOC could improve the invariance by using stronger regularizer. Concretely, Figures 2-(c) and 3-(c) show D-Acc tends to become low (invariance becomes high) for all the models when the regularizer becomes strong (such as $\gamma = 1$ or 10) except that IFLOC-Abl has high D-Acc with $\gamma = 1 0$ in Figure 3-(c). That high D-Acc might be because the validation accuracy achieved the highest value before the domain-invariance matured. (Recall that the more the representation becomes invariant, the lower the accurary becomes under the trade-off). (2) The training of IFLOC tends to be more stable than that of DAN when the regularizer becomes strong. Figures 2-(a,b) and 3-(a,b) show that IFLOC and IFLOC-Abl could achieve higher Y-Acc than DAN when $\gamma = 1$ or 10, i.e., the regularization is strong, except for IFLOC-Abl with $\gamma = 1$ in Figure 3-(b). This tendency might be because the regularizer of IFLOC is KLD and thus bounded by 0, in contrast to that of DAN that can increase to infinity and destabilize the traininig. (3) IFLOC, as it was designed, does not tend to decrease classification accuracy with strong regularizer, and thus IFLOC is robust toward hyperparameter choice. Figures 2-(b) and 3-(a,b) show that while Y-Acc of IFLOC-Abl decreases with strong regularization (such as when $\gamma = 1$ or 10), that of IFLOC does not decrease as much.
+
+Finally, we compared mean accuracies (with standard errors) in both synthetic (BMNISTR) and standard benchmark (PACS and WISDM) datasets (Table 3). Note that the $H ( d ) / H ( d | y )$ column is estimated from source data, which indicates the strength of domain-class dependency. IFLOC outperformed IFLOC-Abl in BMNISTR with all the target domains; PACS with photo, art painting, and sketch target domains; and WISDM with 26- and 20-target-user domains. Also, IFLOC outperformed DAN in BMNISTR with all the target domains; PACS with photo and art painting target domains; and WISDM with 26- and 10-target-user domains. This supports the importance of considering domain-class dependency in real-world datasets and the efficacy of the proposed model.
+
+Table 3: Accuracies for each dataset and target domain   
+
+<table><tr><td>Dataset</td><td>Target</td><td>H(d)/H(dly) （%）</td><td>CNN</td><td>DAN</td><td>IFLOC-Abl</td><td>IFLOC</td></tr><tr><td>BMNISTR-1</td><td>M0 M15</td><td>101.2</td><td>83.9 ± 0.4</td><td>85.0± 0.4</td><td>87.0 ± 0.4</td><td>89.3± 0.4</td></tr><tr><td></td><td>M30</td><td>101.5</td><td>98.5± 0.2 97.5 ± 0.1</td><td>98.5± 0.1</td><td>98.3 ± 0.2</td><td>98.8 ± 0.1</td></tr><tr><td></td><td>M45</td><td>101.6</td><td></td><td>97.4± 0.1</td><td>97.6± 0.1</td><td>98.3 ± 0.2</td></tr><tr><td></td><td></td><td>101.6</td><td>89.9 ± 0.9</td><td>90.2 ± 0.6</td><td>92.8 ± 0.5</td><td>93.3 ± 0.6</td></tr><tr><td></td><td>M60</td><td>101.3</td><td>96.7 ± 0.3</td><td>97.0 ± 0.2</td><td>96.6± 0.2</td><td>97.4 ± 0.2</td></tr><tr><td></td><td>M75</td><td>100.7</td><td>87.1 ± 0.5</td><td>87.3± 0.4</td><td>87.7± 0.5</td><td>88.1 ± 0.4</td></tr><tr><td>BMNISTR-2</td><td>Avg</td><td></td><td>92.3</td><td>92.6</td><td>93.3</td><td>94.2</td></tr><tr><td>BMNISTR-3</td><td>Avg Avg</td><td></td><td>92.3</td><td>92.2</td><td>93</td><td>94.2</td></tr><tr><td>BMNISTR-4</td><td></td><td></td><td>92.2</td><td>92.7</td><td>94</td><td>94.5</td></tr><tr><td>PACS</td><td>Avg</td><td></td><td>90.6</td><td>91.7</td><td>91.6</td><td>92.9</td></tr><tr><td></td><td>photo</td><td>107.2</td><td>80.6± 0.3</td><td>81.1 ± 0.3</td><td>81.6± 0.3</td><td>82.9 ± 0.2</td></tr><tr><td></td><td>art_painting</td><td>108.5</td><td>59.2± 0.4</td><td>60.1± 0.3</td><td>60.5± 0.4</td><td>61.2 ± 0.2</td></tr><tr><td></td><td>cartoon</td><td>109.7</td><td>63.2± 0.3</td><td>64.3 ± 0.3</td><td>64.4 ± 0.4</td><td>63.8± 0.3</td></tr><tr><td></td><td>sketch</td><td>101.5</td><td>58.2± 0.5</td><td>58.9± 0.4</td><td>58.1± 0.6</td><td>59.0 ± 0.5</td></tr><tr><td>WISDM</td><td>Avg</td><td></td><td>65.3</td><td>66.1</td><td>66.2</td><td>66.7</td></tr><tr><td></td><td>26users</td><td>107.1</td><td>78.3± 0.3</td><td>78.2± 0.3</td><td>78.4± 0.2</td><td>78.9 ± 0.3</td></tr><tr><td></td><td>20 users</td><td>104.2</td><td>79.7 ± 0.2</td><td>80.2 ± 0.3</td><td>79.7 ± 0.3</td><td>80.0± 0.3</td></tr><tr><td></td><td>10 users</td><td>103.5</td><td>80.6± 0.2</td><td>80.6± 0.2</td><td>81.2 ± 0.3</td><td>81.2 ± 0.3</td></tr></table>
+
+Table 3 also shows that when the number of source domains increased from 10 to 26 in WISDM, the improvement of IFLOC from IFLOC-Abl became insignificant. One possible reason is that WISDM with 10 target users has low domain-class dependency than with 26 target users as shown in the $H ( d ) / H ( d | y )$ column. Another possible reason is the optimization difficulty. As Moyer et al. (2018) reported, in adversarial invariant feature learning, an encoder often overfits to the discriminator trained alongside that encoder, and does not provide truly invariant representation (the same problem can be observed in Figures 2-(c) and 3-(c)). We suspect that when the number of source domains increases, the optimization of the domain discriminator becomes difficult, which makes the encoder overfit to that poor discriminator and worsen the problem. Also, the improvement of IFLOC from IFLOC-Abl is less significant in WISDM than that in BMNISTR and PACS, which could be related to the same problem since the number of source domains for BMNISTR and PACS is smaller than that for WISDM. If the optimization difficulty prevents IFLOC from working properly, we might be able to mitigate it by using ideas from the studies that investigate the convergence and optimization difficulty in adversarial training (e.g., Nagarajan & Kolter (2017); Heusel et al. (2017); Balduzzi et al. (2018)).
+
+# 6 CONCLUSION
+
+In this paper, we addressed domain generalization under domain-class dependency, which was overlooked by most prior domain generalization methods relying on domain-invariant representation. We theoretically showed the importance of considering the dependency and the way to overcome the problem by expanding the analysis of Xie et al. (2017). We then proposed a novel method IFLOC, which maximizes domain-invariance within a range that does not interfere with classification accuracy. Empirical validations show the superior performance of IFLOC to the baseline methods, supporting the importance of the domain-class dependency in domain generalization tasks and the efficacy of the proposed method for overcoming the issue. Future work includes applying the reguralization idea of making $\_$ to other methods, e.g., Conditional VAE (Louizos et al. (2016)) or CrossGrad (Shankar et al. (2018)) because they have clear and tractable data generating process but assume the independence of and $\cdot$ . Also intended is to use it for transfer learning tasks $\cdot$   
+in a few-shot setting (e.g., life-long learning) where domain-class dependency is likely to occur due to scarce sample size.
+
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+Anuroop Sriram, Heewoo Jun, Yashesh Gaur, and Sanjeev Satheesh. Robust speech recognition using generative adversarial networks. In The IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018.
+
+A. Torralba and A. A. Efros. Unbiased look at dataset bias. In Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition, 2011.
+
+Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maximizing for domain invariance. CoRR, abs/1412.3474, 2014. URL http://arxiv.org/abs/1412.3474.
+
+Eric Tzeng, Judy Hoffman, Trevor Darrell, and Kate Saenko. Simultaneous deep transfer across domains and tasks. In Proc. of the IEEE International Conference on Computer Vision (ICCV), 2015.
+
+Qizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, and Graham Neubig. Controllable invariance through adversarial feature learning. In Proc. of the 30th International Conference on Neural Information Processing Systems. 2017.
+
+Jianbo Yang, Minh Nhut Nguyen, Phyo Phyo San, Xiaoli Li, and Shonali Krishnaswamy. Deep convolutional neural networks on multichannel time series for human activity recognition. In Proc. of the 24th International Joint Conference on Artificial Intelligence, 2015.
+
+Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In Proc. of the 30th International Conference on Machine Learning, 2013.
+
+Kun Zhang, Bernhard Schlkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation under target and conditional shift. In Proc. of the 30th International Conference on Machine Learning, 2013.
+
+# A DOMAIN-CLASS DEPENDENCY IN PACS AND WISDM
+
+Table 4: Sample sizes for each $<$ <domain, class $>$ pair in PACS dataset. The column shows category name while and index shows style.   
+
+<table><tr><td></td><td>Guitar</td><td>House</td><td>Giraffe</td><td>Person</td><td>Horse</td><td>Dog</td><td>Elephant</td></tr><tr><td>Art Painting</td><td>184</td><td>295</td><td>285</td><td>449</td><td>201</td><td>379</td><td>255</td></tr><tr><td>Cartoon</td><td>135</td><td>288</td><td>346</td><td>405</td><td>324</td><td>389</td><td>457</td></tr><tr><td>Photo</td><td>186</td><td>280</td><td>182</td><td>432</td><td>199</td><td>189</td><td>202</td></tr><tr><td>Sketch</td><td>608</td><td>80</td><td>753</td><td>160</td><td>816</td><td>772</td><td>740</td></tr></table>
+
+Table 5: Sample sizes for each <domain, class $>$ pair in WISDM dataset. The column shows activity name and the index shows user id.   
+
+<table><tr><td></td><td>Jogging</td><td>Walking</td><td>Upstairs</td><td>Downstairs</td><td>Sitting</td><td>Standing</td></tr><tr><td>User 1</td><td>145</td><td>742</td><td>108</td><td>224</td><td>160</td><td>78</td></tr><tr><td>User 2</td><td>142</td><td>481</td><td>282</td><td>186</td><td>0</td><td>0</td></tr><tr><td>User 3</td><td>645</td><td>654</td><td>240</td><td>231</td><td>780</td><td>267</td></tr><tr><td>User 4</td><td>637</td><td>619</td><td>237</td><td>214</td><td>113</td><td>78</td></tr><tr><td>User 5</td><td>614</td><td>650</td><td>229</td><td>210</td><td>56</td><td>80</td></tr><tr><td>User 6</td><td>637</td><td>574</td><td>101</td><td>86</td><td>0</td><td>0</td></tr><tr><td>User 7</td><td>588</td><td>617</td><td>81</td><td>69</td><td>81</td><td>33</td></tr><tr><td>User 8</td><td>599</td><td>621</td><td>160</td><td>171</td><td>102</td><td>79</td></tr><tr><td>User 9</td><td>599</td><td>308</td><td>269</td><td>206</td><td>123</td><td>94</td></tr><tr><td>User 10</td><td>597</td><td>625</td><td>119</td><td>118</td><td>71</td><td>95</td></tr><tr><td>User 11</td><td>610</td><td>616</td><td>188</td><td>115</td><td>150</td><td>81</td></tr><tr><td>User 12</td><td>626</td><td>356</td><td>0</td><td>0</td><td>77</td><td>51</td></tr><tr><td>User 13</td><td>620</td><td>604</td><td>217</td><td>131</td><td>0</td><td>0</td></tr><tr><td>User 14</td><td>0</td><td>624</td><td>68</td><td>76</td><td>147</td><td>96</td></tr><tr><td>User 15</td><td>318</td><td>610</td><td>167</td><td>162</td><td>81</td><td>73</td></tr><tr><td>User 16</td><td>602</td><td>650</td><td>212</td><td>187</td><td>0</td><td>81</td></tr><tr><td>User 17</td><td>0</td><td>706</td><td>142</td><td>147</td><td>0</td><td>63</td></tr><tr><td>User 18</td><td>593</td><td>658</td><td>178</td><td>189</td><td>0</td><td>0</td></tr><tr><td>User 19</td><td>661</td><td>690</td><td>406</td><td>141</td><td>0</td><td>0</td></tr><tr><td>User 20</td><td>611</td><td>310</td><td>149</td><td>144</td><td>32</td><td>25</td></tr><tr><td>User 21</td><td>616</td><td>537</td><td>130</td><td>141</td><td>112</td><td>81</td></tr><tr><td>User 22</td><td>613</td><td>327</td><td>239</td><td>94</td><td>0</td><td>0</td></tr><tr><td>User 23</td><td>42</td><td>301</td><td>66</td><td>86</td><td>60</td><td>0</td></tr><tr><td>User 24</td><td>0</td><td>626</td><td>209</td><td>191</td><td>75</td><td>152</td></tr><tr><td>User 25</td><td>641</td><td>666</td><td>194</td><td>140</td><td>76</td><td>65</td></tr><tr><td>User 26</td><td>513</td><td>853</td><td>220</td><td>165</td><td>132</td><td>161</td></tr><tr><td>User 27</td><td>701</td><td>841</td><td>231</td><td>192</td><td>105</td><td>128</td></tr><tr><td>User 28</td><td>477</td><td>622</td><td>240</td><td>199</td><td>78</td><td>140</td></tr><tr><td>User 29</td><td>548</td><td>646</td><td>168</td><td>164</td><td>78</td><td>139</td></tr><tr><td>User 30</td><td>309</td><td>349</td><td>269</td><td>179</td><td>0</td><td>0</td></tr><tr><td>User 31</td><td>550</td><td>641</td><td>154</td><td>145</td><td>0</td><td>0</td></tr><tr><td>User 32</td><td>0</td><td>644</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 33</td><td>322</td><td>346</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 34</td><td>587</td><td>584</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 35</td><td>457</td><td>549</td><td>178</td><td>110</td><td>124</td><td>116</td></tr><tr><td>User 36</td><td>808</td><td>879</td><td>212</td><td>128</td><td>124</td><td>104</td></tr></table>

parse/train/HJx38iC5KX/HJx38iC5KX_content_list.json

@@ -0,0 +1,1572 @@
+[
+    {
+        "type": "text",
+        "text": "DOMAIN GENERALIZATION VIA INVARIANT REPRESENTATION UNDER DOMAIN-CLASS DEPENDENCY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning domain-invariant representation is a dominant approach for domain generalization, where we need to build a classifier that is robust toward domain shifts induced by change of users, acoustic or lighting conditions, etc. However, prior domain-invariance-based methods overlooked the underlying dependency of classes (target variable) on source domains during optimization, which causes the trade-off between classification accuracy and domain-invariance, and often interferes with the domain generalization performance. This study first provides the notion of domain generalization under domain-class dependency and elaborates on the importance of considering the dependency by expanding the analysis of Xie et al. (2017). We then propose a method, invariant feature learning under optimal classifier constrains (IFLOC), which explicitly considers the dependency and maintains accuracy while improving domain-invariance. Specifically, the proposed method regularizes the representation so that it has as much domain information as the class labels, unlike prior methods that remove all domain information. Empirical validations show the superior performance of IFLOC to baseline methods, supporting the importance of the domain-class dependency in domain generalization and the efficacy of the proposed method for overcoming the issue. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In supervised learning problems we typically assume that samples are obtained from the same distribution in training and testing; however, such an assumption does not hold in many practical situations, depressing the classification accuracy for the test data (Torralba & Efros (2011)). One typical situation is domain generalization (e.g., Blanchard et al. (2011)): we have labeled data from several source domains and collectively exploit them so that the trained system generalizes to other, unseen but somewhat similar, target domains. Such challenges arise in many applications, e.g., hand-writing recognition (Shankar et al. (2018)), robust speech recognition (Sriram et al. (2018)), and sensor data interpretation (Erfani et al. (2016)). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To address domain generalization, many methods take advantage of invariant feature learning (Muandet et al. (2013); Erfani et al. (2016); Ghifary et al. (2017); Xie et al. (2017)). Such methods assume that learning the representation $( h )$ that is invariant to domains $( d )$ from input data $( x )$ prevents $h$ to overfit to source domains and leads to higher classification accuracy for unseen domains. To obtain such $h$ , we used various methods to measure the invariance of $h$ to $d$ and imposed some regularization on the measurement. For example, domain adversarial networks (DAN) (Ganin et al. (2016); Xie et al. (2017)) measure the invariance using a domain classifier (also called a discriminator) parameterized by deep neural networks and impose regularization by deceiving it. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Most prior works, however, overlooked the underlying dependency of classes on source domains, which we refer to as domain-class dependency. More specifically, we define domain-class dependency as the situation where domain and class labels are statistically dependent due to some common latent factor $( z )$ of $y$ and $d$ (Figure 1-right). Under the domain-class dependency, merely forcing the optimal domain-invariance harms the classification accuracy, as shown in Figure 1-(c). Intuitively speaking, since $y$ contains information about $d$ under domain-class dependency, $h$ must keep at least as much domain information as $y$ to achieve the optimal classification accuracy; however, invariant feature learning attempts to remove all domain information from $h$ , which causes the trade-off. It might be similar to the situation where $p ( y | x )$ and $p ( x )$ change across domains due to the causal structure $y  x$ (Zhang et al. (2013); Gong et al. (2016) in domain adaptation and Li et al. (2018c) in domain generalization), which we call conditional probability shift. However, the shift does not cause the trade-off as long as $y$ and $\\cdot$ are independent (Figure 1-left), so it is necessary to focus on the relationship between $\\cdot$ and $\\cdot$ . ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1c8f47a0c3956668c94bc462a79e08ab22c86612b5a33a6144b2afac2070354.jpg",
+        "image_caption": [
+            "Figure 1: The illustration of the domain-class dependency problem. While Li et al. (2018c) focused on the causal relationship between $x$ and $y$ , we focus on the relationship between $\\cdot$ and $\\cdot$ because it causes the following trade-off problem. (a) When domain and class are independent, domain invariance and classification accuracy can be optimized at the same time. (b,c) In domain-class dependency, there is a trade-off between these two: (b) optimal invariance cannot be achieved when optimal classification accuracy is achieved, and (c) vice versa. We propose a method to lead explicitly to (b) rather than (c), because the primary purpose for domain generalization is classification, not domain-invariance itself. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, domain-class dependency is common in real-world datasets as shown in Zhang et al. (2013). The dependency can be caused by both the characteristics of data and errors in collecting data. For example, the WISDM Activity Prediction dataset (Kwapisz et al. (2011)), where classes and domains correspond to activities and users, exhibits the dependency because (1) some activities (jogging and climbing stairs) are strenuous (data characteristics) and (2) other activities (sitting and standing) and some users were added only after the study began (data-collection errors). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we address domain generalization under domain-class dependency. We first expand the analysis about DAN by Xie et al. (2017), show that domain-class dependency causes the tradeoff problem, and then derive a way to evade the trade-off. Specifically, we investigate the condition where the domain-invariance is maximized under the constraint that it does not interfere with classification accuracy (Figure 1 (b)), because the primary purpose of domain generalization is classification rather than domain-invariance itself. We then propose a novel method invariant feature learning under optimal classifier constraint (IFLOC), modifying DAN’s regularization term to make the learned representation have as much domain information as the class labels, i.e., $H ( d | h ) = H ( d | y )$ holds (here $H$ denotes entropy). Like DAN, IFLOC has an encoder, classifier, and domain discriminator, and also takes over the good properties of DAN: it does not depend on pre-defined metrics (e.g., maximum mean discrepancy (Tzeng et al. (2014))), and it can be trained in an end-to-end manner. Empirical validations show the superior performance of IFLOC to baseline methods, supporting the importance of considering domain-class dependency in domain generalization tasks and the efficacy of the proposed approach for overcoming the issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main contributions of this paper can be summarized as follows. Firstly, we elaborate on the trade-off problem under domain-class dependency, both theoretically and experimentally, for the first time in domain generalization context. Secondly, to address the issue we provide theoretical analysis, which shows to what extent latent representations can become invariant to domains without interfering with classification accuracy. Finally, based on the analysis we propose a novel method IFLOC, and validated its efficacy by the experiments on both synthetic and real world datasets. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Invariant feature learning is a general-purpose method applicable to domain generalization as well as to domain adaptation (e.g., Tzeng et al. (2014); Ganin et al. (2016)), style transfer (e.g., Lample et al. (2017); Chou et al. (2018)), and fairness-aware classification (e.g., Zemel et al. (2013); ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Louizos et al. (2016); Madras et al. (2018)). However, it is likely that adjusting it to each specific task can improve performance. For example, in the fairness-aware classification task Madras et al. (2018) proposed to optimize the fairness criterion directly instead of applying invariance to sensitive variables. By analogy, we adapted invariant feature learning for domain generalization so as to address the domain-class dependency problem. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Domain generalization has been attracting considerable attention in recent years (Blanchard et al. (2011); Muandet et al. (2013); Shankar et al. (2018)). Note that it is different from domain adaptation in that we cannot obtain input and label data from target domain(s). Although the efficacy of domain-invariance-based methods had been known, Li et al. (2017) showed that non end-to-end methods such as DICA (Muandet et al. (2013)) and MTAE (Ghifary et al. (2015)) do not tend to outperform even vanilla CNN. Thus, end-to-end methods are desirable and can be divided into two categories: adversarial-learning-based methods such as DAN (Ganin et al. (2016); Xie et al. (2017)) and pre-defined-metric-based methods (e.g., Ghifary et al. (2017); Li et al. (2018b)). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In particular, IFLOC closely relates to DAN. Although DAN was originally invented for domain adaptation, Xie et al. (2017) showed its efficacy in domain generalization. Also, Xie et al. (2017) provided the intuitive explanation of the trade-off between classification accuracy and domain invariance. However, they did not provide any way to deal with the problem because its focus is invariant feature learning itself. Louppe et al. (2017) provided the similar analysis with Xie et al. (2017), but differs in that they focused on the relation between nuisance parameters (domains) and output distribution of a domain classifier. IFLOC also relates to domain confusion loss (Tzeng et al. (2015)) in that their encoders attempt to directly minimize Kullback-Leibler divergence (KLD) between output distribution of the discriminators and some domain distribution $( p ( d | y )$ in IFLOC and uniform distribution in domain confusion loss), rather than deceive the discriminator as DAN. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are several studies that address domain generalization without utilizing invariant feature learning. For example, Motiian et al. (2017); Li et al. (2018c) proposed to make use of semantic alignment, which attempts to make latent representation given class label $( p ( h | y ) )$ identical within source domains. This approach was originally proposed in Gong et al. (2016) in domain adaptation context, but its efficacy for domain-class dependency is not obvious because it focuses on conditional probability shift. CrossGrad (Shankar et al. (2018)) is one of the recent state-of-the-art domain generalization methods, which utilizes data augmentation with adversarial examples. However, since the method relies on the assumption that $y$ and $d$ are independent, it might not be directly applicable to our setting. MLDG (Li et al. (2018a)), also one of the state-of-the-art methods, utilizes metalearning. Since it makes no assumption about the relation between $y$ and $d$ , it could be combined with our proposed method, though we have not experimentally confirmed it. ",
+        "bbox": [
+            173,
+            444,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are several kinds of distributional shifts other than domain-class dependency, such as conditional probability shift. Although the distinction between that shift and domain-class dependency is important, it has been received less attention. For example, Li et al. (2018c) claimed that conditional probability shift might harm the performance of domain-invariance-based methods, but our analysis in Sec.4.1.1 suggests that the root cause of the performance degradation is not it but domain-class dependency. They also proposed to correct the shift of $\\cdot$ across source domains by aligning sampling frequency of each class across domains. However, this approach is not applicable to when some classes are rarely or never appear in some domains (e.g., as in WISDM dataset). In domain adaptation, Zhang et al. (2013); Gong et al. (2016) address the situation where $p ( y )$ changes across source and target domains by estimating $p ( y )$ change using unlabeled target data. However, this approach is not applicable (or necessary) to domain generalization because our problem setting is different from theirs in that we are agnostic on target domain and aim to care about $\\cdot$ change within source domains instead. ",
+        "bbox": [
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+            785
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 PROBLEM STATEMENT OF DOMAIN GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            839,
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+            854
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Denote $\\mathcal { X } , \\mathcal { y }$ , and $\\mathcal { D }$ as the input feature, label, and domain spaces, respectively. With random variables $x \\in { \\mathcal { X } } , \\ y \\in { \\mathcal { Y } }$ , and $d \\in \\mathcal { D }$ ，we can define the probability distribution for each domain $d$ as $p ( x , y | d )$ . Here, we assume that $y$ and $d$ are discrete variables for simplicity. In domain generalization, we are given a training dataset consisting of $D _ { s } = \\{ x _ { i } ^ { s } , y _ { i } ^ { s } \\} _ { i = 1 } ^ { n ^ { s } }$ for all $s \\in \\{ 1 , 2 , . . . , m \\}$ . Here, each $D _ { s }$ corresponds to samples drawn from the source domain $p ( x , y | d = s )$ . Using the training dataset, we train a classifier $f : \\mathcal { X }  \\mathcal { Y }$ , and use the classifier to predict labels of samples drawn from the unknown target domain $p ( x , y | d = t )$ . ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 DOMAIN ADVERSARIAL NETWORKS FOR DOMAIN GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+            161,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we give a brief overview of DAN (Ganin et al. (2016)) given that our proposed method is an extension of it. DAN trains a domain discriminator that attempts to predict domains from latent representations encoded by an encoder, while simultaneously trains the encoder to remove domain information by deceiving the discriminator. This procedure ensures that there is no or little domain information in the representations, so a label classifier attached to the encoder can make robust predictions regarding unseen target domains. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Formally, we denote $f _ { E } ( x ) , q _ { M } ( y | h )$ , and $q _ { D } ( d | h ) \\ ( E , M$ , and $D$ are the parameters) as deterministic encoder, probabilistic model of label classifier, and that of domain discriminator, respectively. Then, the objective function of DAN is described as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5a6222ca1f938b22af979076447dda592401a160da787ac5b2d053438e32753.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { E , M } \\operatorname* { m a x } _ { D } J ( E , M , D ) = \\mathbb { E } _ { x , d , y \\sim p ( x , d , y ) } [ \\gamma \\log q _ { D } ( d | h = f _ { E } ( x ) ) - \\log q _ { M } ( y | h = f _ { E } ( x ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            328,
+            784,
+            353
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, the second term in Eq.1 simply maximizes the log likelihood of $q _ { M }$ as well as in standard classification problems. On the other hand, the first term corresponds to a minimax game between the encoder and discriminator, where the decoder $q _ { D } ( d | h )$ tries to predict $d$ from $h$ and the encoder $f _ { E } ( x )$ tries to fool $q _ { D } ( d | h )$ . ",
+        "bbox": [
+            174,
+            358,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As Xie et al. (2017) originally showed, the minmax game ensures that the learned representation has no or little domain information, i.e., the representation becomes domain-invariant. Such invariance makes a prediction from $h$ to $y$ independent from $d$ , and therefore hopefully helps to build a classifier that correctly handle samples drawn from unknown domains. Below is a brief explanation. ",
+        "bbox": [
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+            421,
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+            478
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since $h$ is a deterministic mapping of $x$ , the joint probability distribution of $h , d$ and $y$ can be defined as follows: ",
+        "bbox": [
+            174,
+            484,
+            823,
+            512
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c16e784a336d2abe8e9e1672562da507682b35c80b1d61a2eac63849d14b5bc1.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { p } _ { E } ( h , d , y ) = \\int _ { x } \\tilde { p } _ { E } ( x , d , h , y ) d x } \\\\ { \\displaystyle = \\int _ { x } p ( x , d , y ) \\delta ( f _ { E } ( x ) = h ) d x } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            517,
+            643,
+            585
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, we use the notation of $\\tilde { p } _ { E }$ for the true probability distribution that depends on the encoder’s parameter $E$ . Using Eq.2, Eq.1 can be replaced as follows: ",
+        "bbox": [
+            171,
+            589,
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+            617
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9185804b0f3ff023813966dfe8aa2bf06c4418e7a5eb250ca6e24c3083476b13.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { E , M } \\operatorname* { m a x } _ { D } J ( E , M , D ) = \\mathbb { E } _ { h , d , y \\sim \\tilde { p } _ { E } ( h , d , y ) } [ \\gamma \\log q _ { D } ( d | h ) - \\log q _ { M } ( y | h ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            623,
+            736,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assuming $E$ is fixed, the solutions $M ^ { * }$ and $D ^ { * }$ to Eq.3 obviously satisfy $q _ { M ^ { * } } ( y | h ) = \\tilde { p } _ { E } ( y | h )$ and $q _ { D ^ { * } } ( d | h ) = \\tilde { p } _ { E } ( d | h )$ . Then, substituting $q _ { M ^ { * } }$ and $q _ { D ^ { * } }$ into Eq.3, we can obtain the following optimization problem depending only on $E$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8fdb7aa106731aaabf66f1b68108c93c1920529c9dc7a330afbd3c7337727793.jpg",
+        "text": "$$\n\\displaystyle \\operatorname* { m i n } _ { E } J ( E ) = - \\gamma H _ { \\tilde { p } _ { E } } ( d | h ) + H _ { \\tilde { p } _ { E } } ( y | h )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Solving Eq.4, we can obtain the solutions $M ^ { * } , D ^ { * }$ , and $E ^ { * }$ , which are in Nash equilibrium. Here, $H _ { \\tilde { p } _ { E } } ( d | h )$ means conditional entropy with joint probability distribution $\\tilde { p } _ { E } ( d , h )$ . Thus, minimizing the second term in Eq.4 intuitively means learning (the mapping function $f _ { E }$ to) the latent representation $h$ which contains as much information about $y$ as possible. On the other hand, the first term can be regarded as a regularizer that attempts to learn $h$ which is invariant to $d$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 OUR APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 ANALYSIS OF DOMAIN-CLASS DEPENDENCY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We address domain generalization under domain-class dependency, i.e., the situation where $p ( y | d ) \\neq p ( y )$ holds. Although the issue had been overlooked, it is common in real-world datasets given that they can have the dependency in nature, e.g., nocturnal annimals (class) do not tend to appear in daylight (domain), and the dependency in such datasets is often not corrected unlike in standard benchmark datasets. To address the problem, we expand the analysis of Xie et al. (2017) to theoretically show that domain-class dependency causes the trade-off between accuracy and invariance, and to consider to what extent the latent representation should become invariant. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 TRADE-OFF CAUSED BY DOMAIN-CLASS DEPENDENCY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We first show that the performance of DAN explained in the previous section suffers from the existence of domain-class dependency. The following analysis also suggests that all of the methods that utilize domain-invariant representation suffer from the dependency. Concretely, we show that the domain-class dependency causes the trade-off between classification accuracy and domain invariance: when $d$ and $y$ are not statistically independent, any $E$ cannot optimize the first and second term in Eq.4 at the same time. In this analysis, for simplicity, we assume that we can obtain any $\\tilde { p } _ { E } ( y | h ) , \\tilde { p } _ { E } ( d | h )$ , i.e., the models have enough capacity and there are no optimization difficulties. ",
+        "bbox": [
+            173,
+            200,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To begin with, we consider only the first term in Eq.4 and address the optimization problem: ",
+        "bbox": [
+            171,
+            306,
+            776,
+            321
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/334c13a9e778e949a94c1412515d7f5d2ec5c9d728ef94824032db5931e96595.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { E } J _ { 1 } ( E ) = - \\gamma H _ { \\tilde { p } _ { E } } ( d | h )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            329,
+            593,
+            352
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using the property of entropy, $H _ { \\tilde { p } _ { E } } ( d | h )$ is bounded as follows: ",
+        "bbox": [
+            173,
+            361,
+            593,
+            377
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9a91db62f810528e65638fd15aaa277da9a78d53c7b6977c0caa9d9e789d51f.jpg",
+        "text": "$$\nH _ { \\tilde { p } _ { E } } ( d | h ) \\leq H ( d )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            383,
+            562,
+            401
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, $H _ { \\tilde { p } _ { E } } ( d | h ) = H ( d )$ holds only if $h$ and $d$ are independent. Thus, Eq.5 has the solution $E ^ { 1 * }$ , which satisfies the following condition: ",
+        "bbox": [
+            173,
+            411,
+            825,
+            440
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce034cf5be5c3eaa58f937545cf6686ebad508bce4cd16e92761321aa0b2d674.jpg",
+        "text": "$$\nH _ { \\tilde { p } _ { E ^ { 1 * } } } ( d | h ) = H ( d )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            449,
+            570,
+            467
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Eq.7 suggests that the regularizer in DAN is intended to remove all information about domains from latent variables, thereby making domains and latent variables independent. ",
+        "bbox": [
+            174,
+            473,
+            825,
+            503
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we analogically consider only the second term in Eq.4, thereby addressing the following optimization problem: ",
+        "bbox": [
+            173,
+            508,
+            825,
+            537
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56784c6f51fb40a41052cdfae0f2be1a7bdae0e33d39e492656df87803766059.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { E } J _ { 2 } ( E ) = H _ { \\tilde { p } _ { E } } ( y | h )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            546,
+            581,
+            569
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since conditional entropy $H ( a | b )$ has a minimum value when $b$ contains all information about $a$ , Eq.8 has the solution $E ^ { 2 * }$ , which satisfies the following equation: ",
+        "bbox": [
+            171,
+            578,
+            823,
+            607
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1afe492c0bd917c587b7fd744181bfce035fb07a92cf24e2b535c9e0d79b4af3.jpg",
+        "text": "$$\nH _ { \\tilde { p } _ { E ^ { 2 * } } } ( d | h ) = H _ { \\tilde { p } _ { E ^ { 2 * } } } ( d | h , y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            614,
+            598,
+            633
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using Eq.9 and the property of entropy: $H ( a | b , c ) \\leq H ( a | b )$ , we can obtain the following condition: ",
+        "bbox": [
+            178,
+            641,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3717eda92a4ddc923f39f7d1dd42679aa6b6a38b9b966856471b93b8f4418621.jpg",
+        "text": "$$\nH _ { \\tilde { p } _ { E ^ { 2 * } } } ( d | h ) = H _ { \\tilde { p } _ { E ^ { 2 * } } } ( d | h , y ) \\leq H ( d | y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            665,
+            633,
+            684
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Eq.10 implies that $h$ has at least as much information about $d$ as $y$ does. Now, we assume that $y$ and $d$ are not independent, i.e., domain-class dependency exists, and obtain the following condition: ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/da0b0730c1df7c21db8b7a41aaf08e166b2313f85a4b2f6364a4b6ff98c1bc07.jpg",
+        "text": "$$\nH _ { \\tilde { p } _ { E ^ { 2 * } } } ( d | h ) \\leq H ( d | y ) < H ( d )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            728,
+            604,
+            746
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Considering Eq.7 and Eq.11, $E ^ { 1 * } \\neq E ^ { 2 * }$ holds. This means that when $y$ and $d$ are not independent, there is no solution $E$ that optimizes Eq.5 and Eq.8 at the same time, i.e., there is a trade-off between classification accuracy and domain invariance. ",
+        "bbox": [
+            174,
+            762,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is worth noting that although Li et al. (2018c) claimed that conditional probability shift (the causal structure $\\cdot$ ) could harm the domain generalization performance of invariance-based methods, this analysis suggests that it does not harm DAN as long as domain and class are independent. It can be confirmed by considering Eq.7 and Eq.10; even when the shift occurs, i.e., $H ( y | x , d ) < H ( y | x )$ holds and then $H _ { \\tilde { p } _ { E } } ( y | h , d ) \\le H _ { \\tilde { p } _ { E } } ( y | h )$ holds, it does not conflict with $H _ { \\tilde { p } _ { E ^ { * } } } ( d | h ) = H ( d | y ) =$ $\\cdot$ as long as $H ( d | y ) = H ( d )$ holds. In other words, we only need to infer latent variable $\\cdot$ that satisfies the causal structure $y  h  x$ to avoid the trade-off. Although Gong et al. (2016) showed the similar result in domain adaptation context, it has been overlooked in domain generalization. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 OPTIMAL DOMAIN-INVARIANCE UNDER DOMAIN-CLASS DEPENDENCY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "If we cannot avoid the trade-off, the next question is how to deal with it, i.e., to what extent the representation should become domain-invariant for domain generalization tasks. We propose to maximize domain-invariance within a range that does not interfere with classification accuracy, rather than merely enforcing domain-invariance without any constraint. The reason for the constraint is that the primary purpose of domain generalization is classification for unseen domains rather than domain-invariance itself, and the improvement of the invariance could harm the classification performance for them. For example, in WISDM, if we know the target activity (class) was performed by not an old but yound man (domain), we can predict it was jogging with higher probability, so we should avoid removing such domain information that is useful in the classification task. As another example, if the target domain has the similar characteristics as a certain source domain (or as an extreme case, $p ( x , \\bar { y } | d = s ) = p ( x , y | d = t )$ holds), giving priority to domain-invariance obviously interferes with the domain generalization performance. ",
+        "bbox": [
+            174,
+            128,
+            825,
+            296
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Given that Eq.10 is the necessary condition where we can build an optimal classifier, we can write the optimization problem of maximizing domain-invariance within a range that does not interfere with classification accuracy as follows: ",
+        "bbox": [
+            176,
+            303,
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+            344
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d5201aa4074d92dde7fbcf54732424d7e841ca8774e12a316eec8f7c8f82935.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { E } { \\operatorname* { m i n } } J ( E ) = - \\gamma H _ { \\tilde { p } _ { E } } ( d | h ) } \\\\ & { \\qquad s u b j e c t t o \\ : H _ { \\tilde { p } _ { E } } ( d | h ) \\leq H ( d | y ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            353,
+            620,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Continuing, we can obtain the solution $E ^ { * }$ , which obviously satisfies $H _ { \\tilde { p } _ { E ^ { * } } } ( d | h ) = H ( d | y )$ . More specifically, when we want to maximize domain-invariance (Eq.12) within the range that does not interfere with accuracy (Eq.13), the solution satisfies $H _ { \\tilde { p } _ { E ^ { * } } } ( d | h ) \\ : = \\ : H ( d | y )$ . So without interfering with classification accuracy we can remove domain information from $h$ to the extent that $\\dot { H } _ { \\tilde { p } _ { E } } ( d | h ) = H ( \\dot { d } | y )$ holds, i.e., $h$ has as much information about $d$ as $y$ does. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            493,
+            356,
+            508
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on the above analysis, the remaining challenge is how to impose such regularization that makes $H _ { \\tilde { p } _ { E } } ( d | h ) = H ( d | y )$ hold. Although DAN might be able to achive that condition by carefully tuning the regularizer ( $\\langle \\gamma$ in Eq.1), such tuning is time-consuming and impracticable as suggested in our experiments. Alternatively, we propose a novel method called IFLOC, modifying DAN’s regularization term: while the encoder of DAN attempts to fool a discriminator, that of IFLOC attempts to directly minimize KLD between $p ( d | y )$ and $q _ { D } ( d | h )$ . Formally, IFLOC solves the following joint optimization problem by alternating gradient descent. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bccd5a0c3176c325e891439bd0f7809431ace67a5905258c5141ad2be8c4bc7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { E , M } { \\mathop { \\operatorname* { m i n } } } J ( E , M ) = \\mathbb { E } _ { x , d , y \\sim p ( x , d , y ) } [ \\gamma D _ { K L } [ p ( d | y ) | q _ { D } ( d | h - f _ { E } ( x ) ) ] - \\log q _ { M } ( y | h - f _ { E } ( x ) ) ] } \\\\ & { } \\\\ & { \\underset { D } { \\mathop { \\operatorname* { m i n } } } J ( E , D ) = \\mathbb { E } _ { x , d \\sim p ( x , d ) } [ - \\log q _ { D } ( d | h = f _ { E } ( x ) ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            626,
+            787,
+            678
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The second term in Eq.14 and Eq.15 respectively means maximization of log-likelihood of $q _ { M }$ and $q _ { D }$ as well as DAN. However, the first term in Eq.14 differs from DAN in that it is intended to satisfy $\\bar { q } _ { D } ( d | h ) = p ( d | y )$ for almost every $( y , h )$ pair. ",
+        "bbox": [
+            176,
+            684,
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+            728
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next we show that the regularization of IFLOC is intended to achieve $H _ { \\tilde { p } _ { E } } ( d | h ) = H ( d | y )$ . Similarly to Section 3.2, $D ^ { * }$ and $M ^ { * }$ , which are the solutions to Eq.14 and Eq.15 with fixed $E$ , obviously satisfy $q _ { D } ^ { * } = \\tilde { p } _ { E } ( d | h ) , q _ { M } ^ { * } = \\tilde { p } _ { E } ( y | h )$ . Thus Eq.14 can be written as follows: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff5b38408e0f880f249c8c04b6adc7d4984126b5b9e3eab3566d7b64a5b83270.jpg",
+        "text": "$$\n\\displaystyle \\operatorname* { m i n } _ { E } J ( E ) = \\mathbb { E } _ { h , y \\sim \\tilde { p } _ { E } ( h , y ) } [ \\gamma D _ { K L } [ p ( d | y ) | \\tilde { p } _ { E } ( d | h ) ] ] + H _ { \\tilde { p } _ { E } } ( y | h )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            785,
+            714,
+            808
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since the minimization of the KLD term does not interfere with the second term optimization, $E ^ { * }$ , which is the solution to Eq.16 and in Nash equilibrium, satisfies $\\mathbb { E } _ { h , y \\sim \\tilde { p } _ { E ^ { * } } ( h , y ) } [ D _ { K L } [ p ( d | y ) | \\tilde { p } _ { E ^ { * } } ( d | h ) ] ] = 0$ . Then, $\\tilde { H _ { \\tilde { p } _ { E ^ { * } } } } ( d | h ) = H ( d | y )$ obviously holds. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that we cannot obtain true $p ( d | y )$ , but we can use a maximum likelihood or maximum a posteriori estimator for it. Also, we could use some divergences other than $D _ { K L } [ p ( d | y ) | q _ { D } ( d | h ) ]$ in Eq.14, e.g., $D _ { K L } [ q _ { D } ( d | h ) | p ( d | y ) ]$ , but in doing so, we could not observe performance gain, so we discontinued testing them. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4a1d15d3c8fe19bfd1200c10f47d5e8ad7d8bc453fb77fd398a4499e88d3d042.jpg",
+        "table_caption": [
+            "Table 1: Sample sizes for each domain-class pair in BMNISTR. Those for the classes $0 { \\sim } 4$ are variable across domains, whereas the classes $5 { \\sim } 9$ have identical sample sizes across domains. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Class</td><td>M0</td><td>M15</td><td>M30</td><td>M45</td><td>M60</td><td>M75</td></tr><tr><td>BMNISTR-1</td><td>0~4 5~9</td><td>100 100</td><td>85 100</td><td>70 100</td><td>55 100</td><td>40 100</td><td>25 100</td></tr><tr><td>BMNISTR-2</td><td>0~4 5~9</td><td>100 100</td><td>80 100</td><td>60 100</td><td>40 100</td><td>20 100</td><td>0 100</td></tr><tr><td>BMNISTR-3</td><td>0~4 5~9</td><td>100 100</td><td>90 100</td><td>80 100</td><td>70 100</td><td>60 100</td><td>50 100</td></tr><tr><td>BMNISTR-4</td><td>0~4 5~9</td><td>100 100</td><td>25 100</td><td>100 100</td><td>25 100</td><td>100 100</td><td>25 100</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 DATASETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "BMNISTR We created the Biased Rotated MNIST dataset (BMNISTR) by modifying the sample size of MNISTR (Ghifary et al. (2015)) so that class distribution differs among the domains. Specifically, we created four variants of MNISTR that have different types of domain-class dependency, referred to as BMNISTR-1 through BMNISTR-4. As shown in Table 1, BMNISTR-1, -2, and -3 have similar trends but different degrees of dependency; BMNISTR-1 and BMNISTR-4 differ in trends. In MNISTR, each class is represented by 10 digits. Each domain was created by rotating images by 15 degree increments: 0, 15, 30, 45, 60, and 75 (referred to as M0, ..., M75). Each image is cropped to $1 6 \\mathrm { ~ x ~ } 1 6$ in accordance with Ghifary et al. (2015). In training, we employed one-domainleave-out setting: trained on five of the six domains and then tested using the remaining one. We used two convolution layers and two fully-connected (FC) layers (with nonlinear activations) as the encoder, three FC layers as the classifier, and two FC layers as the discriminator. ",
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+    },
+    {
+        "type": "text",
+        "text": "PACS The PACS dataset (Li et al. (2017)) has 9991 images across 7 categories (dog, elephant, giraffe, guitar, house, horse, and person) and 4 domains comprising different stylistic depictions (Photo, Art painting, Cartoon, and Sketch). It has domain-class dependency probably because samples in some $<$ <domain, class $>$ pairs are difficult to obtain. For example, $-$ ) is much higher than $-$ , which indicates that photos of person are easier to obtain than those of animals, but sketches of persons are more difficult to obtain than those of animals in the wild. The concrete sample sizes for each category and style is shown in Table 4 in appendix. In training, we employed one-domain-leave-out setting as well as in BMNISTR, and used the ImageNet pre-trained AlexNet CNN (Krizhevsky et al. (2012)) as the base network, following previous studies (Li et al. (2017; 2018a)). The two-FC-layer discriminator was connected to the last FC layer, following Ganin et al. (2016). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "WISDM The WISDM Activity Prediction dataset contains sensor data of accelerometers for six human activities (walking, jogging, upstairs, downstairs, sitting, and standing) performed by 36 users (domains). WISDM suffers from the dependency due to the reason noted in Sec.1. The concrete sample sizes for each user and activity is shown in Table 5 in appendix. Referring to Andrey (2017), we use the sliding-window procedure with 60 frames $\\left( = 3 \\right.$ seconds) and 20-frame overlap. The total number of samples was 54455. In training, we used randomly chosen ${ < } 1 0 / 2 6 >$ , ${ < } 1 6 / 2 0 >$ , and ${ < } 2 6 / 1 0 >$ users as $<$ source / target $>$ domains. We parameterized the encoder using three convolution layers followed by one FC layer and the classifier by logistic regression, following previous studies (Yang et al. (2015); Iwasawa et al. (2017)). The two-FC-layer discriminator was connected to the output of the encoder. ",
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+    {
+        "type": "text",
+        "text": "5.2 BASELINES",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the efficacy of the proposed method IFLOC, we compared it with the following methods. (1) CNN is a vanilla convolutional networks trained on the aggregation of data from all source domains. Although CNN has no special treatments for domain generalization, Li et al. (2017) reports that it outperforms many traditional domain generalization methods. (2) DAN (Xie et al. ",
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+    {
+        "type": "table",
+        "img_path": "images/596554684dfe3c1e13fa0ee7b74e060e6c56388732f5e240a162fb7c7f9e45dc.jpg",
+        "table_caption": [
+            "Table 2: Mean F scores for the classes $0 { \\sim } 4$ and classes $5 { \\sim } 9$ with the target domain M0. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Class</td><td>CNN</td><td>DAN</td><td>IFLOC-Abl</td><td>IFLOC</td><td>Relative Improvement of IFLOC to IFLOC-Abl</td></tr><tr><td>BMNISTR-1</td><td>0~4</td><td>83.86</td><td>84.54</td><td>87.46</td><td>90.62</td><td>3.6%</td></tr><tr><td></td><td>5~9</td><td>83.90</td><td>85.24</td><td>86.46</td><td>88.10</td><td>1.9%</td></tr><tr><td>BMNISTR-2</td><td>0~4</td><td>84.76</td><td>86.20</td><td>86.42</td><td>89.58</td><td>3.7%</td></tr><tr><td></td><td>5~9</td><td>83.36</td><td>85.22</td><td>85.62</td><td>86.86</td><td>1.4%</td></tr><tr><td>BMNISTR-3</td><td>0~4</td><td>82.54</td><td>85.30</td><td>88.60</td><td>89.64</td><td>1.2%</td></tr><tr><td></td><td>5~9</td><td>82.18</td><td>85.80</td><td>87.60</td><td>89.04</td><td>1.6%</td></tr><tr><td>BMNISTR-4</td><td>0~4</td><td>71.26</td><td>79.22</td><td>76.56</td><td>80.02</td><td>4.5%</td></tr><tr><td></td><td>5~9</td><td>78.62</td><td>83.14</td><td>82.94</td><td>82.80</td><td>-0.2%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(2017)) is expected to generalize across domains via invariant feature learning, but it has the tradeoff between domain invariance and classification accuracy as explained in Section 4.1.1. We trained DAN with a gradient reverse layer following Ganin et al. (2016); Xie et al. (2017). Also, we used (3) IFLOC-Abl, which is a version of IFLOC modified for ablation studies. IFLOC-Abl replaces $D _ { K L } [ p ( d | y ) | \\tilde { p } _ { E } ( d | h ) ]$ in Eq.14 of $D _ { K L } [ p ( d ) | \\tilde { p } _ { E } ( d | h ) ]$ , so it attempts to learn the representation that is completely invariant to domains or make $\\begin{array} { r } { H ( d | h ) = H ( d ) } \\end{array}$ hold as well as DAN. Comparing IFLOC and IFLOC-Abl, we measured the genuine effect of taking domain-class dependency into account. In training IFLOC and IFLOC-Abl, we cannot obtain true $p ( d | y )$ and $p ( \\bar { d } )$ , so we used maximum likelihood estimators of them for calculating the KLD terms. ",
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+    {
+        "type": "text",
+        "text": "5.3 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For all the datasets and methods, we used RMSprop for optimization. And we set the learning rate, batch size, and the number of iterations as 5e-4, 128, and 10k for BMNISTR; 5e-5, 64, and $1 0 \\mathrm { k }$ for PACS; 1e-4, 128, and $3 0 \\mathrm { k }$ for WISDM, respectively. For DAN, IFLOC-Abl, and IFLOC we optimized the weighting parameter $\\gamma$ from $\\{ 0 . 0 \\bar { 0 0 } 1 , 0 . 0 \\bar { 0 1 } , 0 . 0 1 , 0 . 1 , 1 , 1 0 \\}$ , and used the $\\gamma$ annealing following Ganin et al. (2016). In all the experiments, we split source data into $80 \\%$ of training data and $20 \\%$ of validation data, assuming that target data are not absolutely available in the training phase. We conducted experiments multiple times with different seeds. Specifically, we trained on 10 and 25 seeds in BMNISTR and WISDM, chose the best hyperparameter that achieved the highest validation accuracies measured in each epoch, and reported the mean scores (accuracies and f-values) for the hyperparameter. In PACS, since it requires a long time to train on, we chose the best $\\gamma$ from $\\{ 0 . 0 0 \\dot { 0 } \\dot { 1 } , \\dot { 0 } . 0 0 1 , 0 . 0 1 , 0 . 1 \\}$ with three experiment, and reported the mean scores in experiments with 20 seeds in total. Also, we empirically measured the level of domain-invariance by training a post-hoc classifier that is intended to predict $d$ over learned representation, following previous studies (Xie et al. (2017); Iwasawa et al. (2017); Moyer et al. (2018)). Specifically, we trained the classifier with 400 hidden units on $1 0 \\mathrm { k }$ iterations (by RMSprop optimizer with a 0.001 learning rate and 128 batch size) with the data that is used for training the models. We then evaluated the domain classification accuracy (referred to as D-Acc) 10 times at equal intervals during training, and reported D-Acc in the nearest time when the validation accuracy is maximized. ",
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+    {
+        "type": "text",
+        "text": "5.4 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first investigated how domain-class dependency affects the performance of domain-invariancebased methods. In Table 2, we compared mean f-scores for the classes 0 through 4 and classes 5 through 9 in BMNISTR with the target domain M0. Recall that sample sizes for the classes $0 { \\sim } 4$ are variable across domains, whereas the classes $5 { \\sim } 9$ has identical sample sizes across domains (Table 1). The f-scores show that IFLOC outperformed DAN and IFLOC-Abl in most datasetclass pairs, which supports that domain-class dependency depresses the performance of domaininvariance-based methods and that IFLOC can mitigate the problem. Futher, relative improvement of IFLOC to IFLOC-Abl is more significant for the classes $0 { \\sim } 4$ than $5 { \\sim } 9$ in BMNISTR-1, BMNISTR2, and BMNISTR-4, suggesting that IFLOC tends to increase performance more significantly for classes where the domain-class dependency occurs. Also, the improvement is more significant in ",
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+    {
+        "type": "image",
+        "img_path": "images/a6a9970777ab61ce57842bcd699cce01520f999dd90594796c444b68de6cd6fc.jpg",
+        "image_caption": [
+            "Figure 2: Class accuracy (Y-Acc) and domain accuracy (D-Acc) with various $\\gamma$ in BMNISTR-1. Each caption shows the metric name (Y-Acc or D-Acc) and target domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/6748a095442cbef294cb517e35607deabed7150afa87ba2160db7597f3f864e5.jpg",
+        "image_caption": [
+            "Figure 3: Class accuracy (Y-Acc) and domain accuracy (D-Acc) with various $\\gamma$ in WISDM. Each caption shows the metric name (Y-Acc or D-Acc) and target domain. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "BMNISTR-1 than in BMNISTR-3, suggesting that the stronger the domain-class dependency is, the lower the performance of domain-invariance-based methods becomes. Finally, although the dependencies of BMNISTR-1 and BMNISTR-4 have different trends as described in Table 1, IFLOC improved f-scores in both datasets. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Next we investigated the relationship between the strength of regularization and performance. Figures 2 and 3 show the hyperparameter sensitivity of class accuracies (Y-Acc) and domain accuracies (D-Acc) for DAN, IFLOC-Abl, and IFLOC. Note that the gray line in Figures 2-(c) and 3-(c) shows the trivial baseline predicting the majority label. From these figures, we can make the following observations. (1) All the methods including IFLOC could improve the invariance by using stronger regularizer. Concretely, Figures 2-(c) and 3-(c) show D-Acc tends to become low (invariance becomes high) for all the models when the regularizer becomes strong (such as $\\gamma = 1$ or 10) except that IFLOC-Abl has high D-Acc with $\\gamma = 1 0$ in Figure 3-(c). That high D-Acc might be because the validation accuracy achieved the highest value before the domain-invariance matured. (Recall that the more the representation becomes invariant, the lower the accurary becomes under the trade-off). (2) The training of IFLOC tends to be more stable than that of DAN when the regularizer becomes strong. Figures 2-(a,b) and 3-(a,b) show that IFLOC and IFLOC-Abl could achieve higher Y-Acc than DAN when $\\gamma = 1$ or 10, i.e., the regularization is strong, except for IFLOC-Abl with $\\gamma = 1$ in Figure 3-(b). This tendency might be because the regularizer of IFLOC is KLD and thus bounded by 0, in contrast to that of DAN that can increase to infinity and destabilize the traininig. (3) IFLOC, as it was designed, does not tend to decrease classification accuracy with strong regularizer, and thus IFLOC is robust toward hyperparameter choice. Figures 2-(b) and 3-(a,b) show that while Y-Acc of IFLOC-Abl decreases with strong regularization (such as when $\\gamma = 1$ or 10), that of IFLOC does not decrease as much. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, we compared mean accuracies (with standard errors) in both synthetic (BMNISTR) and standard benchmark (PACS and WISDM) datasets (Table 3). Note that the $H ( d ) / H ( d | y )$ column is estimated from source data, which indicates the strength of domain-class dependency. IFLOC outperformed IFLOC-Abl in BMNISTR with all the target domains; PACS with photo, art painting, and sketch target domains; and WISDM with 26- and 20-target-user domains. Also, IFLOC outperformed DAN in BMNISTR with all the target domains; PACS with photo and art painting target domains; and WISDM with 26- and 10-target-user domains. This supports the importance of considering domain-class dependency in real-world datasets and the efficacy of the proposed model. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7d4300e12153c6c15676dbcbee595260cf3c9c5b5d06c9ffab5d792ce9e12b6a.jpg",
+        "table_caption": [
+            "Table 3: Accuracies for each dataset and target domain "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Target</td><td>H(d)/H(dly) （%）</td><td>CNN</td><td>DAN</td><td>IFLOC-Abl</td><td>IFLOC</td></tr><tr><td>BMNISTR-1</td><td>M0 M15</td><td>101.2</td><td>83.9 ± 0.4</td><td>85.0± 0.4</td><td>87.0 ± 0.4</td><td>89.3± 0.4</td></tr><tr><td></td><td>M30</td><td>101.5</td><td>98.5± 0.2 97.5 ± 0.1</td><td>98.5± 0.1</td><td>98.3 ± 0.2</td><td>98.8 ± 0.1</td></tr><tr><td></td><td>M45</td><td>101.6</td><td></td><td>97.4± 0.1</td><td>97.6± 0.1</td><td>98.3 ± 0.2</td></tr><tr><td></td><td></td><td>101.6</td><td>89.9 ± 0.9</td><td>90.2 ± 0.6</td><td>92.8 ± 0.5</td><td>93.3 ± 0.6</td></tr><tr><td></td><td>M60</td><td>101.3</td><td>96.7 ± 0.3</td><td>97.0 ± 0.2</td><td>96.6± 0.2</td><td>97.4 ± 0.2</td></tr><tr><td></td><td>M75</td><td>100.7</td><td>87.1 ± 0.5</td><td>87.3± 0.4</td><td>87.7± 0.5</td><td>88.1 ± 0.4</td></tr><tr><td>BMNISTR-2</td><td>Avg</td><td></td><td>92.3</td><td>92.6</td><td>93.3</td><td>94.2</td></tr><tr><td>BMNISTR-3</td><td>Avg Avg</td><td></td><td>92.3</td><td>92.2</td><td>93</td><td>94.2</td></tr><tr><td>BMNISTR-4</td><td></td><td></td><td>92.2</td><td>92.7</td><td>94</td><td>94.5</td></tr><tr><td>PACS</td><td>Avg</td><td></td><td>90.6</td><td>91.7</td><td>91.6</td><td>92.9</td></tr><tr><td></td><td>photo</td><td>107.2</td><td>80.6± 0.3</td><td>81.1 ± 0.3</td><td>81.6± 0.3</td><td>82.9 ± 0.2</td></tr><tr><td></td><td>art_painting</td><td>108.5</td><td>59.2± 0.4</td><td>60.1± 0.3</td><td>60.5± 0.4</td><td>61.2 ± 0.2</td></tr><tr><td></td><td>cartoon</td><td>109.7</td><td>63.2± 0.3</td><td>64.3 ± 0.3</td><td>64.4 ± 0.4</td><td>63.8± 0.3</td></tr><tr><td></td><td>sketch</td><td>101.5</td><td>58.2± 0.5</td><td>58.9± 0.4</td><td>58.1± 0.6</td><td>59.0 ± 0.5</td></tr><tr><td>WISDM</td><td>Avg</td><td></td><td>65.3</td><td>66.1</td><td>66.2</td><td>66.7</td></tr><tr><td></td><td>26users</td><td>107.1</td><td>78.3± 0.3</td><td>78.2± 0.3</td><td>78.4± 0.2</td><td>78.9 ± 0.3</td></tr><tr><td></td><td>20 users</td><td>104.2</td><td>79.7 ± 0.2</td><td>80.2 ± 0.3</td><td>79.7 ± 0.3</td><td>80.0± 0.3</td></tr><tr><td></td><td>10 users</td><td>103.5</td><td>80.6± 0.2</td><td>80.6± 0.2</td><td>81.2 ± 0.3</td><td>81.2 ± 0.3</td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 3 also shows that when the number of source domains increased from 10 to 26 in WISDM, the improvement of IFLOC from IFLOC-Abl became insignificant. One possible reason is that WISDM with 10 target users has low domain-class dependency than with 26 target users as shown in the $H ( d ) / H ( d | y )$ column. Another possible reason is the optimization difficulty. As Moyer et al. (2018) reported, in adversarial invariant feature learning, an encoder often overfits to the discriminator trained alongside that encoder, and does not provide truly invariant representation (the same problem can be observed in Figures 2-(c) and 3-(c)). We suspect that when the number of source domains increases, the optimization of the domain discriminator becomes difficult, which makes the encoder overfit to that poor discriminator and worsen the problem. Also, the improvement of IFLOC from IFLOC-Abl is less significant in WISDM than that in BMNISTR and PACS, which could be related to the same problem since the number of source domains for BMNISTR and PACS is smaller than that for WISDM. If the optimization difficulty prevents IFLOC from working properly, we might be able to mitigate it by using ideas from the studies that investigate the convergence and optimization difficulty in adversarial training (e.g., Nagarajan & Kolter (2017); Heusel et al. (2017); Balduzzi et al. (2018)). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we addressed domain generalization under domain-class dependency, which was overlooked by most prior domain generalization methods relying on domain-invariant representation. We theoretically showed the importance of considering the dependency and the way to overcome the problem by expanding the analysis of Xie et al. (2017). We then proposed a novel method IFLOC, which maximizes domain-invariance within a range that does not interfere with classification accuracy. Empirical validations show the superior performance of IFLOC to the baseline methods, supporting the importance of the domain-class dependency in domain generalization tasks and the efficacy of the proposed method for overcoming the issue. Future work includes applying the reguralization idea of making $\\_$ to other methods, e.g., Conditional VAE (Louizos et al. (2016)) or CrossGrad (Shankar et al. (2018)) because they have clear and tractable data generating process but assume the independence of and $\\cdot$ . Also intended is to use it for transfer learning tasks $\\cdot$   \nin a few-shot setting (e.g., life-long learning) where domain-class dependency is likely to occur due to scarce sample size. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ignatov Andrey. Real-time human activity recognition from accelerometer data using convolutional neural networks. Applied Soft Computing, 2017. ",
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+        "type": "text",
+        "text": "A DOMAIN-CLASS DEPENDENCY IN PACS AND WISDM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            660,
+            118
+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6bfcdfba00acfd2c9489283cfe87fab21264b49dd66733ee4c69abad05729ec7.jpg",
+        "table_caption": [
+            "Table 4: Sample sizes for each $<$ <domain, class $>$ pair in PACS dataset. The column shows category name while and index shows style. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Guitar</td><td>House</td><td>Giraffe</td><td>Person</td><td>Horse</td><td>Dog</td><td>Elephant</td></tr><tr><td>Art Painting</td><td>184</td><td>295</td><td>285</td><td>449</td><td>201</td><td>379</td><td>255</td></tr><tr><td>Cartoon</td><td>135</td><td>288</td><td>346</td><td>405</td><td>324</td><td>389</td><td>457</td></tr><tr><td>Photo</td><td>186</td><td>280</td><td>182</td><td>432</td><td>199</td><td>189</td><td>202</td></tr><tr><td>Sketch</td><td>608</td><td>80</td><td>753</td><td>160</td><td>816</td><td>772</td><td>740</td></tr></table>",
+        "bbox": [
+            228,
+            183,
+            769,
+            268
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/ff494f96e52a7c037d8e378f66848d917140fc14a3db73ce98c66157fb9af43c.jpg",
+        "table_caption": [
+            "Table 5: Sample sizes for each <domain, class $>$ pair in WISDM dataset. The column shows activity name and the index shows user id. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Jogging</td><td>Walking</td><td>Upstairs</td><td>Downstairs</td><td>Sitting</td><td>Standing</td></tr><tr><td>User 1</td><td>145</td><td>742</td><td>108</td><td>224</td><td>160</td><td>78</td></tr><tr><td>User 2</td><td>142</td><td>481</td><td>282</td><td>186</td><td>0</td><td>0</td></tr><tr><td>User 3</td><td>645</td><td>654</td><td>240</td><td>231</td><td>780</td><td>267</td></tr><tr><td>User 4</td><td>637</td><td>619</td><td>237</td><td>214</td><td>113</td><td>78</td></tr><tr><td>User 5</td><td>614</td><td>650</td><td>229</td><td>210</td><td>56</td><td>80</td></tr><tr><td>User 6</td><td>637</td><td>574</td><td>101</td><td>86</td><td>0</td><td>0</td></tr><tr><td>User 7</td><td>588</td><td>617</td><td>81</td><td>69</td><td>81</td><td>33</td></tr><tr><td>User 8</td><td>599</td><td>621</td><td>160</td><td>171</td><td>102</td><td>79</td></tr><tr><td>User 9</td><td>599</td><td>308</td><td>269</td><td>206</td><td>123</td><td>94</td></tr><tr><td>User 10</td><td>597</td><td>625</td><td>119</td><td>118</td><td>71</td><td>95</td></tr><tr><td>User 11</td><td>610</td><td>616</td><td>188</td><td>115</td><td>150</td><td>81</td></tr><tr><td>User 12</td><td>626</td><td>356</td><td>0</td><td>0</td><td>77</td><td>51</td></tr><tr><td>User 13</td><td>620</td><td>604</td><td>217</td><td>131</td><td>0</td><td>0</td></tr><tr><td>User 14</td><td>0</td><td>624</td><td>68</td><td>76</td><td>147</td><td>96</td></tr><tr><td>User 15</td><td>318</td><td>610</td><td>167</td><td>162</td><td>81</td><td>73</td></tr><tr><td>User 16</td><td>602</td><td>650</td><td>212</td><td>187</td><td>0</td><td>81</td></tr><tr><td>User 17</td><td>0</td><td>706</td><td>142</td><td>147</td><td>0</td><td>63</td></tr><tr><td>User 18</td><td>593</td><td>658</td><td>178</td><td>189</td><td>0</td><td>0</td></tr><tr><td>User 19</td><td>661</td><td>690</td><td>406</td><td>141</td><td>0</td><td>0</td></tr><tr><td>User 20</td><td>611</td><td>310</td><td>149</td><td>144</td><td>32</td><td>25</td></tr><tr><td>User 21</td><td>616</td><td>537</td><td>130</td><td>141</td><td>112</td><td>81</td></tr><tr><td>User 22</td><td>613</td><td>327</td><td>239</td><td>94</td><td>0</td><td>0</td></tr><tr><td>User 23</td><td>42</td><td>301</td><td>66</td><td>86</td><td>60</td><td>0</td></tr><tr><td>User 24</td><td>0</td><td>626</td><td>209</td><td>191</td><td>75</td><td>152</td></tr><tr><td>User 25</td><td>641</td><td>666</td><td>194</td><td>140</td><td>76</td><td>65</td></tr><tr><td>User 26</td><td>513</td><td>853</td><td>220</td><td>165</td><td>132</td><td>161</td></tr><tr><td>User 27</td><td>701</td><td>841</td><td>231</td><td>192</td><td>105</td><td>128</td></tr><tr><td>User 28</td><td>477</td><td>622</td><td>240</td><td>199</td><td>78</td><td>140</td></tr><tr><td>User 29</td><td>548</td><td>646</td><td>168</td><td>164</td><td>78</td><td>139</td></tr><tr><td>User 30</td><td>309</td><td>349</td><td>269</td><td>179</td><td>0</td><td>0</td></tr><tr><td>User 31</td><td>550</td><td>641</td><td>154</td><td>145</td><td>0</td><td>0</td></tr><tr><td>User 32</td><td>0</td><td>644</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 33</td><td>322</td><td>346</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 34</td><td>587</td><td>584</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>User 35</td><td>457</td><td>549</td><td>178</td><td>110</td><td>124</td><td>116</td></tr><tr><td>User 36</td><td>808</td><td>879</td><td>212</td><td>128</td><td>124</td><td>104</td></tr></table>",
+        "bbox": [
+            243,
+            330,
+            758,
+            862
+        ],
+        "page_idx": 12
+    }
+]

parse/train/HyxKIiAqYQ/HyxKIiAqYQ.md

@@ -0,0 +1,254 @@
+# CONTEXT-ADAPTIVE ENTROPY MODEL FOR END-TOEND OPTIMIZED IMAGE COMPRESSION
+
+Jooyoung Lee, Seunghyun Cho & Seung-Kwon Beack
+
+Broadcasting Media Research Laboratory Electronics and Telecommunications Research Institute Daejeon, Korea {leejy1003,shcho,skbeack}@etri.re.kr
+
+# ABSTRACT
+
+We propose a context-adaptive entropy model for use in end-to-end optimized image compression. Our model exploits two types of contexts, bit-consuming contexts and bit-free contexts, distinguished based upon whether additional bit allocation is required. Based on these contexts, we allow the model to more accurately estimate the distribution of each latent representation with a more generalized form of the approximation models, which accordingly leads to an enhanced compression performance. Based on the experimental results, the proposed method outperforms the traditional image codecs, such as BPG and JPEG2000, as well as other previous artificial-neural-network (ANN) based approaches, in terms of the peak signal-to-noise ratio (PSNR) and multi-scale structural similarity (MS-SSIM) index. The test code is publicly available at https://github.com/JooyoungLeeETRI/CA_Entropy_Model.
+
+# 1 INTRODUCTION
+
+Recently, artificial neural networks (ANNs) have been applied in various areas and have achieved a number of breakthroughs resulting from their superior optimization and representation learning performance. In particular, for various problems that are sufficiently straightforward that they can be solved within a short period of time by hand, a number of ANN-based studies have been conducted and significant progress has been made. With regard to image compression, however, relatively slow progress has been made owing to its complicated target problems. A number of works, focusing on the quality enhancement of reconstructed images, were proposed. For instance, certain approaches (Dong et al., 2015; Svoboda et al., 2016; Zhang et al., 2017) have been proposed to reduce artifacts caused by image compression, relying on the superior image restoration capability of an ANN. Although it is indisputable that artifact reduction is one of the most promising areas exploiting the advantages of ANNs, such approaches can be viewed as a type of post-processing, rather than image compression itself.
+
+Regarding ANN-based image compression, the previous methods can be divided into two types. First, as a consequence of the recent success of generative models, some image compression approaches targeting the superior perceptual quality (Agustsson et al., 2018; Santurkar et al., 2018; Rippel & Bourdev, 2017) have been proposed. The basic idea here is that learning the distribution of natural images enables a very high compression level without severe perceptual loss by allowing the generation of image components, such as textures, which do not highly affect the structure or the perceptual quality of the reconstructed images. Although the generated images are very realistic, the acceptability of the machine-created image components eventually becomes somewhat applicationdependent. Meanwhile, a few end-to-end optimized ANN-based approaches (Toderici et al., 2017; Johnston et al., 2018; Balle et al., 2017; Theis et al., 2017; Ball ´ e et al., 2018), without generative ´ models, have been proposed. In these approaches, unlike traditional codecs comprising separate tools, such as prediction, transform, and quantization, a comprehensive solution covering all functions has been sought after using end-to-end optimization. Toderici et al. (2017)’s approach exploits a small number of latent binary representations to contain the compressed information in every step, and each step increasingly stacks the additional latent representations to achieve a progressive improvement in quality of the reconstructed images. Johnston et al. (2018) improved the compression performance by enhancing operation methods of the networks developed by Toderici et al. (2017). Although Toderici et al. (2017); Johnston et al. (2018) provided novel frameworks suitable to quality control using a single trained network, the increasing number of iteration steps to obtain higher image quality can be a burden to certain applications. In contrast to the approaches developed by Toderici et al. (2017) and Johnston et al. (2018), which extract binary representations with as high an entropy as possible, Balle et al. (2017), Theis et al. (2017), and Ball ´ e et al. (2018) regard the im- ´ age compression problem as being how to retrieve discrete latent representations having as low an entropy as possible. In other words, the target problem of the former methods can be viewed as how to include as much information as possible in a fixed number of representations, whereas the latter is simply how to reduce the expected bit-rate when a sufficient number of representations are given, assuming that the low entropy corresponds to small number of bits from the entropy coder. To solve the second target problem, Balle et al. (2017), Theis et al. (2017), and Ball ´ e et al. (2018) adopt their ´ own entropy models to approximate the actual distributions of the discrete latent representations. More specifically, Balle et al. (2017) and Theis et al. (2017) proposed novel frameworks that exploit ´ the entropy models, and proved their performance capabilities by comparing the results with those of conventional codecs such as JPEG2000. Whereas Balle et al. (2017) and Theis et al. (2017) assume ´ that each representation has a fixed distribution, Balle et al. (2018) introduced an input-adaptive ´ entropy model that estimates the scale of the distribution for each representation. This idea is based on the characteristics of natural images in which the scales of the representations vary together in adjacent areas. They provided test results that outperform all previous ANN-based approaches, and reach very close to those of BPG (Bellard, 2014), which is known as a subset of HEVC (ISO/IEC 23008-2, ITU-T H.265), used for image compression.
+
+![](images/34dbc9c19daf0a08cec1170f1febb5b86680b2102aaa92fa61b9e61d7324816e.jpg)  
+Figure 1: Comparison of sample test results including the ground truth, our method, Balle et al. ´ (2018)’s approach, BPG, and JPEG2000.
+
+One of the principle elements in end-to-end optimized image compression is the trainable entropy model used for the latent representations. Because the actual distributions of latent representations are unknown, the entropy models provide the means to estimate the required bits for encoding the latent representations by approximating their distributions. When an input image $_ { \textbf { \em x } }$ is transformed into a latent representation $\textbf {  { y } }$ and then uniformly quantized into $\hat { y }$ , the simple entropy model can be represented by $p _ { \hat { \pmb { y } } } ( \hat { \pmb { y } } )$ , as described by Balle et al. (2018). When the actual marginal distribution ´ of $\hat { y }$ is denoted as $m ( \hat { \pmb y } )$ , the rate estimation, calculated through cross entropy using the entropy model, $p _ { \hat { \pmb { y } } } ( \hat { \pmb { y } } )$ , can be represented as shown in equation (1), and can be decomposed into the actual entropy of $\hat { y }$ and the additional bits owing to a mismatch between the actual distributions and their approximations. Therefore, decreasing the rate term $R$ during the training process allows the entropy model $p _ { \hat { \pmb { y } } } ( \hat { \pmb { y } } )$ to approximate $m ( \hat { \pmb y } )$ as closely as possible, and let the other parameters transform $x$ into $y$ properly such that the actual entropy of $\hat { y }$ becomes small.
+
+$$
+R = \mathbb { E } _ { \hat { \pmb { y } } \sim m } [ - \log _ { 2 } p _ { \hat { \pmb { y } } } ( \hat { \pmb { y } } ) ] = H ( m ) + D _ { K L } ( m \vert \vert p _ { \hat { \pmb { y } } } ) .
+$$
+
+In terms of KL-divergence, $R$ is minimized when $p _ { \hat { \pmb { y } } } ( \hat { \pmb { y } } )$ becomes perfectly matched with the actual distribution $m ( \hat { \pmb y } )$ . This means that the compression performance of the methods essentially depends on the capacity of the entropy model. To enhance the capacity, we propose a new entropy model that exploits two types of contexts, bit-consuming and bit-free contexts, distinguished according to whether additional bit allocation is required. Utilizing these two contexts, we allow the model to more accurately estimate the distribution of each latent representation through the use of a more generalized form of the entropy models, and thus more effectively reduce the spatial dependencies among the adjacent latent representations. Figure 1 demonstrates a comparison of the compression results of our method to those of other previous approaches. The contributions of our work are as follows:
+
+• We propose a new context-adaptive entropy model framework that incorporates the two different types of contexts.   
+• We provide the test results that outperform the widely used conventional image codec BPG in terms of the PSNR and MS-SSIM.   
+• We discuss the directions of improvement in the proposed methods in terms of the model capacity and the level of the contexts.
+
+Note that we follow a number of notations given by Balle et al. (2018) because our approach can ´ be viewed as an extension of their work, in that we exploit the same rate-distortion (R-D) optimization framework. The rest of this paper is organized as follows. In Section 2, we introduce the key approaches of end-to-end optimized image compression and propose the context-adaptive entropy model. Section 3 demonstrates the structure of the encoder and decoder models used, and the experimental setup and results are then given in section 4. Finally, in Section 5, we discuss the current state of our work and directions for improvement.
+
+# 2 END-TO-END OPTIMIZATION BASED ON CONTEXT-ADAPTIVE ENTROPY MODELS
+
+# 2.1 PREVIOUS ENTROPY MODELS
+
+Since they were first proposed by Balle et al. (2017) and Theis et al. (2017), entropy models, which ´ approximate the distribution of discrete latent representations, have noticeably improved the image compression performance of ANN-based approaches. Balle et al. (2017) assumes the entropy mod-´ els of the latent representations as non-parametric models, whereas Theis et al. (2017) adopted a Gaussian scale mixture model composed of six weighted zero-mean Gaussian models per representation. Although they assume different forms of entropy models, they have a common feature in that both concentrate on learning the distributions of the representations without considering input adaptivity. In other words, once the entropy models are trained, the trained model parameters for the representations are fixed for any input during the test time. Balle et al. (2018), in contrast, in- ´ troduced a novel entropy model that adaptively estimates the scales of the representations based on input. They assume that the scales of the latent representations from the natural images tend to move together within an adjacent area. To reduce this redundancy, they use a small amount of additional information by which the proper scale parameters (standard deviations) of the latent representations are estimated. In addition to the scale estimation, Balle et al. (2018) have also shown that when ´ the prior probability density function (PDF) for each representation in a continuous domain is convolved with a standard uniform density function, it approximates the prior probability mass function (PMF) of the discrete latent representation, which is uniformly quantized by rounding, much more closely. For training, a uniform noise is added to each latent representation so as to fit the distribution of these noisy representations into the mentioned PMF-approximating functions. Using these approaches, Balle et al. (2018) achieved a state-of-the-art compression performance, close to that of ´ BPG.
+
+# 2.2 SPATIAL DEPENDENCIES OF THE LATENT VARIABLES
+
+The latent representations, when transformed through a convolutional neural network, essentially contain spatial dependencies because the same convolutional filters are shared across the spatial regions, and natural images have various factors in common within adjacent regions. Balle et al. ´ (2018) successfully captured these spatial dependencies and enhanced the compression performance by input-adaptively estimating standard deviations of the latent representations. Taking a step forward, we generalize the form of the estimated distribution by allowing, in addition to the standard deviation, the mean estimation utilizing the contexts. For instance, assuming that certain representations tend to have similar values within a spatially adjacent area, when all neighborhood representations have a value of 10, we can intuitively guess that, for the current representation, the chance of having a value equal or similar to 10 is relatively high. This simple estimation will consequently reduce the entropy. Likewise, our method utilizes the given contexts for estimating the mean, as well as the standard deviation, of each latent representation. Note that Toderici et al. (2017), Johnston et al. (2018), and Rippel & Bourdev (2017) also apply context-adaptive entropy coding by estimating the probability of each binary representation. However, these context-adaptive entropy-coding methods can be viewed as separate components, rather than one end-to-end optimization component because their probability estimation does not directly contribute to the rate term of the R-D optimization framework. Figure 2 visualizes the latent variables $\hat { y }$ and their normalized versions of the two different approaches, one estimating only the standard deviation parameters and the other estimating both the mu and standard deviation parameters with the two types of mentioned contexts. The visualization shows that the spatial dependency can be removed more effectively when the mu is estimated along with the given contexts.
+
+![](images/04976c4594e77d9e31aedc6865f52df2ccb7de18e96a773e22551c6fd7a93b96.jpg)  
+Figure 2: Examples of latent representations and their normalized versions for the two cases (the first two images show the results in which only the standard deviations are estimated using side information, whereas the last two images show the results in which the mu and standard deviation are estimated using our method). For a clear demonstration, the latent representations having the highest covariance between the spatially adjacent variables are extracted. Left: the latent representations of $\hat { y }$ from the first case. Middle left: the normalized versions of $\hat { y }$ from the first case, divided by the estimated standard deviation. Middle right: the latent variable of $\hat { y }$ from the second case. Right: the normalized versions of $\hat { y }$ from the second case, shifted and divided by the estimated mu and the standard deviation.
+
+# 2.3 CONTEXT-ADAPTIVE ENTROPY MODEL
+
+The optimization problem described in this paper is similar with Balle et al. (2018), in that the input ´ $_ { \textbf { \em x } }$ is transformed into $\textbf {  { y } }$ having a low entropy, and the spatial dependencies of $\textbf {  { y } }$ are captured into $\hat { z }$ . Therefore, we also use four fundamental parametric transform functions: an analysis transform $g _ { a } ( { \pmb x } ; \phi _ { g } )$ to transform $_ { \textbf { \em x } }$ into a latent representation $\textbf {  { y } }$ , a synthesis transform $g _ { s } ( \hat { y } ; \pmb { \theta } _ { g } )$ to reconstruct image $\hat { \pmb x }$ , an analysis transform $h _ { a } ( \hat { \pmb { y } } ; \phi _ { h } )$ to capture the spatial redundancies of $\hat { y }$ into a latent representation $_ { z }$ , and a synthesis transform $h _ { s } ( \hat { z } ; \pmb { \theta } _ { h } )$ used to generate the contexts for the model estimation. Note that $h _ { s }$ does not estimate the standard deviations of the representations directly as in Balle et al. (2018)’s approach. In our method, instead, ´ $h _ { s }$ generates the context $c ^ { \prime }$ , one of the two types of contexts for estimating the distribution. These two types of contexts are described in this section.
+
+Balle et al. (2018) analyzed the optimization problem from the viewpoint of the variational autoen- ´ coder (Kingma & Welling (2014); Rezende et al. (2014)), and showed that the minimization of the KL-divergence is the same problem as the R-D optimization of image compression. Basically, we follow the same concept; however, for training, we use the discrete representations on the conditions instead of the noisy representations, and thus the noisy representations are only used as the inputs to the entropy models. Empirically, we found that using discrete representations on the conditions show better results, as shown in appendix 6.2. These results might come from removing the mismatches of the conditions between the training and testing, thereby enhancing the training capacity by limiting the affect of the uniform noise only to help the approximation to the probability mass functions. We use the gradient overriding method with the identity function, as in Theis et al. (2017), to deal with the discontinuities from the uniform quantization. The resulting objective function used in this paper is given in equation (2). The total loss consists of two terms representing the rates and distortions, and the coefficient $\lambda$ controls the balance between the rate and distortion during the R-D optimization. Note that $\lambda$ is not an optimization target, but a manually configured condition that determines which to focus on between rate and distortion:
+
+$$
+\begin{array} { r l } & { \mathcal { L } = R + \lambda D } \\ & { \mathrm { w i t h } \ : R = \mathbb { E } _ { { \pmb x } \sim p _ { \pmb x } } \mathbb { E } _ { { \pmb \tilde { y } } , { \boldsymbol \tilde { z } } \sim q } \Big [ - \log p _ { { \pmb { \tilde { y } } } | { \hat { z } } } ( { \pmb \tilde { y } } \mid { \hat { z } } ) - \log p _ { { \boldsymbol \tilde { z } } } ( { \boldsymbol \tilde { z } } ) \Big ] , } \\ & { \quad \quad D = \mathbb { E } _ { { \pmb x } \sim p _ { \infty } } \Big [ - \log p _ { { \pmb x } | { \hat { y } } } ( { \pmb x } \mid { \hat { y } } ) \Big ] } \end{array}
+$$
+
+Here, the noisy representations of $\tilde { y }$ and $\tilde { z }$ follow the standard uniform distribution, the mean values of which are $\textbf {  { y } }$ and $_ { z }$ , respectively, when $\textbf {  { y } }$ and $_ z$ are the result of the transforms $g _ { a }$ and $h _ { a }$ , repectively. Note that the input to $h _ { a }$ is $\hat { y }$ , which is a uniformly quantized representation of $\textbf {  { y } }$ , rather than the noisy representation ${ \tilde { y } } . \ Q$ denotes the uniform quantization function, for which we simply use a rounding function:
+
+$$
+\begin{array} { l l l } { { q ( \tilde { y } , \tilde { z } \mid x , \phi _ { g } , \phi _ { h } ) } } & { { = } } & { { \displaystyle \prod _ { i } \mathcal { U } \big ( \tilde { y } _ { i } \mid y _ { i } - \frac { 1 } { 2 } , y _ { i } + \frac { 1 } { 2 } \big ) \cdot \prod _ { j } \mathcal { U } \big ( \tilde { z } _ { j } \mid z _ { j } - \frac { 1 } { 2 } , z _ { j } + \frac { 1 } { 2 } \big ) } } \\ { { } } & { { } } & { { \mathrm { w i t h } ~ y = g _ { a } ( { \bf x } ; \phi _ { g } ) , \hat { y } = Q ( y ) , z = h _ { a } ( \hat { y } ; \phi _ { h } ) . } } \end{array}
+$$
+
+The rate term represents the expected bits calculated using the entropy models of $p _ { \tilde { y } | \hat { z } }$ and $p _ { \tilde { z } }$ . Note that $p _ { \tilde { y } | \hat { z } }$ and $p _ { \tilde { z } }$ are eventually the approximations of $p _ { \hat { y } | \hat { z } }$ and $p _ { \hat { z } }$ , respectively. Equation (4) represents the entropy model for approximating the required bits for $\hat { y }$ . The model is based on the Gaussian model, which not only has the standard deviation parameter $\sigma _ { i }$ , but also the mu parameter $\mu _ { i }$ . The values of $\mu _ { i }$ and $\sigma _ { i }$ are estimated from the two types of given contexts based on the function $f$ , the distribution estimator, in a deterministic manner. The two types of contexts, bit-consuming and bit-free contexts, for estimating the distribution of a certain representation are denoted as $ { \boldsymbol { c } } _ { i } ^ { \prime }$ and $ { c _ { i } ^ { \prime \prime } }$ . $E ^ { \prime }$ extracts $ { \boldsymbol { c } } _ { i } ^ { \prime }$ from $c ^ { \prime }$ , the result of transform $h _ { s }$ . In contrast to $\pmb { c } _ { i } ^ { \prime }$ , no additional bit allocation is required for $ { c _ { i } ^ { \prime \prime } }$ . Instead, we simply utilize the known (already entropy-coded or decoded) subset of $\hat { y }$ , denoted as $\langle \hat { y } \rangle$ . Here, $ { \boldsymbol { c } } _ { i } ^ { \prime \prime }$ is extracted from $\langle \hat { y } \rangle$ by the extractor $E ^ { \prime \prime }$ . We assume that the entropy coder and the decoder sequentially process $\hat { y } _ { i }$ in the same specific order, such as with raster scanning, and thus $\langle \hat { y } \rangle$ given to the encoder and decoder can always be identical when processing the same $\hat { y } _ { i }$ . A formal expression of this is as follows:
+
+$$
+\begin{array} { c } { { p _ { \tilde { y } | \hat { z } } ( \tilde { y } \mid \hat { z } , \theta _ { h } ) = { \displaystyle \prod _ { i } \biggl ( \mathcal { N } \bigl ( \mu _ { i } , \sigma _ { i } ^ { 2 } \bigr ) * \mathcal { U } \bigl ( - { \textstyle \frac { 1 } { 2 } } , { \textstyle \frac { 1 } { 2 } } \bigr ) \biggr ) \bigl ( \tilde { y } _ { i } \bigr ) } } } \\ { { \mathrm { w i t h } \ \mu _ { i } , \sigma _ { i } = f ( c _ { i } ^ { \prime } , c _ { i } ^ { \prime \prime } ) , } }  \\ { { c _ { i } ^ { \prime } = E ^ { \prime } \bigl ( h _ { s } \bigl ( \hat { z } ; \theta _ { h } \bigr ) , i \bigr ) , } }  \\ { { c _ { i } ^ { \prime \prime } = E ^ { \prime \prime } \bigl ( \langle \hat { y } \rangle , i \bigr ) , } }  \\ { { \hat { z } = Q ( z ) } } \end{array}
+$$
+
+In the case of $\hat { z }$ , a simple entropy model is used. We assumed that the model follows zero-mean Gaussian distributions which have a trainable $\sigma$ . Note that $\hat { z }$ is regarded as side information and it contributes a very small amount of the total bit-rate, as described by Balle et al. (2018), and thus ´ we use this simpler version of the entropy model, rather than a more complex model, for end-to-end optimization over all parameters of the proposed method:
+
+$$
+p _ { \tilde { z } } ( \tilde { z } ) = \prod _ { j } \Bigl ( \mathcal { N } \bigl ( 0 , \sigma _ { j } ^ { 2 } \bigr ) * \mathcal { U } \bigl ( - \textstyle \frac { 1 } { 2 } , \frac { 1 } { 2 } \bigr ) \Bigr ) ( \tilde { z } _ { j } )
+$$
+
+![](images/6b597b00e569a802b4bbf4c715d6d688ed082f96708d4bdad36bdc133d4e00db.jpg)  
+Figure 3: Encoder-decoder model of the proposed method. The left block represents the encoder side, whereas the right block represents the deocder side. The small icons between them represent the entropy-coded bitstreams. EC and ED represent entropy coding and entropy decoding, and U | Q represents the addition of uniform noise to $\textbf {  { y } }$ or a uniform quantization of $\textbf {  { y } }$ . Noisy representations are used only for training as inputs to the entropy models, and are illustrated with the dotted lines.
+
+Note that actual entropy coding or decoding processes are not necessarily required for training or encoding because the rate term is not the amount of real bits, but an estimation calculated from the entropy models, as mentioned previously. We calculate the distortion term using the mean squared error (MSE)1, assuming that $p _ { \mathbf { \hat { \mathbf { \theta } } } \mathbf { \hat { \mathbf { x } } } | \hat { \mathbf { \theta } } }$ follows a Gaussian distribution as a widely used distortion metric.
+
+# 3 ENCODER-DECODER MODEL
+
+This section describes the basic structure of the proposed encoder-decoder model. On the encoder side, an input image is transformed into latent representations, quantized, and then entropy-coded using the trained entropy models. In contrast, the decoder first applies entropy decoding with the same entropy models used for the encoder, and reconstructs the image from the latent representations, as illustrated in figure 3. It is assumed that all parameters that appear in this section were already trained. The structure of the encoder-decoder model basically includes $\scriptstyle { \pmb { g } } _ { a }$ and $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ in charge of the transform of $_ { \textbf { \em x } }$ into $\textbf {  { y } }$ and its inverse transform, respectively. The transformed $\textbf {  { y } }$ is uniformly quantized into $\hat { y }$ by rounding. Note that, in the case of approaches based on the entropy models, unlike traditional codecs, tuning the quantization steps is usually unnecessary because the scales of the representations are optimized together through training. The other components between $\scriptstyle { \pmb { g } } _ { a }$ and $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ carry out the role of entropy coding (or decoding) with the shared entropy models and underlying context preparation processes. More specifically, the entropy model estimates the distribution of each $\hat { y } _ { i }$ individually, in which $\mu _ { i }$ and $\sigma _ { i }$ are estimated with the two types of given contexts, $\pmb { c } _ { i } ^ { \prime }$ and $ { c _ { i } ^ { \prime \prime } }$ . Among these contexts, $c ^ { \prime }$ can be viewed as side information, which requires an additional bit allocation. To reduce the required bit-rate for carrying $c ^ { \prime }$ , the latent representation $_ z$ , transformed from $\hat { y }$ , is quantized and entropy-coded by its own entropy model, as specified in section 2.3. On the other hand, $ { \boldsymbol { c } } _ { i } ^ { \prime \prime }$ is extracted from $\langle \hat { y } \rangle$ , without any additional bit allocation. Note that $\langle \hat { y } \rangle$ varies as the entropy coding or decoding progresses, but is always identical for processing the same $\hat { y } _ { i }$ in both the encoder and decoder, as described in 2.3. The parameters of $h _ { s }$ and the entropy models are simply shared by both the encoder and the decoder. Note that the inputs to the entropy models during training are the noisy representations, as illustrated with the dotted line in figure 3, to allow the entropy model to approximate the probability mass functions of the discrete representations.
+
+![](images/7c54fa6307bcb8e20b007b9479ec8ee54178f761d5cb621d014f8c013f93bc6a.jpg)  
+Figure 4: Implementation of the proposed method. We basically use the convolutional autoencoder structure, and the distribution estimator $f$ is also implemented using convolutional neural networks. The notations of the convolutional layer follow Balle et al. (2018): the number of filters ´ $\times$ filter height $\times$ filter width / the downscale or upscale factor, where $\uparrow$ and $\downarrow$ denote the up and downscaling, respectively. For up or downscaling, we use the transposed convolution. For the networks, input images are normalized into a scale between $^ { - 1 }$ and 1.
+
+# 4 EXPERIMENTS
+
+# 4.1 IMPLEMENTATION
+
+We use a convolutional neural networks to implement the analysis transform and the synthesis transform functions, $g _ { a } , g _ { s }$ , $h _ { a }$ , and $h _ { s }$ . The structures of the implemented networks follow the same structures of Balle et al. (2018), except that we use the exponentiation operator instead of an abso- ´ lute operator at the end of $h _ { s }$ . Based on Balle et al. (2018)’s structure, we added the components to ´ estimate the distribution of each $\hat { y } _ { i }$ , as shown in figure 4. Herein, we represent a uniform quantization (round) as $\because Q , \mathrm { { } }$ entropy coding as ”EC,” and entropy decoding as ”ED.” The distribution estimator is denoted as $f$ , and is also implemented using the convolutional layers which takes channel-wise concatenated $ { \mathbf { c } } _ { i } ^ { \prime }$ and ${ c } _ { i } ^ { \prime \prime }$ as inputs and provides estimated $\mu _ { i }$ and $\sigma _ { i }$ as results. Note that the same $\mathbf { c } _ { i } ^ { \prime }$ and ${ c } _ { i } ^ { \prime \prime }$ are shared for all ${ \hat { y } } _ { i } \mathbf { s }$ located at the same spatial position. In other words, we let $E ^ { \prime }$ extract all spatially adjacent elements from $c ^ { \prime }$ across the channels to retrieve $ { \boldsymbol { c } } _ { i } ^ { \prime }$ and likewise let $E ^ { \prime \prime }$ extract all adjacent known elements from $\langle \hat { y } \rangle$ for $ { \boldsymbol { c } } _ { i } ^ { \prime \prime }$ . This could have the effect of capturing the remaining correlations among the different channels. In short, when $M$ is the total number of channels of $\textbf {  { y } }$ , we let $f$ estimate all $M$ distributions of ${ \hat { y } } _ { i } \mathbf { s }$ , which are located at the same spatial position, using only a single step, thereby allowing the total number of estimations to be reduced. Furthermore, the parameters of $f$ are shared for all spatial positions of $\hat { y }$ , and thus only one trained $f$ per $\lambda$ is necessary to process any sized images. In the case of training, however, collecting the results from the all spatial positions to calculate the rate term becomes a significant burden, despite the simplifications mentioned above. To reduce this burden, we designate a certain number (32 and 16 for the base model and the hybrid model, respectively) of random spatial points as the representatives per training step, to calculate the rate term easily. Note that we let these random points contribute solely to the rate term, whereas the distortion is still calculated over all of the images.
+
+Because $\textbf {  { y } }$ is a three-dimensional array in our implementation, index $i$ can be represented as three indexes, $k , l ,$ and $m$ , representing the horizontal index, the vertical index, and the channel index, respectively. When the current position is given as $( k , l , m )$ , $E ^ { \prime }$ extracts $\pmb { c } _ { [ k - 2 . . . k + 1 ] , [ l - 3 . . . l ] , [ 1 . . . M ] } ^ { \prime }$ as $\mathbf { c } _ { i } ^ { \prime }$ , and $E ^ { \prime \prime }$ extracts $\left. \hat { \pmb y } \right. _ { [ k - 2 . . . k + 1 ] , [ l - 3 . . . l ] , [ 1 . . . M ] }$ as $ { c _ { i } ^ { \prime \prime } }$ , when $\langle \hat { y } \rangle$ represents the known area of $\hat { y }$ . Note that we filled in the unknown area of $\langle \hat { y } \rangle$ with zeros, to maintain the dimensions of $\langle \hat { y } \rangle$ identical to those of $\hat { y }$ . Consequently, $\pmb { c } _ { i \ [ 3 . . . 4 ] , 4 , [ 1 . . . M ] } ^ { \prime \prime }$ are always padded with zeros. To keep the dimensions of the estimation results to the inputs, the marginal areas of $c ^ { \prime }$ and $\langle \hat { y } \rangle$ are also set to zeros. Note that when training or encoding, $\pmb { c } _ { i } ^ { \prime \prime }$ can be extracted using simple $4 { \times } 4 { \times } M$ windows and binary masks, thereby enabling parallel processing, whereas a sequential reconstruction is inevitable for decoding.
+
+Another implementation technique used to reduce the implementation cost is combining the lightweight entropy model with the proposed model. The lightweight entropy model assumes that the representations follow a zero-mean Gaussian model with the estimated standard deviations, which is very similar with Balle et al. (2018)’s approach. We utilize this hybrid approach for the top ´ four cases, in bit-rate descending order, of the nine $\lambda$ configurations, based on the assumption that for the higher-quality compression, the number of sparse representations having a very low spatial dependency increases, and thus a direct scale estimation provides sufficient performance for these added representations. For implementation, we separate the latent representation $\textbf {  { y } }$ into two parts, $\mathbf { \pmb { y _ { 1 } } }$ and $\mathbf { \pmb { y } _ { 2 } }$ , and two different entropy models are applied for them. Note that the parameters of $g _ { a }$ , $g _ { s }$ , $h _ { a }$ , and $h _ { s }$ are shared, and all parameters are still trained together. The detailed structure and experimental settings are described in appendix 6.1.
+
+The number of parameters $N$ and $M$ are set to 128 and 192, respectively, for the five $\lambda$ configurations for lower bit-rates, whereas 2-3 times more parameters, described in appendix 6.1, are used for the four $\lambda$ configurations for higher bit-rates. Tensorflow and Python were used to setup the overall network structures, and for the actual entropy coding and decoding using the estimated model parameters, we implemented an arithmetic coder and decoder, for which the source code of the ”Reference arithmetic coding” project2 was used as the base code.
+
+# 4.2 EXPERIMENTAL ENVIRONMENTS
+
+We optimized the networks using two different types of distortion terms, one with MSE and the other with MS-SSIM. For each distortion type, the average bits per pixel (BPP) and the distortion, PSNR and MS-SSIM, over the test set are measured for each of the nine $\lambda$ configurations. Therefore, a total of 18 networks are trained and evaluated within the experimental environments, as explained below:
+
+• For training, we used $2 5 6 \times 2 5 6$ patches extracted from 32,420 randomly selected YFCC100m (Thomee et al. (2016)) images. We extracted one patch per image, and the extracted regions were randomly chosen. Each batch consists of eight images, and 1M iterations of the training steps were conducted, applying the ADAM optimizer (Kingma & Ba (2015)). We set the initial learning rate to $5 \times 1 0 ^ { - } 5$ , and reduced the rate by half every 50,000 iterations for the last 200,000 iterations. Note that, in the case of the four $\lambda$ configurations for high bpp, in which the hybrid entropy model is used, 1M iterations of pre-training steps were conducted using the learning rate of $1 \times 1 0 ^ { - } 5$ . Although we previously indicated that the total loss is the sum of $R$ and $\lambda D$ for a simple explanation, we tuned the balancing parameter $\lambda$ in a similar way as Theis et al. (2017), as indicated in equation (6). We used the $\lambda$ parameters ranging from 0.01 to 0.5.
+
+$$
+\mathcal { L } = \frac { \lambda } { W _ { y } \cdot H _ { y } \cdot 2 5 6 } R + \frac { 1 - \lambda } { 1 0 0 0 } D .
+$$
+
+• For the evaluation, we measured the average BPP and average quality of the reconstructed images in terms of the PSNR and MS-SSIM over 24 PNG images of the Kodak PhotoCD image dataset (Kodak, 1993). Note that we represent the MS-SSIM results in the form of decibels, as in Balle et al. (2018), to increase the discrimination.´
+
+# 4.3 EXPERIMENTAL RESULTS
+
+We compared the test results with other previous methods, including traditional codecs such as BPG and JPEG2000, as well as previous ANN-based approaches such as Theis et al. (2017) and Balle´ et al. (2018). Because two different quality metrics are used, the results are presented with two separate plots. As shown in figure 5, our methods outperform all other previous methods in both metrics. In particular, our models not only outperform Balle et al. (2018)’s method, which is believed ´ to be a state-of-the-art ANN-based approach, but we also obtain better results than the widely used conventional image codec, BPG.
+
+![](images/f8a3b96ef70c7b5691e918165aef6fead9b832dc0412c6876f9f37b9f74769da.jpg)  
+Figure 5: Rate–distortion curves of the proposed method and competitive methods. The top plot represents the PSNR values as a result of changes in bpp, whereas the bottom plot shows MS-SSIM values in the same manner. Note that MS-SSIM values are converted into decibels $( - 1 0 \log _ { 1 0 } ( 1 -$ MS-SSIM)) for differentiating the quality levels, in the same manner as in Balle et al. (2018). ´
+
+More specifically, the compression gains in terms of the BD-rate of PSNR over JPEG2000, Balle´ et al. (2018)’s approach (MSE-optimized), and BPG are $3 4 . 0 8 \%$ , $1 1 . 9 7 \%$ , and $6 . 8 5 \%$ , respectively. In the case of MS-SSIM, we found wider gaps of $6 8 . 8 2 \%$ , $1 3 . 9 3 \%$ , and $4 9 . 6 8 \%$ , respectively. Note that we achieved significant gains over traditional codecs in terms of MS-SSIM, although this might be because the dominant target metric of the traditional codec developments have been the PSNR.
+
+In other words, they can be viewed as a type of MSE-optimized codec. Even when setting aside the case of MS-SSIM, our results can be viewed as one concrete evidence supporting that ANN-based image compression can outperform the existing traditional image codecs in terms of the compression performance. Supplemental image samples are provided in appendix 6.3.
+
+# 5 DISCUSSION
+
+Based on previous ANN-based image compression approaches utilizing entropy models (Balle et al., ´ 2017; Theis et al., 2017; Balle et al., 2018), we extended the entropy model to exploit two different ´ types of contexts. These contexts allow the entropy models to more accurately estimate the distribution of the representations with a more generalized form having both mean and standard deviation parameters. Based on the evaluation results, we showed the superiority of the proposed method. The contexts we utilized are divided into two types. One is a sort of free context, containing the part of the latent variables known to both the encoder and the decoder, whereas the other is the context, which requires additional bit allocation. Because the former is a generally used context in a variety of codecs, and the latter was already verified to help compression using Balle et al. (2018)’s approach, ´ our contributions are not the contexts themselves, but can be viewed as providing a framework of entropy models utilizing these contexts.
+
+Although the experiments showed the best results in the ANN-based image compression domain, we still have various studies to conduct to further improve the performance. One possible way is generalizing the distribution models underlying the entropy model. Although we enhanced the performance by generalizing the previous entropy models, and have achieved quite acceptable results, the Gaussian-based entropy models apparently have a limited expression power. If more elaborate models, such as the non-parametric models of Balle et al. (2018) or Oord et al. (2016), are combined ´ with the context-adaptivity proposed in this paper, they would provide better results by reducing the mismatch between the actual distributions and the approximation models. Another possible way is improving the level of the contexts. Currently, our methods only use low-level representations within very limited adjacent areas. However, if the sufficient capacity of the networks and higher-level contexts are given, a much more accurate estimation could be possible. For instance, if an entropy model understands the structures of human faces, in that they usually have two eyes, between which a symmetry exists, the entropy model could approximate the distributions more accurately when encoding the second eye of a human face by referencing the shape and position of the first given eye. As is widely known, various generative models (Goodfellow et al., 2014; Radford et al., 2016; Zhao et al., 2017) learn the distribution $p ( { \pmb x } )$ of the images within a specific domain, such as human faces or bedrooms. In addition, various in-painting methods (Pathak et al., 2016; Yang et al., 2017; Yeh et al., 2017) learn the conditional distribution $p ( \pmb { x } \mid c o n t e x t )$ when the viewed areas are given as context. Although these methods have not been developed for image compression, hopefully such high-level understandings can be utilized sooner or later. Furthermore, the contexts carried using side information can also be extended to some high-level information such as segmentation maps or any other information that helps with compression. Segmentation maps, for instance, may be able to help the entropy models estimate the distribution of a representation discriminatively according to the segment class the representation belongs to.
+
+Traditional codecs have a long development history, and a vast number of hand-crafted heuristics have been stacked thus far, not only for enhancing compression performance, but also for compromising computational complexities. Therefore, ANN-based image compression approaches may not provide satisfactory solutions as of yet, when taking their high complexity into account. However, considering its much shorter history, we believe that ANN-based image compression has much more potential and possibility in terms of future extension. Although we remain a long way from completion, we hope the proposed context-adaptive entropy model will provide an useful contribution to this area.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government (MSIT) (No. 2017-0-00072, Development of Audio/Video Coding and Light Field Media Fundamental Technologies for Ultra Realistic Teramedia).
+
+# REFERENCES
+
+Eirikur Agustsson, Michael Tschannen, Fabian Mentzer, Radu Timofte, and Luc Van Gool. Generative adversarial networks for extreme learned image compression. arXiv preprint arXiv:1804.02958, 2018. URL http://arxiv.org/abs/1804.02958.
+
+Johannes Balle, Valero Laparra, and Eero P. Simoncelli. End-to-end optimized image compression. ´ In the 5th Int. Conf. on Learning Representations, 2017. URL http://arxiv.org/abs/ 1611.01704.
+
+Johannes Balle, David Minnen, Saurabh Singh, Sung Jin Hwang, and Nick Johnston. Variational ´ image compression with a scale hyperprior. In the 6th Int. Conf. on Learning Representations, 2018. URL http://arxiv.org/abs/1802.01436.
+
+Fabrice Bellard. Bpg image format, 2014. URL http://bellard.org/bpg/.
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+Chao Dong, Yubin Deng, Chen Change Loy, and Xiaoou Tang Tang. Compression artifacts reduction by a deep convolutional network. In IEEE International Conference on Computer Vision (ICCV), 2015. URL http://arxiv.org/abs/1504.06993.
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+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 2672–2680. Curran Associates, Inc., 2014. URL http://papers. nips.cc/paper/5423-generative-adversarial-nets.pdf.
+
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+
+Nick Johnston, Damien Vincent, David Minnen, Michele Covell, Saurabh Singh, Troy Chinen, Sung Jin Hwang, Joel Shor, and George Toderici. Improved lossy image compression with priming and spatially adaptive bit rates for recurrent networks. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
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+
+Eastman Kodak. Kodak lossless true color image suite (photocd pcd0992), 1993. URL http: //r0k.us/graphics/kodak/.
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+Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In the 33 rd International Conference on Machine Learning, 2016. URL http://arxiv. org/abs/1601.06759.
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+Deepak Pathak, Philipp Krahenb ¨ uhl, Jeff Donahue, Trevor Darrell, and Alexei A. Efros. Context ¨ encoders: Feature learning by inpainting. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In the 4th Int. Conf. on Learning Representations, 2016. URL http://arxiv.org/abs/1511.06434.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In the 31st Int. Conf. on Machine Learning, 2014. URL http://arxiv.org/abs/1401.4082.
+
+Oren Rippel and Lubomir Bourdev. Real-time adaptive image compression. In International Conference on Machine Learning, 2017. URL http://arxiv.org/abs/1705.05823.
+
+Shibani Santurkar, David M. Budden, and Nir Shavit. Generative compression. In The 33rd Picture Coding Symposium, 2018. URL http://arxiv.org/abs/1703.01467.
+
+Pavel Svoboda, Michal Hradis, David Barina, and Pavel Zemc´ık. Compression artifacts removal using convolutional neural networks. In WSCG 24th International Conference on Computer Graphics, Visualization and Computer Vision, 2016. URL http://arxiv.org/abs/1605. 00366.
+
+Lucas Theis, Wenzhe Shi, Andrew Cunningham, and Ferenc Huszar. Lossy image compression ´ with compressive autoencoders. In the 5th Int. Conf. on Learning Representations, 2017. URL http://arxiv.org/abs/1703.00395.
+
+Bart Thomee, David A. Shamma, Gerald Friedland, Benjamin Elizalde, Karl Ni, Douglas Poland, Damian Borth, and Li-Jia Li. The new data in multimedia research. Communications of the ACM, 59(2):64–73, 2016.
+
+George Toderici, Damien Vincent, Nick Johnston, Sung Jin Hwang, David Minnen, Joel Shor, and Michele Covell. Full resolution image compression with recurrent neural networks. In IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2017. doi: 10.1109/CVPR.2017.577. URL http://arxiv.org/abs/1608.05148.
+
+Zhou Wang, Eero P. Simoncelli, and Alan C. Bovik. Multiscale structural similarity for image quality assessment. In The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2003. doi: 10.1109/ACSSC.2003.1292216.
+
+Chao Yang, Xin Lu, Zhe Lin, Eli Shechtman, Oliver Wang, and Hao Li. High-resolution image inpainting using multi-scale neural patch synthesis. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.
+
+Raymond A. Yeh, Chen Chen, Teck-Yian Lim, Alexander G. Schwing, Mark Hasegawa-Johnson, and Minh N. Do. Semantic image inpainting with deep generative models. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 6882–6890. IEEE Computer Society, 2017.
+
+Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and Lei Zhang. Beyond a gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Transactions on Image Processing, 26(7):3142–3155, 2017. doi: 10.1109/TIP.2017.2662206. URL http://arxiv.org/abs/ 1608.03981.
+
+Junbo Jake Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. ¨ In the 5th Int. Conf. on Learning Representations, 2017. URL http://arxiv.org/abs/ 1609.03126.
+
+![](images/4c108c0eb986ca11778b67a98a0735890f479ebfe9ba3409b68d2a9dfb198e82.jpg)  
+Figure 6: The structure of the hybrid network for higher bit-rate environments. The same notations as in the figure 4 are used. The representation $\textbf {  { y } }$ is divided into two parts and quantized. One of the resulting parts, $\hat { y } _ { 1 }$ , is encoded using the proposed model, whereas the other, $\hat { y } _ { 2 }$ , is encoded using a simpler model in which only the standard deviations are estimated using side information. The detailed structure of the proposed model is illustrated in figure 4. All concatenation and split operations are performed in a channel-wise manner.
+
+# 6 APPENDIX
+
+# 6.1 HYBRID NETWORK FOR HIGHER BIT-RATE COMPRESSIONS
+
+We combined the lightweight entropy model with the context-adaptive entropy model to reduce the implementation costs for high-bpp configurations. The lightweight model exploits the scale (standard deviation) estimation, assuming that the PMF approximations of the quantized representations follow zero-mean Gaussian distributions convolved with a standard uniform distribution.
+
+Figure 6 illustrates the network structure of this hybrid network. The representation $\textbf {  { y } }$ is split channel-wise into two parts, $\mathbf { \pmb { y _ { 1 } } }$ and $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { 2 } }$ , which have $M _ { 1 }$ and $M _ { 2 }$ channels, respectively, and is then quantized. Here, $\hat { y } _ { 1 }$ is entropy coded using the proposed entropy model, whereas $\hat { y } _ { 2 }$ is coded with the lightweight entropy model. The standard deviations of $\hat { y } _ { 2 }$ are estimated using $h _ { a }$ and $h _ { s }$ . Unlike the context-adaptive entropy model, which uses the results of $h _ { a }$ $( \hat { c ^ { \prime } } )$ as the input source to the estimator $f$ , the lightweight entropy model retrieves the estimated standard deviations from $h _ { a }$ directly. Note that $h _ { a }$ takes the concatenated $\hat { y } _ { 1 }$ and $\hat { y } _ { 2 }$ as input, and $h _ { s }$ generates $\hat { c ^ { \prime } }$ as well as $\pmb { \sigma } _ { 2 }$ , at the same time.
+
+The total loss function also consists of the rate and distortion terms, although the rate is divided into three parts, each of which is for $\hat { y } _ { 1 } , \hat { y } _ { 2 }$ , and $\hat { z }$ , respectively. The distortion term is the same as before, but note that $\hat { y }$ is the channel-wise concatenated representation of $\hat { y } _ { 1 }$ and $\hat { y } _ { 2 }$ :
+
+$$
+\begin{array} { r l } & { \mathcal { L } = R + \lambda D } \\ & { \mathrm { w i t h } \ R = \mathbb { E } _ { { \mathbf { x } } \sim p _ { \mathbf { x } } } \mathbb { E } _ { \mathcal { \tilde { I } } _ { 1 } , \tilde { y _ { 2 } } , \tilde { z } \sim q } \Big [ - \log p _ { \tilde { y _ { 1 } } \mid \hat { z } } ( \tilde { y _ { 1 } } \mid \hat { z } ) - \log p _ { \tilde { y _ { 2 } } \mid \hat { z } } ( \tilde { y _ { 2 } } \mid \hat { z } ) - \log p _ { \tilde { z } } ( \tilde { z } ) \Big ] , } \\ & { \qquad D = \mathbb { E } _ { \mathbf { x } \sim p _ { \mathbf { x } } } \Big [ - \log p _ { \mathbf { x } \mid \hat { y } } ( \boldsymbol { x } \mid \hat { y } ) \big ] } \end{array}
+$$
+
+Here, the noisy representations of ${ \tilde { y } } _ { 1 } , { \tilde { y } } _ { 2 }$ , and $\tilde { z }$ follow a standard uniform distribution, the mean values of which are ${ \bf { y 1 } } , ~ { \bf { y 2 } }$ , and $_ { z }$ , respectively. In addition, $\mathbf { \pmb { y _ { 1 } } }$ and $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { 2 } }$ are channel-wise split representations from $\textbf {  { y } }$ , the results of the transform $g _ { a }$ , and have $M _ { 1 }$ and $M _ { 2 }$ channels, respectively:
+
+$$
+\begin{array} { l c l } { \iota ( \tilde { y } _ { 1 } , \tilde { y } _ { 2 } , \tilde { z } \mid x , \phi _ { g } , \phi _ { h } ) } & { = } & { \displaystyle \prod _ { i } U \big ( \tilde { y } _ { 1 i } \mid y _ { 1 i } - \frac { 1 } { 2 } , y _ { 1 i } + \frac { 1 } { 2 } \big ) \cdot } \\ & & { \displaystyle \prod _ { j } U \big ( \tilde { y } _ { 2 j } \mid y _ { 2 j } - \frac { 1 } { 2 } , y _ { 2 j } + \frac { 1 } { 2 } \big ) \cdot } \\ & & { \displaystyle \prod _ { k } U \big ( \tilde { z } _ { k } \mid z _ { k } - \frac { 1 } { 2 } , z _ { k } + \frac { 1 } { 2 } \big ) } \\ & & { \mathrm { w i t h } ~ y _ { 1 } , y _ { 2 } = S \big ( g _ { a } ( x ; \phi _ { g } ) \big ) , \hat { y } = Q ( y _ { 1 } ) \oplus Q ( y _ { 2 } ) , z = h _ { a } ( \hat { y } ; \phi _ { h } ) . } \end{array}
+$$
+
+The rate term for $\hat { y } _ { 1 }$ is the same model as that of equation (4). Note that $\hat { \pmb { \sigma _ { 2 } } }$ does not contribute here, but does contribute to the model for $\hat { y } _ { 2 }$ :
+
+$$
+\begin{array} { c } { \displaystyle p _ { \tilde { y } _ { 1 } | \hat { z } } ( \tilde { y } _ { \mathbf { 1 \rho } } | \hat { z } , \pmb { \theta } _ { h } ) = \prod _ { i } \Bigl ( \mathcal { N } \bigl ( \mu _ { 1 i } , { \sigma _ { 1 i } } ^ { 2 } \bigr ) * \mathcal { U } \bigl ( - { \textstyle \frac { 1 } { 2 } } , { \textstyle \frac { 1 } { 2 } } \bigr ) \Bigr ) ( \tilde { y } _ { 1 i } ) } \\ { \mathrm { w i t h } \ \mu _ { 1 i } , \sigma _ { 1 i } = f ( c _ { i } ^ { \prime } , c _ { i } ^ { \prime \prime } ) , } \\ { c _ { i } ^ { \prime } = E ^ { \prime } ( c ^ { \prime } , i ) , } \\ { c _ { i } ^ { \prime \prime } = E ^ { \prime \prime } ( \langle { \hat { y } _ { \mathbf { 1 } } } \rangle , i ) , } \\ { c ^ { \prime } , \sigma _ { 2 } = S ( h _ { s } ( \hat { z } ; \pmb { \theta } _ { h } ) ) } \end{array}
+$$
+
+The rate term for $\hat { y } _ { 2 }$ is almost the same as Balle et al. (2018), except that noisy representations are ´ only used as the inputs to the entropy models for training, and not for the conditions of the models.
+
+$$
+\begin{array} { c c l } { { p _ { \tilde { y } _ { 2 } | \hat { z } } ( \tilde { y } _ { 2 } \mid \hat { z } , \theta _ { h } ) } } & { { = } } & { { \displaystyle { \prod _ { j } } \Big ( \mathcal { N } \big ( 0 , { \sigma _ { 2 j } } ^ { 2 } \big ) * \mathcal { U } \big ( - { \textstyle { \frac { 1 } { 2 } } } , { \textstyle { \frac { 1 } { 2 } } } \big ) \Big ) ( y _ { 2 j } ^ { \sim } ) } } \end{array}
+$$
+
+The model of $_ z$ is the same as in equation (5). For implementation, we used this hybrid structure for the top-four configurations in bit-rate descending order. We set $N$ , $M _ { 1 }$ , and $M _ { 2 }$ to 400, 192, and 408, respectively, for the top-two configurations, and to 320, 192, and 228, respectively, for the next two configurations.
+
+In addition, we measured average execution time per image, spent for encoding and decoding Kodak PhotoCD image dataset (Kodak, 1993), to clarify benefit of the hybrid model. The test was conducted under CPU environments, Intel i9-7900X. Note that we ignored time for actual entropy coding because all models with the same values of $N$ and $M$ spend the same amount of time for entropy coding. As shown in figure 7, the hybrid models clearly reduced execution time of the models. Setting $N$ and $M$ to 320 and 420, respectively, we obtained $4 6 . 8 3 \%$ of speed gain. With the higher number of parameters, 400 of $N$ and 600 of $M$ , we obtained $5 7 . 2 8 \%$ of speed gain.
+
+# 6.2 TEST RESULTS OF THE MODELS TRAINED USING DIFFERENT TYPES OF REPRESENTATIONS
+
+In this section, we provide test results of the two models, the proposed model trained using discrete representations as inputs to the synthesis transforms, $g _ { s }$ and $h _ { s }$ , and the same model but trained using noisy representations following the training process of Balle et al. (2018)’s approach. In detail, in ´ training phase of the proposed model, we used quantized representations $\hat { y }$ and $\hat { z }$ as inputs to the transforms $g _ { s }$ and $h _ { s }$ , respectively, to ensure the same conditions of training and testing phases. On the other hand, for training the compared model, representations $\tilde { y }$ and $\tilde { z }$ are used as inputs to the transforms. An additional change of the proposed model is using $\hat { y }$ , instead of $\textbf {  { y } }$ , as inputs to $h _ { a }$ , but note that this has nothing to do with the mismatches between training and testing. We used them to match inputs to $h _ { a }$ to targets of model estimation via $f$ . As shown in figure 8, the proposed model, trained using discrete representations, was $5 . 9 4 \%$ better than the model trained using noisy representations, in terms of the BD-rate of PSNR. Compared with Balle et al. (2018)’s ´ approach, the performance gains of the two models, trained using discrete representations and noisy representations, were $1 1 . 9 7 \%$ and $7 . 2 0 \%$ , respectively.
+
+![](images/ad16d6deb93f05c4fd680fcaf7c1216fd2ad4ebb4c9186baee6570d644069a16.jpg)  
+Figure 7: Comparison results of execution time between the base model and hybrid model
+
+![](images/ed9e6a4334d57f769881d715721542ff7ade100ef184f0e7ae6665296026473f.jpg)  
+Figure 8: Evaluation results of the models trained using noisy/discrete representations as input to synthesis transforms
+
+# 6.3 SAMPLES OF THE EXPERIMENTS
+
+In this section, we provide a few more supplemental test results. Figures 9, 10, and 11 show the results of the MSE optimized version of our method, whereas figures 12 and 13 show the results of the MS-SSIM optimized version. All figures include a ground truth image and the reconstructed images for our method, BPG, and Balle et al. (2018)’s approach, in the clockwise direction. ´
+
+![](images/ba2f9a16c46925bfff9d984bd4dc3089d4566cfadafebd6d0ff24fe11009f1d5.jpg)  
+Figure 9: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.2040; PSNR, 32.2063); bottom left, BPG (bpp, 0.2078; PSNR, 32.0406); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized; bpp, 0.2101; PSNR, 31.7054)
+
+![](images/a5e9b3cbad34be270b63c4cafef16802502750d29fcf6cb3990519eb62328372.jpg)  
+Figure 10: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.1236; PSNR, 32.4284); bottom left, BPG (bpp, 0.1285; PSNR, 32.0444); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized, bpp, 0.1229; PSNR, 31.0596)
+
+![](images/09f09357256830a7717ead8f505597746cafb9c2361a6ea1cc49df3dbaebc2d9.jpg)  
+Figure 11: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.1501; PSNR, 34.7103); bottom left, BPG (bpp, 0.1477; PSNR, 33.9623); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized; bpp, 0.1520; PSNR, 34.0465)
+
+![](images/f025e2841b317bd6d157749c48311e6a9e6242e4fad8e4a0925b1bb085157b90.jpg)  
+Figure 12: Sample test results. Top left, ground truth; top right, our method (MS-SSIM optimized; bpp, 0.2507; MS-SSIM, 0.9740); bottom left, BPG (bpp, 0.2441; MS-SSIM, 0.9555); bottom right, Balle et al. (2018)’s approach (MS-SSIM optimized; bpp, 0.2101; MS-SSIM, 0.9705) ´
+
+![](images/e8c3d59a92ff621eb003f73d83559d7541aba5b7293047e44bbeb1260e180a17.jpg)  
+Figure 13: Sample test results. Top left, ground truth; top right, our method (MS-SSIM optimized; bpp, 0.2269; MS-SSIM, 0.9810); bottom left, BPG (bpp, 0.2316; MS-SSIM, 0.9658); bottom right, Balle et al. (2018)’s approach (MS-SSIM optimized; bpp, 0.2291; MS-SSIM, 0.9786) ´

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+        "text": "We propose a context-adaptive entropy model for use in end-to-end optimized image compression. Our model exploits two types of contexts, bit-consuming contexts and bit-free contexts, distinguished based upon whether additional bit allocation is required. Based on these contexts, we allow the model to more accurately estimate the distribution of each latent representation with a more generalized form of the approximation models, which accordingly leads to an enhanced compression performance. Based on the experimental results, the proposed method outperforms the traditional image codecs, such as BPG and JPEG2000, as well as other previous artificial-neural-network (ANN) based approaches, in terms of the peak signal-to-noise ratio (PSNR) and multi-scale structural similarity (MS-SSIM) index. The test code is publicly available at https://github.com/JooyoungLeeETRI/CA_Entropy_Model. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Recently, artificial neural networks (ANNs) have been applied in various areas and have achieved a number of breakthroughs resulting from their superior optimization and representation learning performance. In particular, for various problems that are sufficiently straightforward that they can be solved within a short period of time by hand, a number of ANN-based studies have been conducted and significant progress has been made. With regard to image compression, however, relatively slow progress has been made owing to its complicated target problems. A number of works, focusing on the quality enhancement of reconstructed images, were proposed. For instance, certain approaches (Dong et al., 2015; Svoboda et al., 2016; Zhang et al., 2017) have been proposed to reduce artifacts caused by image compression, relying on the superior image restoration capability of an ANN. Although it is indisputable that artifact reduction is one of the most promising areas exploiting the advantages of ANNs, such approaches can be viewed as a type of post-processing, rather than image compression itself. ",
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+        "text": "Regarding ANN-based image compression, the previous methods can be divided into two types. First, as a consequence of the recent success of generative models, some image compression approaches targeting the superior perceptual quality (Agustsson et al., 2018; Santurkar et al., 2018; Rippel & Bourdev, 2017) have been proposed. The basic idea here is that learning the distribution of natural images enables a very high compression level without severe perceptual loss by allowing the generation of image components, such as textures, which do not highly affect the structure or the perceptual quality of the reconstructed images. Although the generated images are very realistic, the acceptability of the machine-created image components eventually becomes somewhat applicationdependent. Meanwhile, a few end-to-end optimized ANN-based approaches (Toderici et al., 2017; Johnston et al., 2018; Balle et al., 2017; Theis et al., 2017; Ball ´ e et al., 2018), without generative ´ models, have been proposed. In these approaches, unlike traditional codecs comprising separate tools, such as prediction, transform, and quantization, a comprehensive solution covering all functions has been sought after using end-to-end optimization. Toderici et al. (2017)’s approach exploits a small number of latent binary representations to contain the compressed information in every step, and each step increasingly stacks the additional latent representations to achieve a progressive improvement in quality of the reconstructed images. Johnston et al. (2018) improved the compression performance by enhancing operation methods of the networks developed by Toderici et al. (2017). Although Toderici et al. (2017); Johnston et al. (2018) provided novel frameworks suitable to quality control using a single trained network, the increasing number of iteration steps to obtain higher image quality can be a burden to certain applications. In contrast to the approaches developed by Toderici et al. (2017) and Johnston et al. (2018), which extract binary representations with as high an entropy as possible, Balle et al. (2017), Theis et al. (2017), and Ball ´ e et al. (2018) regard the im- ´ age compression problem as being how to retrieve discrete latent representations having as low an entropy as possible. In other words, the target problem of the former methods can be viewed as how to include as much information as possible in a fixed number of representations, whereas the latter is simply how to reduce the expected bit-rate when a sufficient number of representations are given, assuming that the low entropy corresponds to small number of bits from the entropy coder. To solve the second target problem, Balle et al. (2017), Theis et al. (2017), and Ball ´ e et al. (2018) adopt their ´ own entropy models to approximate the actual distributions of the discrete latent representations. More specifically, Balle et al. (2017) and Theis et al. (2017) proposed novel frameworks that exploit ´ the entropy models, and proved their performance capabilities by comparing the results with those of conventional codecs such as JPEG2000. Whereas Balle et al. (2017) and Theis et al. (2017) assume ´ that each representation has a fixed distribution, Balle et al. (2018) introduced an input-adaptive ´ entropy model that estimates the scale of the distribution for each representation. This idea is based on the characteristics of natural images in which the scales of the representations vary together in adjacent areas. They provided test results that outperform all previous ANN-based approaches, and reach very close to those of BPG (Bellard, 2014), which is known as a subset of HEVC (ISO/IEC 23008-2, ITU-T H.265), used for image compression. ",
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+        "img_path": "images/34dbc9c19daf0a08cec1170f1febb5b86680b2102aaa92fa61b9e61d7324816e.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of sample test results including the ground truth, our method, Balle et al. ´ (2018)’s approach, BPG, and JPEG2000. "
+        ],
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+        "text": "One of the principle elements in end-to-end optimized image compression is the trainable entropy model used for the latent representations. Because the actual distributions of latent representations are unknown, the entropy models provide the means to estimate the required bits for encoding the latent representations by approximating their distributions. When an input image $_ { \\textbf { \\em x } }$ is transformed into a latent representation $\\textbf {  { y } }$ and then uniformly quantized into $\\hat { y }$ , the simple entropy model can be represented by $p _ { \\hat { \\pmb { y } } } ( \\hat { \\pmb { y } } )$ , as described by Balle et al. (2018). When the actual marginal distribution ´ of $\\hat { y }$ is denoted as $m ( \\hat { \\pmb y } )$ , the rate estimation, calculated through cross entropy using the entropy model, $p _ { \\hat { \\pmb { y } } } ( \\hat { \\pmb { y } } )$ , can be represented as shown in equation (1), and can be decomposed into the actual entropy of $\\hat { y }$ and the additional bits owing to a mismatch between the actual distributions and their approximations. Therefore, decreasing the rate term $R$ during the training process allows the entropy model $p _ { \\hat { \\pmb { y } } } ( \\hat { \\pmb { y } } )$ to approximate $m ( \\hat { \\pmb y } )$ as closely as possible, and let the other parameters transform $x$ into $y$ properly such that the actual entropy of $\\hat { y }$ becomes small. ",
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+        "img_path": "images/027760161490479f16d851e3092bf7d0ab43888190c60ddd89cb47e5cd0dabf7.jpg",
+        "text": "$$\nR = \\mathbb { E } _ { \\hat { \\pmb { y } } \\sim m } [ - \\log _ { 2 } p _ { \\hat { \\pmb { y } } } ( \\hat { \\pmb { y } } ) ] = H ( m ) + D _ { K L } ( m \\vert \\vert p _ { \\hat { \\pmb { y } } } ) .\n$$",
+        "text_format": "latex",
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+        "text": "In terms of KL-divergence, $R$ is minimized when $p _ { \\hat { \\pmb { y } } } ( \\hat { \\pmb { y } } )$ becomes perfectly matched with the actual distribution $m ( \\hat { \\pmb y } )$ . This means that the compression performance of the methods essentially depends on the capacity of the entropy model. To enhance the capacity, we propose a new entropy model that exploits two types of contexts, bit-consuming and bit-free contexts, distinguished according to whether additional bit allocation is required. Utilizing these two contexts, we allow the model to more accurately estimate the distribution of each latent representation through the use of a more generalized form of the entropy models, and thus more effectively reduce the spatial dependencies among the adjacent latent representations. Figure 1 demonstrates a comparison of the compression results of our method to those of other previous approaches. The contributions of our work are as follows: ",
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+        "text": "• We propose a new context-adaptive entropy model framework that incorporates the two different types of contexts.   \n• We provide the test results that outperform the widely used conventional image codec BPG in terms of the PSNR and MS-SSIM.   \n• We discuss the directions of improvement in the proposed methods in terms of the model capacity and the level of the contexts. ",
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+        "text": "Note that we follow a number of notations given by Balle et al. (2018) because our approach can ´ be viewed as an extension of their work, in that we exploit the same rate-distortion (R-D) optimization framework. The rest of this paper is organized as follows. In Section 2, we introduce the key approaches of end-to-end optimized image compression and propose the context-adaptive entropy model. Section 3 demonstrates the structure of the encoder and decoder models used, and the experimental setup and results are then given in section 4. Finally, in Section 5, we discuss the current state of our work and directions for improvement. ",
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+        "type": "text",
+        "text": "2 END-TO-END OPTIMIZATION BASED ON CONTEXT-ADAPTIVE ENTROPY MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 PREVIOUS ENTROPY MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Since they were first proposed by Balle et al. (2017) and Theis et al. (2017), entropy models, which ´ approximate the distribution of discrete latent representations, have noticeably improved the image compression performance of ANN-based approaches. Balle et al. (2017) assumes the entropy mod-´ els of the latent representations as non-parametric models, whereas Theis et al. (2017) adopted a Gaussian scale mixture model composed of six weighted zero-mean Gaussian models per representation. Although they assume different forms of entropy models, they have a common feature in that both concentrate on learning the distributions of the representations without considering input adaptivity. In other words, once the entropy models are trained, the trained model parameters for the representations are fixed for any input during the test time. Balle et al. (2018), in contrast, in- ´ troduced a novel entropy model that adaptively estimates the scales of the representations based on input. They assume that the scales of the latent representations from the natural images tend to move together within an adjacent area. To reduce this redundancy, they use a small amount of additional information by which the proper scale parameters (standard deviations) of the latent representations are estimated. In addition to the scale estimation, Balle et al. (2018) have also shown that when ´ the prior probability density function (PDF) for each representation in a continuous domain is convolved with a standard uniform density function, it approximates the prior probability mass function (PMF) of the discrete latent representation, which is uniformly quantized by rounding, much more closely. For training, a uniform noise is added to each latent representation so as to fit the distribution of these noisy representations into the mentioned PMF-approximating functions. Using these approaches, Balle et al. (2018) achieved a state-of-the-art compression performance, close to that of ´ BPG. ",
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+        "type": "text",
+        "text": "2.2 SPATIAL DEPENDENCIES OF THE LATENT VARIABLES ",
+        "text_level": 1,
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+        "text": "The latent representations, when transformed through a convolutional neural network, essentially contain spatial dependencies because the same convolutional filters are shared across the spatial regions, and natural images have various factors in common within adjacent regions. Balle et al. ´ (2018) successfully captured these spatial dependencies and enhanced the compression performance by input-adaptively estimating standard deviations of the latent representations. Taking a step forward, we generalize the form of the estimated distribution by allowing, in addition to the standard deviation, the mean estimation utilizing the contexts. For instance, assuming that certain representations tend to have similar values within a spatially adjacent area, when all neighborhood representations have a value of 10, we can intuitively guess that, for the current representation, the chance of having a value equal or similar to 10 is relatively high. This simple estimation will consequently reduce the entropy. Likewise, our method utilizes the given contexts for estimating the mean, as well as the standard deviation, of each latent representation. Note that Toderici et al. (2017), Johnston et al. (2018), and Rippel & Bourdev (2017) also apply context-adaptive entropy coding by estimating the probability of each binary representation. However, these context-adaptive entropy-coding methods can be viewed as separate components, rather than one end-to-end optimization component because their probability estimation does not directly contribute to the rate term of the R-D optimization framework. Figure 2 visualizes the latent variables $\\hat { y }$ and their normalized versions of the two different approaches, one estimating only the standard deviation parameters and the other estimating both the mu and standard deviation parameters with the two types of mentioned contexts. The visualization shows that the spatial dependency can be removed more effectively when the mu is estimated along with the given contexts. ",
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+        "img_path": "images/04976c4594e77d9e31aedc6865f52df2ccb7de18e96a773e22551c6fd7a93b96.jpg",
+        "image_caption": [
+            "Figure 2: Examples of latent representations and their normalized versions for the two cases (the first two images show the results in which only the standard deviations are estimated using side information, whereas the last two images show the results in which the mu and standard deviation are estimated using our method). For a clear demonstration, the latent representations having the highest covariance between the spatially adjacent variables are extracted. Left: the latent representations of $\\hat { y }$ from the first case. Middle left: the normalized versions of $\\hat { y }$ from the first case, divided by the estimated standard deviation. Middle right: the latent variable of $\\hat { y }$ from the second case. Right: the normalized versions of $\\hat { y }$ from the second case, shifted and divided by the estimated mu and the standard deviation. "
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+        "text": "2.3 CONTEXT-ADAPTIVE ENTROPY MODEL ",
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+        "text": "The optimization problem described in this paper is similar with Balle et al. (2018), in that the input ´ $_ { \\textbf { \\em x } }$ is transformed into $\\textbf {  { y } }$ having a low entropy, and the spatial dependencies of $\\textbf {  { y } }$ are captured into $\\hat { z }$ . Therefore, we also use four fundamental parametric transform functions: an analysis transform $g _ { a } ( { \\pmb x } ; \\phi _ { g } )$ to transform $_ { \\textbf { \\em x } }$ into a latent representation $\\textbf {  { y } }$ , a synthesis transform $g _ { s } ( \\hat { y } ; \\pmb { \\theta } _ { g } )$ to reconstruct image $\\hat { \\pmb x }$ , an analysis transform $h _ { a } ( \\hat { \\pmb { y } } ; \\phi _ { h } )$ to capture the spatial redundancies of $\\hat { y }$ into a latent representation $_ { z }$ , and a synthesis transform $h _ { s } ( \\hat { z } ; \\pmb { \\theta } _ { h } )$ used to generate the contexts for the model estimation. Note that $h _ { s }$ does not estimate the standard deviations of the representations directly as in Balle et al. (2018)’s approach. In our method, instead, ´ $h _ { s }$ generates the context $c ^ { \\prime }$ , one of the two types of contexts for estimating the distribution. These two types of contexts are described in this section. ",
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+        "type": "text",
+        "text": "Balle et al. (2018) analyzed the optimization problem from the viewpoint of the variational autoen- ´ coder (Kingma & Welling (2014); Rezende et al. (2014)), and showed that the minimization of the KL-divergence is the same problem as the R-D optimization of image compression. Basically, we follow the same concept; however, for training, we use the discrete representations on the conditions instead of the noisy representations, and thus the noisy representations are only used as the inputs to the entropy models. Empirically, we found that using discrete representations on the conditions show better results, as shown in appendix 6.2. These results might come from removing the mismatches of the conditions between the training and testing, thereby enhancing the training capacity by limiting the affect of the uniform noise only to help the approximation to the probability mass functions. We use the gradient overriding method with the identity function, as in Theis et al. (2017), to deal with the discontinuities from the uniform quantization. The resulting objective function used in this paper is given in equation (2). The total loss consists of two terms representing the rates and distortions, and the coefficient $\\lambda$ controls the balance between the rate and distortion during the R-D optimization. Note that $\\lambda$ is not an optimization target, but a manually configured condition that determines which to focus on between rate and distortion: ",
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+        "img_path": "images/496a0cff77428dc0ab4a27392a31da9119bd4679522410cf9428dc829ff2ae70.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } = R + \\lambda D } \\\\ & { \\mathrm { w i t h } \\ : R = \\mathbb { E } _ { { \\pmb x } \\sim p _ { \\pmb x } } \\mathbb { E } _ { { \\pmb \\tilde { y } } , { \\boldsymbol \\tilde { z } } \\sim q } \\Big [ - \\log p _ { { \\pmb { \\tilde { y } } } | { \\hat { z } } } ( { \\pmb \\tilde { y } } \\mid { \\hat { z } } ) - \\log p _ { { \\boldsymbol \\tilde { z } } } ( { \\boldsymbol \\tilde { z } } ) \\Big ] , } \\\\ & { \\quad \\quad D = \\mathbb { E } _ { { \\pmb x } \\sim p _ { \\infty } } \\Big [ - \\log p _ { { \\pmb x } | { \\hat { y } } } ( { \\pmb x } \\mid { \\hat { y } } ) \\Big ] } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, the noisy representations of $\\tilde { y }$ and $\\tilde { z }$ follow the standard uniform distribution, the mean values of which are $\\textbf {  { y } }$ and $_ { z }$ , respectively, when $\\textbf {  { y } }$ and $_ z$ are the result of the transforms $g _ { a }$ and $h _ { a }$ , repectively. Note that the input to $h _ { a }$ is $\\hat { y }$ , which is a uniformly quantized representation of $\\textbf {  { y } }$ , rather than the noisy representation ${ \\tilde { y } } . \\ Q$ denotes the uniform quantization function, for which we simply use a rounding function: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3246172ba8ddd426e3fe077bf5e6106e76e86f9bb725aec72f563bbfeeb18316.jpg",
+        "text": "$$\n\\begin{array} { l l l } { { q ( \\tilde { y } , \\tilde { z } \\mid x , \\phi _ { g } , \\phi _ { h } ) } } & { { = } } & { { \\displaystyle \\prod _ { i } \\mathcal { U } \\big ( \\tilde { y } _ { i } \\mid y _ { i } - \\frac { 1 } { 2 } , y _ { i } + \\frac { 1 } { 2 } \\big ) \\cdot \\prod _ { j } \\mathcal { U } \\big ( \\tilde { z } _ { j } \\mid z _ { j } - \\frac { 1 } { 2 } , z _ { j } + \\frac { 1 } { 2 } \\big ) } } \\\\ { { } } & { { } } & { { \\mathrm { w i t h } ~ y = g _ { a } ( { \\bf x } ; \\phi _ { g } ) , \\hat { y } = Q ( y ) , z = h _ { a } ( \\hat { y } ; \\phi _ { h } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The rate term represents the expected bits calculated using the entropy models of $p _ { \\tilde { y } | \\hat { z } }$ and $p _ { \\tilde { z } }$ . Note that $p _ { \\tilde { y } | \\hat { z } }$ and $p _ { \\tilde { z } }$ are eventually the approximations of $p _ { \\hat { y } | \\hat { z } }$ and $p _ { \\hat { z } }$ , respectively. Equation (4) represents the entropy model for approximating the required bits for $\\hat { y }$ . The model is based on the Gaussian model, which not only has the standard deviation parameter $\\sigma _ { i }$ , but also the mu parameter $\\mu _ { i }$ . The values of $\\mu _ { i }$ and $\\sigma _ { i }$ are estimated from the two types of given contexts based on the function $f$ , the distribution estimator, in a deterministic manner. The two types of contexts, bit-consuming and bit-free contexts, for estimating the distribution of a certain representation are denoted as $ { \\boldsymbol { c } } _ { i } ^ { \\prime }$ and $ { c _ { i } ^ { \\prime \\prime } }$ . $E ^ { \\prime }$ extracts $ { \\boldsymbol { c } } _ { i } ^ { \\prime }$ from $c ^ { \\prime }$ , the result of transform $h _ { s }$ . In contrast to $\\pmb { c } _ { i } ^ { \\prime }$ , no additional bit allocation is required for $ { c _ { i } ^ { \\prime \\prime } }$ . Instead, we simply utilize the known (already entropy-coded or decoded) subset of $\\hat { y }$ , denoted as $\\langle \\hat { y } \\rangle$ . Here, $ { \\boldsymbol { c } } _ { i } ^ { \\prime \\prime }$ is extracted from $\\langle \\hat { y } \\rangle$ by the extractor $E ^ { \\prime \\prime }$ . We assume that the entropy coder and the decoder sequentially process $\\hat { y } _ { i }$ in the same specific order, such as with raster scanning, and thus $\\langle \\hat { y } \\rangle$ given to the encoder and decoder can always be identical when processing the same $\\hat { y } _ { i }$ . A formal expression of this is as follows: ",
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+        "text": "$$\n\\begin{array} { c } { { p _ { \\tilde { y } | \\hat { z } } ( \\tilde { y } \\mid \\hat { z } , \\theta _ { h } ) = { \\displaystyle \\prod _ { i } \\biggl ( \\mathcal { N } \\bigl ( \\mu _ { i } , \\sigma _ { i } ^ { 2 } \\bigr ) * \\mathcal { U } \\bigl ( - { \\textstyle \\frac { 1 } { 2 } } , { \\textstyle \\frac { 1 } { 2 } } \\bigr ) \\biggr ) \\bigl ( \\tilde { y } _ { i } \\bigr ) } } } \\\\ { { \\mathrm { w i t h } \\ \\mu _ { i } , \\sigma _ { i } = f ( c _ { i } ^ { \\prime } , c _ { i } ^ { \\prime \\prime } ) , } }  \\\\ { { c _ { i } ^ { \\prime } = E ^ { \\prime } \\bigl ( h _ { s } \\bigl ( \\hat { z } ; \\theta _ { h } \\bigr ) , i \\bigr ) , } }  \\\\ { { c _ { i } ^ { \\prime \\prime } = E ^ { \\prime \\prime } \\bigl ( \\langle \\hat { y } \\rangle , i \\bigr ) , } }  \\\\ { { \\hat { z } = Q ( z ) } } \\end{array}\n$$",
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+        "type": "text",
+        "text": "In the case of $\\hat { z }$ , a simple entropy model is used. We assumed that the model follows zero-mean Gaussian distributions which have a trainable $\\sigma$ . Note that $\\hat { z }$ is regarded as side information and it contributes a very small amount of the total bit-rate, as described by Balle et al. (2018), and thus ´ we use this simpler version of the entropy model, rather than a more complex model, for end-to-end optimization over all parameters of the proposed method: ",
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+        "text": "$$\np _ { \\tilde { z } } ( \\tilde { z } ) = \\prod _ { j } \\Bigl ( \\mathcal { N } \\bigl ( 0 , \\sigma _ { j } ^ { 2 } \\bigr ) * \\mathcal { U } \\bigl ( - \\textstyle \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\bigr ) \\Bigr ) ( \\tilde { z } _ { j } )\n$$",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6b597b00e569a802b4bbf4c715d6d688ed082f96708d4bdad36bdc133d4e00db.jpg",
+        "image_caption": [
+            "Figure 3: Encoder-decoder model of the proposed method. The left block represents the encoder side, whereas the right block represents the deocder side. The small icons between them represent the entropy-coded bitstreams. EC and ED represent entropy coding and entropy decoding, and U | Q represents the addition of uniform noise to $\\textbf {  { y } }$ or a uniform quantization of $\\textbf {  { y } }$ . Noisy representations are used only for training as inputs to the entropy models, and are illustrated with the dotted lines. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Note that actual entropy coding or decoding processes are not necessarily required for training or encoding because the rate term is not the amount of real bits, but an estimation calculated from the entropy models, as mentioned previously. We calculate the distortion term using the mean squared error (MSE)1, assuming that $p _ { \\mathbf { \\hat { \\mathbf { \\theta } } } \\mathbf { \\hat { \\mathbf { x } } } | \\hat { \\mathbf { \\theta } } }$ follows a Gaussian distribution as a widely used distortion metric. ",
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+    {
+        "type": "text",
+        "text": "3 ENCODER-DECODER MODEL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This section describes the basic structure of the proposed encoder-decoder model. On the encoder side, an input image is transformed into latent representations, quantized, and then entropy-coded using the trained entropy models. In contrast, the decoder first applies entropy decoding with the same entropy models used for the encoder, and reconstructs the image from the latent representations, as illustrated in figure 3. It is assumed that all parameters that appear in this section were already trained. The structure of the encoder-decoder model basically includes $\\scriptstyle { \\pmb { g } } _ { a }$ and $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ in charge of the transform of $_ { \\textbf { \\em x } }$ into $\\textbf {  { y } }$ and its inverse transform, respectively. The transformed $\\textbf {  { y } }$ is uniformly quantized into $\\hat { y }$ by rounding. Note that, in the case of approaches based on the entropy models, unlike traditional codecs, tuning the quantization steps is usually unnecessary because the scales of the representations are optimized together through training. The other components between $\\scriptstyle { \\pmb { g } } _ { a }$ and $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ carry out the role of entropy coding (or decoding) with the shared entropy models and underlying context preparation processes. More specifically, the entropy model estimates the distribution of each $\\hat { y } _ { i }$ individually, in which $\\mu _ { i }$ and $\\sigma _ { i }$ are estimated with the two types of given contexts, $\\pmb { c } _ { i } ^ { \\prime }$ and $ { c _ { i } ^ { \\prime \\prime } }$ . Among these contexts, $c ^ { \\prime }$ can be viewed as side information, which requires an additional bit allocation. To reduce the required bit-rate for carrying $c ^ { \\prime }$ , the latent representation $_ z$ , transformed from $\\hat { y }$ , is quantized and entropy-coded by its own entropy model, as specified in section 2.3. On the other hand, $ { \\boldsymbol { c } } _ { i } ^ { \\prime \\prime }$ is extracted from $\\langle \\hat { y } \\rangle$ , without any additional bit allocation. Note that $\\langle \\hat { y } \\rangle$ varies as the entropy coding or decoding progresses, but is always identical for processing the same $\\hat { y } _ { i }$ in both the encoder and decoder, as described in 2.3. The parameters of $h _ { s }$ and the entropy models are simply shared by both the encoder and the decoder. Note that the inputs to the entropy models during training are the noisy representations, as illustrated with the dotted line in figure 3, to allow the entropy model to approximate the probability mass functions of the discrete representations. ",
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+        "img_path": "images/7c54fa6307bcb8e20b007b9479ec8ee54178f761d5cb621d014f8c013f93bc6a.jpg",
+        "image_caption": [
+            "Figure 4: Implementation of the proposed method. We basically use the convolutional autoencoder structure, and the distribution estimator $f$ is also implemented using convolutional neural networks. The notations of the convolutional layer follow Balle et al. (2018): the number of filters ´ $\\times$ filter height $\\times$ filter width / the downscale or upscale factor, where $\\uparrow$ and $\\downarrow$ denote the up and downscaling, respectively. For up or downscaling, we use the transposed convolution. For the networks, input images are normalized into a scale between $^ { - 1 }$ and 1. "
+        ],
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 IMPLEMENTATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We use a convolutional neural networks to implement the analysis transform and the synthesis transform functions, $g _ { a } , g _ { s }$ , $h _ { a }$ , and $h _ { s }$ . The structures of the implemented networks follow the same structures of Balle et al. (2018), except that we use the exponentiation operator instead of an abso- ´ lute operator at the end of $h _ { s }$ . Based on Balle et al. (2018)’s structure, we added the components to ´ estimate the distribution of each $\\hat { y } _ { i }$ , as shown in figure 4. Herein, we represent a uniform quantization (round) as $\\because Q , \\mathrm { { } }$ entropy coding as ”EC,” and entropy decoding as ”ED.” The distribution estimator is denoted as $f$ , and is also implemented using the convolutional layers which takes channel-wise concatenated $ { \\mathbf { c } } _ { i } ^ { \\prime }$ and ${ c } _ { i } ^ { \\prime \\prime }$ as inputs and provides estimated $\\mu _ { i }$ and $\\sigma _ { i }$ as results. Note that the same $\\mathbf { c } _ { i } ^ { \\prime }$ and ${ c } _ { i } ^ { \\prime \\prime }$ are shared for all ${ \\hat { y } } _ { i } \\mathbf { s }$ located at the same spatial position. In other words, we let $E ^ { \\prime }$ extract all spatially adjacent elements from $c ^ { \\prime }$ across the channels to retrieve $ { \\boldsymbol { c } } _ { i } ^ { \\prime }$ and likewise let $E ^ { \\prime \\prime }$ extract all adjacent known elements from $\\langle \\hat { y } \\rangle$ for $ { \\boldsymbol { c } } _ { i } ^ { \\prime \\prime }$ . This could have the effect of capturing the remaining correlations among the different channels. In short, when $M$ is the total number of channels of $\\textbf {  { y } }$ , we let $f$ estimate all $M$ distributions of ${ \\hat { y } } _ { i } \\mathbf { s }$ , which are located at the same spatial position, using only a single step, thereby allowing the total number of estimations to be reduced. Furthermore, the parameters of $f$ are shared for all spatial positions of $\\hat { y }$ , and thus only one trained $f$ per $\\lambda$ is necessary to process any sized images. In the case of training, however, collecting the results from the all spatial positions to calculate the rate term becomes a significant burden, despite the simplifications mentioned above. To reduce this burden, we designate a certain number (32 and 16 for the base model and the hybrid model, respectively) of random spatial points as the representatives per training step, to calculate the rate term easily. Note that we let these random points contribute solely to the rate term, whereas the distortion is still calculated over all of the images. ",
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+        "type": "text",
+        "text": "Because $\\textbf {  { y } }$ is a three-dimensional array in our implementation, index $i$ can be represented as three indexes, $k , l ,$ and $m$ , representing the horizontal index, the vertical index, and the channel index, respectively. When the current position is given as $( k , l , m )$ , $E ^ { \\prime }$ extracts $\\pmb { c } _ { [ k - 2 . . . k + 1 ] , [ l - 3 . . . l ] , [ 1 . . . M ] } ^ { \\prime }$ as $\\mathbf { c } _ { i } ^ { \\prime }$ , and $E ^ { \\prime \\prime }$ extracts $\\left. \\hat { \\pmb y } \\right. _ { [ k - 2 . . . k + 1 ] , [ l - 3 . . . l ] , [ 1 . . . M ] }$ as $ { c _ { i } ^ { \\prime \\prime } }$ , when $\\langle \\hat { y } \\rangle$ represents the known area of $\\hat { y }$ . Note that we filled in the unknown area of $\\langle \\hat { y } \\rangle$ with zeros, to maintain the dimensions of $\\langle \\hat { y } \\rangle$ identical to those of $\\hat { y }$ . Consequently, $\\pmb { c } _ { i \\ [ 3 . . . 4 ] , 4 , [ 1 . . . M ] } ^ { \\prime \\prime }$ are always padded with zeros. To keep the dimensions of the estimation results to the inputs, the marginal areas of $c ^ { \\prime }$ and $\\langle \\hat { y } \\rangle$ are also set to zeros. Note that when training or encoding, $\\pmb { c } _ { i } ^ { \\prime \\prime }$ can be extracted using simple $4 { \\times } 4 { \\times } M$ windows and binary masks, thereby enabling parallel processing, whereas a sequential reconstruction is inevitable for decoding. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Another implementation technique used to reduce the implementation cost is combining the lightweight entropy model with the proposed model. The lightweight entropy model assumes that the representations follow a zero-mean Gaussian model with the estimated standard deviations, which is very similar with Balle et al. (2018)’s approach. We utilize this hybrid approach for the top ´ four cases, in bit-rate descending order, of the nine $\\lambda$ configurations, based on the assumption that for the higher-quality compression, the number of sparse representations having a very low spatial dependency increases, and thus a direct scale estimation provides sufficient performance for these added representations. For implementation, we separate the latent representation $\\textbf {  { y } }$ into two parts, $\\mathbf { \\pmb { y _ { 1 } } }$ and $\\mathbf { \\pmb { y } _ { 2 } }$ , and two different entropy models are applied for them. Note that the parameters of $g _ { a }$ , $g _ { s }$ , $h _ { a }$ , and $h _ { s }$ are shared, and all parameters are still trained together. The detailed structure and experimental settings are described in appendix 6.1. ",
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+    {
+        "type": "text",
+        "text": "The number of parameters $N$ and $M$ are set to 128 and 192, respectively, for the five $\\lambda$ configurations for lower bit-rates, whereas 2-3 times more parameters, described in appendix 6.1, are used for the four $\\lambda$ configurations for higher bit-rates. Tensorflow and Python were used to setup the overall network structures, and for the actual entropy coding and decoding using the estimated model parameters, we implemented an arithmetic coder and decoder, for which the source code of the ”Reference arithmetic coding” project2 was used as the base code. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL ENVIRONMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We optimized the networks using two different types of distortion terms, one with MSE and the other with MS-SSIM. For each distortion type, the average bits per pixel (BPP) and the distortion, PSNR and MS-SSIM, over the test set are measured for each of the nine $\\lambda$ configurations. Therefore, a total of 18 networks are trained and evaluated within the experimental environments, as explained below: ",
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+        "type": "text",
+        "text": "• For training, we used $2 5 6 \\times 2 5 6$ patches extracted from 32,420 randomly selected YFCC100m (Thomee et al. (2016)) images. We extracted one patch per image, and the extracted regions were randomly chosen. Each batch consists of eight images, and 1M iterations of the training steps were conducted, applying the ADAM optimizer (Kingma & Ba (2015)). We set the initial learning rate to $5 \\times 1 0 ^ { - } 5$ , and reduced the rate by half every 50,000 iterations for the last 200,000 iterations. Note that, in the case of the four $\\lambda$ configurations for high bpp, in which the hybrid entropy model is used, 1M iterations of pre-training steps were conducted using the learning rate of $1 \\times 1 0 ^ { - } 5$ . Although we previously indicated that the total loss is the sum of $R$ and $\\lambda D$ for a simple explanation, we tuned the balancing parameter $\\lambda$ in a similar way as Theis et al. (2017), as indicated in equation (6). We used the $\\lambda$ parameters ranging from 0.01 to 0.5. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "equation",
+        "img_path": "images/59d1652035d510bb088a68d52d40e38c0739064d4d72c5fe70240dfd4b801a3b.jpg",
+        "text": "$$\n\\mathcal { L } = \\frac { \\lambda } { W _ { y } \\cdot H _ { y } \\cdot 2 5 6 } R + \\frac { 1 - \\lambda } { 1 0 0 0 } D .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "• For the evaluation, we measured the average BPP and average quality of the reconstructed images in terms of the PSNR and MS-SSIM over 24 PNG images of the Kodak PhotoCD image dataset (Kodak, 1993). Note that we represent the MS-SSIM results in the form of decibels, as in Balle et al. (2018), to increase the discrimination.´ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compared the test results with other previous methods, including traditional codecs such as BPG and JPEG2000, as well as previous ANN-based approaches such as Theis et al. (2017) and Balle´ et al. (2018). Because two different quality metrics are used, the results are presented with two separate plots. As shown in figure 5, our methods outperform all other previous methods in both metrics. In particular, our models not only outperform Balle et al. (2018)’s method, which is believed ´ to be a state-of-the-art ANN-based approach, but we also obtain better results than the widely used conventional image codec, BPG. ",
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+    {
+        "type": "image",
+        "img_path": "images/f8a3b96ef70c7b5691e918165aef6fead9b832dc0412c6876f9f37b9f74769da.jpg",
+        "image_caption": [
+            "Figure 5: Rate–distortion curves of the proposed method and competitive methods. The top plot represents the PSNR values as a result of changes in bpp, whereas the bottom plot shows MS-SSIM values in the same manner. Note that MS-SSIM values are converted into decibels $( - 1 0 \\log _ { 1 0 } ( 1 -$ MS-SSIM)) for differentiating the quality levels, in the same manner as in Balle et al. (2018). ´ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "More specifically, the compression gains in terms of the BD-rate of PSNR over JPEG2000, Balle´ et al. (2018)’s approach (MSE-optimized), and BPG are $3 4 . 0 8 \\%$ , $1 1 . 9 7 \\%$ , and $6 . 8 5 \\%$ , respectively. In the case of MS-SSIM, we found wider gaps of $6 8 . 8 2 \\%$ , $1 3 . 9 3 \\%$ , and $4 9 . 6 8 \\%$ , respectively. Note that we achieved significant gains over traditional codecs in terms of MS-SSIM, although this might be because the dominant target metric of the traditional codec developments have been the PSNR. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In other words, they can be viewed as a type of MSE-optimized codec. Even when setting aside the case of MS-SSIM, our results can be viewed as one concrete evidence supporting that ANN-based image compression can outperform the existing traditional image codecs in terms of the compression performance. Supplemental image samples are provided in appendix 6.3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Based on previous ANN-based image compression approaches utilizing entropy models (Balle et al., ´ 2017; Theis et al., 2017; Balle et al., 2018), we extended the entropy model to exploit two different ´ types of contexts. These contexts allow the entropy models to more accurately estimate the distribution of the representations with a more generalized form having both mean and standard deviation parameters. Based on the evaluation results, we showed the superiority of the proposed method. The contexts we utilized are divided into two types. One is a sort of free context, containing the part of the latent variables known to both the encoder and the decoder, whereas the other is the context, which requires additional bit allocation. Because the former is a generally used context in a variety of codecs, and the latter was already verified to help compression using Balle et al. (2018)’s approach, ´ our contributions are not the contexts themselves, but can be viewed as providing a framework of entropy models utilizing these contexts. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Although the experiments showed the best results in the ANN-based image compression domain, we still have various studies to conduct to further improve the performance. One possible way is generalizing the distribution models underlying the entropy model. Although we enhanced the performance by generalizing the previous entropy models, and have achieved quite acceptable results, the Gaussian-based entropy models apparently have a limited expression power. If more elaborate models, such as the non-parametric models of Balle et al. (2018) or Oord et al. (2016), are combined ´ with the context-adaptivity proposed in this paper, they would provide better results by reducing the mismatch between the actual distributions and the approximation models. Another possible way is improving the level of the contexts. Currently, our methods only use low-level representations within very limited adjacent areas. However, if the sufficient capacity of the networks and higher-level contexts are given, a much more accurate estimation could be possible. For instance, if an entropy model understands the structures of human faces, in that they usually have two eyes, between which a symmetry exists, the entropy model could approximate the distributions more accurately when encoding the second eye of a human face by referencing the shape and position of the first given eye. As is widely known, various generative models (Goodfellow et al., 2014; Radford et al., 2016; Zhao et al., 2017) learn the distribution $p ( { \\pmb x } )$ of the images within a specific domain, such as human faces or bedrooms. In addition, various in-painting methods (Pathak et al., 2016; Yang et al., 2017; Yeh et al., 2017) learn the conditional distribution $p ( \\pmb { x } \\mid c o n t e x t )$ when the viewed areas are given as context. Although these methods have not been developed for image compression, hopefully such high-level understandings can be utilized sooner or later. Furthermore, the contexts carried using side information can also be extended to some high-level information such as segmentation maps or any other information that helps with compression. Segmentation maps, for instance, may be able to help the entropy models estimate the distribution of a representation discriminatively according to the segment class the representation belongs to. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Traditional codecs have a long development history, and a vast number of hand-crafted heuristics have been stacked thus far, not only for enhancing compression performance, but also for compromising computational complexities. Therefore, ANN-based image compression approaches may not provide satisfactory solutions as of yet, when taking their high complexity into account. However, considering its much shorter history, we believe that ANN-based image compression has much more potential and possibility in terms of future extension. Although we remain a long way from completion, we hope the proposed context-adaptive entropy model will provide an useful contribution to this area. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government (MSIT) (No. 2017-0-00072, Development of Audio/Video Coding and Light Field Media Fundamental Technologies for Ultra Realistic Teramedia). ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "image",
+        "img_path": "images/4c108c0eb986ca11778b67a98a0735890f479ebfe9ba3409b68d2a9dfb198e82.jpg",
+        "image_caption": [
+            "Figure 6: The structure of the hybrid network for higher bit-rate environments. The same notations as in the figure 4 are used. The representation $\\textbf {  { y } }$ is divided into two parts and quantized. One of the resulting parts, $\\hat { y } _ { 1 }$ , is encoded using the proposed model, whereas the other, $\\hat { y } _ { 2 }$ , is encoded using a simpler model in which only the standard deviations are estimated using side information. The detailed structure of the proposed model is illustrated in figure 4. All concatenation and split operations are performed in a channel-wise manner. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            294,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6.1 HYBRID NETWORK FOR HIGHER BIT-RATE COMPRESSIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            460,
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+            473
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We combined the lightweight entropy model with the context-adaptive entropy model to reduce the implementation costs for high-bpp configurations. The lightweight model exploits the scale (standard deviation) estimation, assuming that the PMF approximations of the quantized representations follow zero-mean Gaussian distributions convolved with a standard uniform distribution. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 illustrates the network structure of this hybrid network. The representation $\\textbf {  { y } }$ is split channel-wise into two parts, $\\mathbf { \\pmb { y _ { 1 } } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { 2 } }$ , which have $M _ { 1 }$ and $M _ { 2 }$ channels, respectively, and is then quantized. Here, $\\hat { y } _ { 1 }$ is entropy coded using the proposed entropy model, whereas $\\hat { y } _ { 2 }$ is coded with the lightweight entropy model. The standard deviations of $\\hat { y } _ { 2 }$ are estimated using $h _ { a }$ and $h _ { s }$ . Unlike the context-adaptive entropy model, which uses the results of $h _ { a }$ $( \\hat { c ^ { \\prime } } )$ as the input source to the estimator $f$ , the lightweight entropy model retrieves the estimated standard deviations from $h _ { a }$ directly. Note that $h _ { a }$ takes the concatenated $\\hat { y } _ { 1 }$ and $\\hat { y } _ { 2 }$ as input, and $h _ { s }$ generates $\\hat { c ^ { \\prime } }$ as well as $\\pmb { \\sigma } _ { 2 }$ , at the same time. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The total loss function also consists of the rate and distortion terms, although the rate is divided into three parts, each of which is for $\\hat { y } _ { 1 } , \\hat { y } _ { 2 }$ , and $\\hat { z }$ , respectively. The distortion term is the same as before, but note that $\\hat { y }$ is the channel-wise concatenated representation of $\\hat { y } _ { 1 }$ and $\\hat { y } _ { 2 }$ : ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7c6b01fe37e584d34977be9877ff778298fb961d2dd6a7c531c61b1106924f0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } = R + \\lambda D } \\\\ & { \\mathrm { w i t h } \\ R = \\mathbb { E } _ { { \\mathbf { x } } \\sim p _ { \\mathbf { x } } } \\mathbb { E } _ { \\mathcal { \\tilde { I } } _ { 1 } , \\tilde { y _ { 2 } } , \\tilde { z } \\sim q } \\Big [ - \\log p _ { \\tilde { y _ { 1 } } \\mid \\hat { z } } ( \\tilde { y _ { 1 } } \\mid \\hat { z } ) - \\log p _ { \\tilde { y _ { 2 } } \\mid \\hat { z } } ( \\tilde { y _ { 2 } } \\mid \\hat { z } ) - \\log p _ { \\tilde { z } } ( \\tilde { z } ) \\Big ] , } \\\\ & { \\qquad D = \\mathbb { E } _ { \\mathbf { x } \\sim p _ { \\mathbf { x } } } \\Big [ - \\log p _ { \\mathbf { x } \\mid \\hat { y } } ( \\boldsymbol { x } \\mid \\hat { y } ) \\big ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            770,
+            781,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here, the noisy representations of ${ \\tilde { y } } _ { 1 } , { \\tilde { y } } _ { 2 }$ , and $\\tilde { z }$ follow a standard uniform distribution, the mean values of which are ${ \\bf { y 1 } } , ~ { \\bf { y 2 } }$ , and $_ { z }$ , respectively. In addition, $\\mathbf { \\pmb { y _ { 1 } } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { 2 } }$ are channel-wise split representations from $\\textbf {  { y } }$ , the results of the transform $g _ { a }$ , and have $M _ { 1 }$ and $M _ { 2 }$ channels, respectively: ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/29315c8c053b300ccd04551c21656c159a0073c0b4b2d3450ce58815de2642e0.jpg",
+        "text": "$$\n\\begin{array} { l c l } { \\iota ( \\tilde { y } _ { 1 } , \\tilde { y } _ { 2 } , \\tilde { z } \\mid x , \\phi _ { g } , \\phi _ { h } ) } & { = } & { \\displaystyle \\prod _ { i } U \\big ( \\tilde { y } _ { 1 i } \\mid y _ { 1 i } - \\frac { 1 } { 2 } , y _ { 1 i } + \\frac { 1 } { 2 } \\big ) \\cdot } \\\\ & & { \\displaystyle \\prod _ { j } U \\big ( \\tilde { y } _ { 2 j } \\mid y _ { 2 j } - \\frac { 1 } { 2 } , y _ { 2 j } + \\frac { 1 } { 2 } \\big ) \\cdot } \\\\ & & { \\displaystyle \\prod _ { k } U \\big ( \\tilde { z } _ { k } \\mid z _ { k } - \\frac { 1 } { 2 } , z _ { k } + \\frac { 1 } { 2 } \\big ) } \\\\ & & { \\mathrm { w i t h } ~ y _ { 1 } , y _ { 2 } = S \\big ( g _ { a } ( x ; \\phi _ { g } ) \\big ) , \\hat { y } = Q ( y _ { 1 } ) \\oplus Q ( y _ { 2 } ) , z = h _ { a } ( \\hat { y } ; \\phi _ { h } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The rate term for $\\hat { y } _ { 1 }$ is the same model as that of equation (4). Note that $\\hat { \\pmb { \\sigma _ { 2 } } }$ does not contribute here, but does contribute to the model for $\\hat { y } _ { 2 }$ : ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e6b632c7ba3f439c65684e56ff5c855325172ceaa5103beb14a8422b4fbf758.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle p _ { \\tilde { y } _ { 1 } | \\hat { z } } ( \\tilde { y } _ { \\mathbf { 1 \\rho } } | \\hat { z } , \\pmb { \\theta } _ { h } ) = \\prod _ { i } \\Bigl ( \\mathcal { N } \\bigl ( \\mu _ { 1 i } , { \\sigma _ { 1 i } } ^ { 2 } \\bigr ) * \\mathcal { U } \\bigl ( - { \\textstyle \\frac { 1 } { 2 } } , { \\textstyle \\frac { 1 } { 2 } } \\bigr ) \\Bigr ) ( \\tilde { y } _ { 1 i } ) } \\\\ { \\mathrm { w i t h } \\ \\mu _ { 1 i } , \\sigma _ { 1 i } = f ( c _ { i } ^ { \\prime } , c _ { i } ^ { \\prime \\prime } ) , } \\\\ { c _ { i } ^ { \\prime } = E ^ { \\prime } ( c ^ { \\prime } , i ) , } \\\\ { c _ { i } ^ { \\prime \\prime } = E ^ { \\prime \\prime } ( \\langle { \\hat { y } _ { \\mathbf { 1 } } } \\rangle , i ) , } \\\\ { c ^ { \\prime } , \\sigma _ { 2 } = S ( h _ { s } ( \\hat { z } ; \\pmb { \\theta } _ { h } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
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+            419
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The rate term for $\\hat { y } _ { 2 }$ is almost the same as Balle et al. (2018), except that noisy representations are ´ only used as the inputs to the entropy models for training, and not for the conditions of the models. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/99ae7a4a2be0f67ee44d83b39aff0b9d0c2acb26d9d39dd59cd2c8eef3201e1f.jpg",
+        "text": "$$\n\\begin{array} { c c l } { { p _ { \\tilde { y } _ { 2 } | \\hat { z } } ( \\tilde { y } _ { 2 } \\mid \\hat { z } , \\theta _ { h } ) } } & { { = } } & { { \\displaystyle { \\prod _ { j } } \\Big ( \\mathcal { N } \\big ( 0 , { \\sigma _ { 2 j } } ^ { 2 } \\big ) * \\mathcal { U } \\big ( - { \\textstyle { \\frac { 1 } { 2 } } } , { \\textstyle { \\frac { 1 } { 2 } } } \\big ) \\Big ) ( y _ { 2 j } ^ { \\sim } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            478,
+            697,
+            515
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The model of $_ z$ is the same as in equation (5). For implementation, we used this hybrid structure for the top-four configurations in bit-rate descending order. We set $N$ , $M _ { 1 }$ , and $M _ { 2 }$ to 400, 192, and 408, respectively, for the top-two configurations, and to 320, 192, and 228, respectively, for the next two configurations. ",
+        "bbox": [
+            173,
+            525,
+            825,
+            582
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In addition, we measured average execution time per image, spent for encoding and decoding Kodak PhotoCD image dataset (Kodak, 1993), to clarify benefit of the hybrid model. The test was conducted under CPU environments, Intel i9-7900X. Note that we ignored time for actual entropy coding because all models with the same values of $N$ and $M$ spend the same amount of time for entropy coding. As shown in figure 7, the hybrid models clearly reduced execution time of the models. Setting $N$ and $M$ to 320 and 420, respectively, we obtained $4 6 . 8 3 \\%$ of speed gain. With the higher number of parameters, 400 of $N$ and 600 of $M$ , we obtained $5 7 . 2 8 \\%$ of speed gain. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "6.2 TEST RESULTS OF THE MODELS TRAINED USING DIFFERENT TYPES OF REPRESENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            704,
+            697,
+            731
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide test results of the two models, the proposed model trained using discrete representations as inputs to the synthesis transforms, $g _ { s }$ and $h _ { s }$ , and the same model but trained using noisy representations following the training process of Balle et al. (2018)’s approach. In detail, in ´ training phase of the proposed model, we used quantized representations $\\hat { y }$ and $\\hat { z }$ as inputs to the transforms $g _ { s }$ and $h _ { s }$ , respectively, to ensure the same conditions of training and testing phases. On the other hand, for training the compared model, representations $\\tilde { y }$ and $\\tilde { z }$ are used as inputs to the transforms. An additional change of the proposed model is using $\\hat { y }$ , instead of $\\textbf {  { y } }$ , as inputs to $h _ { a }$ , but note that this has nothing to do with the mismatches between training and testing. We used them to match inputs to $h _ { a }$ to targets of model estimation via $f$ . As shown in figure 8, the proposed model, trained using discrete representations, was $5 . 9 4 \\%$ better than the model trained using noisy representations, in terms of the BD-rate of PSNR. Compared with Balle et al. (2018)’s ´ approach, the performance gains of the two models, trained using discrete representations and noisy representations, were $1 1 . 9 7 \\%$ and $7 . 2 0 \\%$ , respectively. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad16d6deb93f05c4fd680fcaf7c1216fd2ad4ebb4c9186baee6570d644069a16.jpg",
+        "image_caption": [
+            "Figure 7: Comparison results of execution time between the base model and hybrid model "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed9e6a4334d57f769881d715721542ff7ade100ef184f0e7ae6665296026473f.jpg",
+        "image_caption": [
+            "Figure 8: Evaluation results of the models trained using noisy/discrete representations as input to synthesis transforms "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "6.3 SAMPLES OF THE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
+            437,
+            779
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide a few more supplemental test results. Figures 9, 10, and 11 show the results of the MSE optimized version of our method, whereas figures 12 and 13 show the results of the MS-SSIM optimized version. All figures include a ground truth image and the reconstructed images for our method, BPG, and Balle et al. (2018)’s approach, in the clockwise direction. ´ ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba2f9a16c46925bfff9d984bd4dc3089d4566cfadafebd6d0ff24fe11009f1d5.jpg",
+        "image_caption": [
+            "Figure 9: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.2040; PSNR, 32.2063); bottom left, BPG (bpp, 0.2078; PSNR, 32.0406); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized; bpp, 0.2101; PSNR, 31.7054) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a5e9b3cbad34be270b63c4cafef16802502750d29fcf6cb3990519eb62328372.jpg",
+        "image_caption": [
+            "Figure 10: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.1236; PSNR, 32.4284); bottom left, BPG (bpp, 0.1285; PSNR, 32.0444); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized, bpp, 0.1229; PSNR, 31.0596) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            93,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/09f09357256830a7717ead8f505597746cafb9c2361a6ea1cc49df3dbaebc2d9.jpg",
+        "image_caption": [
+            "Figure 11: Sample test results. Top left, ground truth; top right, our method (MSE optimized; bpp, 0.1501; PSNR, 34.7103); bottom left, BPG (bpp, 0.1477; PSNR, 33.9623); bottom right, Balle et al. ´ (2018)’s approach (MSE optimized; bpp, 0.1520; PSNR, 34.0465) "
+        ],
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+        "image_caption": [
+            "Figure 12: Sample test results. Top left, ground truth; top right, our method (MS-SSIM optimized; bpp, 0.2507; MS-SSIM, 0.9740); bottom left, BPG (bpp, 0.2441; MS-SSIM, 0.9555); bottom right, Balle et al. (2018)’s approach (MS-SSIM optimized; bpp, 0.2101; MS-SSIM, 0.9705) ´ "
+        ],
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+        "image_caption": [
+            "Figure 13: Sample test results. Top left, ground truth; top right, our method (MS-SSIM optimized; bpp, 0.2269; MS-SSIM, 0.9810); bottom left, BPG (bpp, 0.2316; MS-SSIM, 0.9658); bottom right, Balle et al. (2018)’s approach (MS-SSIM optimized; bpp, 0.2291; MS-SSIM, 0.9786) ´ "
+        ],
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parse/train/NjF772F4ZZR/NjF772F4ZZR.md

@@ -0,0 +1,398 @@
+# LEARNING THE PARETO FRONT WITH HYPERNETWORKS
+
+Aviv Navon∗ Bar-Ilan University, Israel aviv.navon@biu.ac.il
+
+Aviv Shamsian∗ Bar-Ilan University, Israel aviv.shamsian@biu.ac.il
+
+Ethan Fetaya† Bar-Ilan University, Israel ethan.fetaya@biu.ac.il
+
+Gal Chechik†   
+Bar-Ilan University, Israel   
+NVIDIA, Israel   
+gal.chechik@biu.ac.il
+
+# ABSTRACT
+
+Multi-objective optimization (MOO) problems are prevalent in machine learning. These problems have a set of optimal solutions, called the Pareto front, where each point on the front represents a different trade-off between possibly conflicting objectives. Recent MOO methods can target a specific desired ray in loss space however, most approaches still face two grave limitations: (i) A separate model has to be trained for each point on the front; and (ii) The exact trade-off must be known before the optimization process. Here, we tackle the problem of learning the entire Pareto front, with the capability of selecting a desired operating point on the front after training. We call this new setup Pareto-Front Learning (PFL).
+
+We describe an approach to PFL implemented using HyperNetworks, which we term Pareto HyperNetworks (PHNs). PHN learns the entire Pareto front simultaneously using a single hypernetwork, which receives as input a desired preference vector and returns a Pareto-optimal model whose loss vector is in the desired ray. The unified model is runtime efficient compared to training multiple models and generalizes to new operating points not used during training. We evaluate our method on a wide set of problems, from multi-task regression and classification to fairness. PHNs learn the entire Pareto front at roughly the same time as learning a single point on the front and at the same time reach a better solution set. PFL opens the door to new applications where models are selected based on preferences that are only available at run time.
+
+# 1 INTRODUCTION
+
+Multi-objective optimization (MOO) aims to optimize several possibly conflicting objectives. MOO is abundant in machine learning problems, from multi-task learning (MTL), where the goal is to learn several tasks simultaneously, to constrained problems. In such problems, one aims to learn a single task while finding solutions that satisfy properties like fairness or privacy. It is common to optimize the main task while adding loss terms to encourage the learned model to obtain these properties.
+
+MOO problems have a set of optimal solutions, the Pareto front, each reflecting a different trade-off between objectives. Points on the Pareto front can be viewed as an intersection of the front with a specific direction in loss space (a ray, Figure 1). We refer to this direction as a preference vector, as it represents a single trade-off between objectives. When a direction is known in advance, it is possible to obtain the corresponding solution on the front (Mahapatra & Rajan, 2020).
+
+However, in many cases, we are interested in more than one predefined direction, either because the trade-off is not known before training, or because there are many possible trade-offs of interest. For example, network routing optimization aims to maximize bandwidth, minimize latency, and obey fairness. However, the cost and trade-off vary from one application running in the network to another, or even continuously change in time. The challenge remains to design a model that can be applied at inference time to any given preference direction, even ones not seen during training. We call this problem Pareto front learning (PFL).
+
+![](images/cb897b1cbdcf566c5c6ce00658fef5b296bc2700b5ae6699089c44bda6cf4a50.jpg)  
+Figure 1: Illustrative example using the popular task of Fonseca (1995): demonstrating the relation between Pareto front, preference rays, and solutions. Pareto front (black solid line) for a 2D loss space and several rays (colored dashed lines) which represent various possible preferences. Left: A single PHN-EPO model learns the entire Pareto front, mapping any given preference ray to its corresponding solution on the front. The data and task are detailed in Section 5.1.
+
+Although several recent studies (Yang et al., 2019; Parisi et al., 2016) suggested to learn the trade-off curve in MOO problems, there is no existing scalable and general-purpose MOO approach that provides Pareto-optimal solutions for numerous preferences in objective space. Classical approaches, like genetic algorithms, do not scale to modern high-dimensional problems. It is possible in principle to run a single-direction optimization multiple times, each for a different preference, but this approach faces two major drawbacks: (i) Scalability – the number of models to be trained to cover the objective space grows exponentially with the number of objectives; and (ii) Flexibility – the decision maker cannot switch freely between preferences unless all models are trained and stored in advance.
+
+Here we put forward a new view of PFL as a problem of learning a conditional model, where the conditioning is over the preference direction. During training, a single unified model is trained to produce Pareto optimal solutions while satisfying the given preferences. During inference, the model covers the Pareto front by varying the input preference vector. We further describe an architecture that implements this idea using HyperNetworks. Specifically, we train a hypernetwork, termed Pareto Hypernetwork (PHN), that given a preference vector as an input, produces a deep network model tuned for that objective preference. Training is applied to preferences sampled from the $m -$ dimensional simplex where $m$ represents the number of objectives.
+
+We evaluate PHN on a wide set of problems, from multi-class classification, through fairness and image segmentation to multi-task regression. We find that PHN can achieve superior overall solutions while being $1 0 \sim 5 0$ times faster (see Figure 5). PHN addresses both scalability and flexibility. Training a unified model allows using any objective preference at inference time. Finally, as PHN generates a continuous parametrization of the entire Pareto front, it could open new possibilities to analyze Pareto optimal solutions in large-scale neural networks.
+
+Our paper has the following contributions: (1) We define the Pareto-front learning problem – learn a model that at inference time can operate on any given preference vector, providing a Pareto-optimal solution for that specified objective trade-off. (2) We describe Pareto Hypernetworks (PHN), a unified architecture based on hypernetworks that addresses PFL and shows it can be effectively trained. (3) Empirical evaluations on various tasks and datasets demonstrate the ability of PHNs to generate better objective space coverage compared to multiple baseline models, with significant improvement in training time.
+
+# 2 MULTI-OBJECTIVE OPTIMIZATION
+
+We start by formally defining the MOO problem. An MOO is defined by $m$ losses $\ell _ { i } : \mathbb { R } ^ { d } \to \mathbb { R } _ { + }$ $i = 1 , \ldots , m$ , or in vector form $\ell : \mathbb { R } ^ { \bar { d } }  \mathbb { R } _ { + } ^ { m }$ . We define a partial ordering on the loss space $\mathbb { R } _ { + } ^ { m }$ by setting $\ell ( \theta _ { 1 } ) \preceq \ell ( \theta _ { 2 } )$ if for all $i \in [ m ]$ , $\ell _ { i } ( \theta _ { 1 } ) \leq \ell _ { i } ( \theta _ { 2 } )$ . We define $\ell ( \theta _ { 1 } ) \prec \ell ( \theta _ { 2 } )$ if $\ell ( \theta _ { 1 } ) \preceq \ell ( \theta _ { 2 } )$ and for some $i \in [ m ] , \ell _ { i } ( \theta _ { 1 } ) < \ell _ { i } ( \theta _ { 2 } )$ . We say that a point $\theta _ { 1 } \in \mathbb { R } ^ { d }$ dominates $\theta _ { 2 } \in \mathbb { R } ^ { d }$ if $\ell ( \theta _ { 1 } ) \prec \ell ( \theta _ { 2 } )$ .
+
+If a point $\theta _ { 1 }$ dominates $\theta _ { 2 }$ , then $\theta _ { 1 }$ is clearly preferable, because it improves some objectives and is not worse on any other objective. Otherwise, solutions present a certain trade-off and selecting one specific solution requires additional information about the user preferences. A point that is not dominated by any other point is called Pareto optimal. The set of all Pareto optimal points is called the Pareto front. Since many modern machine learning models, e.g., deep neural networks, use non-convex optimization, one cannot expect global optimality. We call a point local Pareto optimal if it is Pareto optimal in some open neighborhood of the point.
+
+The most straightforward approach to MOO is linear scalarization (LS), where one defines a new single loss $\begin{array} { r } { \ell _ { r } ( \mathbf { \bar { \boldsymbol { \theta } } } ) = \sum _ { i } r _ { i } \ell _ { i } ( \boldsymbol { \theta } ) } \end{array}$ given a vector $\pmb { r } \in \mathbb { R } _ { + } ^ { m }$ of weights. One can then apply standard, single-objective optimization algorithms. LS has two major limitations. First, it can only reach the convex part of the Pareto front (Boyd et al., 2004, Chapter 4.7), as shown empirically in Figure 1. Second, if one wishes to target a specific ray in loss space, specified by a preference vector, it is not clear which linear weights lead to that desired Pareto optimal point.
+
+In the context of the current work, we highlight two properties that an MOO optimization procedure should possess: scale to many objectives and control which solution on the Pareto front is obtained. Lin et al. (2019) described Pareto multi-task learning (PMTL), an algorithm that splits the loss space into separate cones based on the selected reference rays, and returns a solution per cone using a constrained version of Fliege & Svaiter (2000). This approach allows the user to target several points on the Pareto front; However, it scales poorly with the number of cones and does not converge to the exact desired ray on the Pareto front.
+
+Convergence to the desired ray in loss space can be achieved using Exact Pareto Optimal (EPO) (Mahapatra & Rajan, 2020). To find the intersection of the Pareto front with a given preference ray $\mathbfit { \Delta } \mathbf { r }$ EPO balances two goals: Finding a descent direction towards the Pareto front and approaching the desired ray. EPO searches for a point in the convex hull of the gradients, known by Désidéri (2012) to include descent directions, that has a maximal angle with a vector $d _ { b a l }$ which pulls the point to the desired ray. EPO combines gradient descent and controlled ascent enabling it to reach an exact Pareto optimal solution if one exists, or the closest Pareto optimal solution.
+
+# 3 RELATWED WORK
+
+Multitask learning. In multitask learning (MTL) we simultaneously solve several learning problems while sharing information among tasks (Zhang & Yang, 2017; Ruder, 2017). In some cases, MTL-based models outperform their single task counterparts in terms of per-task performance and computational efficiency (Standley et al., 2019). MTL approaches map the loss vector into a single loss term using a fixed or dynamic weighting scheme. The most frequently used approach is Linear Scalarization (LS), in which each loss term’s weight is chosen apriori. A proper set of weights is commonly selected using grid search. Unfortunately, such an approach scales poorly with the number of tasks. Recently, MTL methods propose dynamically balance the loss terms using gradient magnitude (Chen et al., 2018), the rate of change in losses (Liu et al., 2019), task uncertainty (Kendall et al., 2018), or learning non-linear loss combinations by implicit differentiation (Navon et al., 2020). However, those methods seek a balanced solution and are not suitable for modeling task trade-offs.
+
+Multi-objective optimization. The goal of Multi-Objective Optimization (MOO) is to find Pareto optimal solutions corresponding to different trade-offs between objectives (Ehrgott, 2005). MOO has a wide variety of applications in machine learning, spanning Reinforcement Learning (Van Moffaert & Nowé, 2014; Pirotta & Restelli, 2016; Parisi et al., 2014; Pirotta et al., 2015; Parisi et al., 2016; Yang et al., 2019), neural architecture search (Lu et al., 2019; Hsu et al., 2018), fairness (Martínez et al., 2020; Liu & Vicente, 2020), and Bayesian optimization (Shah & Ghahramani, 2016; Hernández-Lobato et al., 2016). Genetic algorithms (GA) are a popular approach for small-scale multi-objective optimization problems. GAs are designed to maintain a set of solutions during optimization. Therefore, they can be extended in natural ways to maintain solutions for different rays. In this area, leading approaches include NSGA-III (Deb & Jain, 2013) and MOEA/D (Zhang & Li,
+
+2007). However, these gradient-free methods scale poorly with the number of parameters and are not suitable for training large-scale neural networks.
+
+Sener & Koltun (2018) proposed a gradient-based MOO algorithm for MTL, based on MDGA (Désidéri, 2012), suitable for training large-scale neural networks. Other recent works include (Lin et al., 2019; Mahapatra & Rajan, 2020) detailed in Section 2. Another recent approach that aims at a more complete view of the Pareto front is Ma et al. (2020). They extend a given Pareto optimal solution in its local neighborhood. In a concurrent work, Lin et al. (2020) extends Lin et al. (2019) for approximating the entire Pareto front. They train a single hypernetwork by constantly changing the reference directions in the PMTL objective. The proposed method is conceptually similar to our approach, but since it builds on PMTL, it may not produce an exact mapping between the preference and the corresponding solution. Similarly, Dosovitskiy & Djolonga (2019) proposed learning a single model conditioning on the objective weight vector. The method uses feature-wise linear modulation Perez et al. (2017), and dynamically weighted LS loss criterion.
+
+Hypernetworks. Ha et al. (2017) introduced the idea of hypernetworks (HNs) inspired by the genotype-phenotype relation in cellular biology. HN presents an approach of using one network (hypernetwork) to generate weights for a second network (target network). In recent years, HN are widely used in various domains such as computer vision (Klocek et al., 2019; Ha et al., 2017), language modeling (Suarez, 2017), sequence decoding (Nachmani & Wolf, 2019), continual learning (Oswald et al., 2020), federated learning (Shamsian et al., 2021) and hyperparameter optimization (Mackay et al., 2019; Lorraine & Duvenaud, 2018). HN dynamically generates models conditioned on a given input, obtaining a set of customized models using a single learnable network.
+
+# 4 PARETO HYPERNETWORKS
+
+<table><tr><td>Algorithm1PHN</td></tr><tr><td>while not converged do r ~ Dir(α)</td></tr><tr><td>0(Φ,r)=h(r;) Sample mini-batch (x1,y1),.,(xB,yB)</td></tr><tr><td>if LS then</td></tr><tr><td>g←B∑i,jriVΦli(xj,yj,0(Φ,r))</td></tr><tr><td>if EPO then</td></tr><tr><td>β=EPO(0(Φ,r),l,r)</td></tr><tr><td>g←B∑i,jβi√li(xj,yj,0(Φ,r))</td></tr><tr><td></td></tr><tr><td>Φ←-ngΦ</td></tr><tr><td></td></tr><tr><td>return $</td></tr></table>
+
+![](images/c9b62b1546101cd36883d200a2a1501c1d2b55c7c15ec0201d0ecb509952d38e.jpg)  
+Figure 2: A PHN $h ( \cdot ; \phi )$ receives an input preference ray $\mathbfit { \Delta } \mathbf { r }$ and outputs the corresponding model weights $\theta _ { r }$ .
+
+We start by describing the basic of hypernetworks architecture (Figure 2). Hypernetworks are deep models that generate the weights of another deep model as their output, named the target network. The weights produced by the hypernetwork depend on its input, and one can view the training of a hypernetwork as training a family of target networks simultaneously, conditioned on the input. Let $\bar { h } ( \cdot ; \phi )$ denote the hypernetwork with parameters $\phi \in \mathbb { R } ^ { n }$ , and let $f$ denote the target network whose parameters we denote by $\theta$ . In our implementation, we parametrize $h$ using a feed-forward network with multiple heads. Each head outputs a different weight tensor of the target network. More precisely, the input is first mapped to a higher dimensional space using an MLP, to construct shared features. These features are passed through fully connected layers to create a weight matrix per layer in the target network.
+
+Consider an $m$ −dimensional MOO problem with an objective vector $\ell$ . Let $\pmb { r } = ( r _ { 1 } , . . . , r _ { m } ) \in S _ { m }$ denote a preference vector representing a desired trade-off over the objectives. Here $S _ { m } = \{ r \in$ $\begin{array} { r } { \mathbb { R } ^ { m } \ | \ \sum _ { j } r _ { j } \ = 1 , r _ { j } \ \ge \ 0 \} } \end{array}$ is the $m -$ dimensional simplex. Our PHN model takes as input the preference vector $\pmb { r }$ to produce the weights $\theta _ { r } = \theta ( \phi , r ) = h ( r ; \phi )$ of the target network. The target network is then applied to the training data in a usual manner. Throughout the training, we randomly sample $\mathbfit { \Delta } \mathbf { r }$ at each iteration from a Dirichlet distribution with parameter $\alpha \in \mathbb { R } ^ { m }$ . At inference time, a given preference direction $\mathbfit { \Delta } \mathbf { r }$ , is provided as input to the $\mathrm { H N } h$ , which outputs the corresponding Pareto optimal solution, $\theta _ { r }$ , namely, a set of model weights for specific given $\pmb { r }$ . We train $\theta$ to be Pareto optimal and corresponds to the preference vector. For that purpose, we explore two alternatives.
+
+PHN-LS: The first, named PHN-LS, uses linear scalarization with the preference vector $\mathbfit { \Delta } \mathbf { r }$ as loss weights, i.e the loss for input $\pmb { r }$ is $\textstyle \sum _ { i } r _ { i } \ell _ { i }$ . While linear scalarization has some theoretical limitations, it is a straightforward and fast approach that tends to work well in practice. One can think of the PHN-LS approach as optimizing the loss $\begin{array} { r } { \ell _ { L S } ( \phi ) = \mathbb { E } _ { r , x , y } [ \sum _ { i } r _ { i } \ell _ { i } ( \bar { x , } y , \theta ( \phi , \pmb { r } ) ) ] } \end{array}$ where we sample a mini-batch and a weighting $\mathbfit { \Delta } \mathbf { r }$ . We note, as a direct result of the implicit function theorem, the following:
+
+Proposition 1. Let $\theta ( \hat { \pmb r } )$ be a local Pareto optimal point corresponding to weights $\hat { r }$ , i.e. $\begin{array} { r } { \sum _ { i j } \overline { { \hat { r } _ { i } } } \nabla \ell _ { i } ( x _ { j } , y _ { j } , \theta ) = \mathrm { 0 } } \end{array}$ and assume the Hessian of $\textstyle \sum _ { i } { \bar { r } } _ { i } \ell _ { i }$ has full rank at $\theta ( \hat { \pmb r } )$ . There exists $a$ neighborhood $U$ of $\scriptstyle { \hat { r } }$ and a smooth mapping $\theta ( r )$ such that $\begin{array} { r } { \forall \pmb { r } \in \pmb { U } : \sum _ { i j } r _ { i } \nabla \ell _ { i } ( x _ { j } , y _ { j } , \theta ( \pmb { r } ) ) = 0 . } \end{array}$
+
+This shows that under the full rank assumption the connection between the LS term $\pmb { r }$ and the optimal model weights $\theta ( r )$ is smooth and can be learned using universal approximation (Cybenko, 1989). We also observe that if for all $\mathbfit { \Delta } \mathbf { r }$ , $\theta ( \phi , \pmb { r } )$ is a local Pareto optimal point corresponding to $\mathbfit { \Delta } \mathbf { r }$ , i.e., $\begin{array} { r } { \sum _ { i j } r _ { i } \nabla \ell _ { i } ( x _ { j } , y _ { j } , \theta ( \pmb { r } ) ) = 0 } \end{array}$ , then $\dot { \nabla } _ { \phi } \ell _ { L S } ( \phi ) = 0$ and this is a stationary point for $\ell _ { L S }$ . However, there could exists stationary points and even local minima such that $\nabla _ { \phi } \ell _ { L S } ( \phi ) = 0$ but in general $\begin{array} { r } { \sum _ { i j } r _ { i } \nabla \ell _ { i } ( x _ { j } , y _ { j } , \theta ( \pmb { r } ) ) \neq \mathbf { \bar { 0 } } } \end{array}$ and the output of our hypernetwork is not a local Pareto optimal point. While these bad local minima exist, as in standard deep neural networks, we empirically find that PHNs avoid these solutions and achieve good results.
+
+PHN-EPO: Our second approach treats the preference $\mathbfit { \Delta } \mathbf { r }$ as a ray in loss space and trains $\theta ( \phi , \pmb { r } )$ to reach a Pareto optimal point on the inverse ray $\scriptstyle \mathbf { \hat { r } } ^ { - 1 }$ , namely, $r _ { 1 } \cdot \ell _ { 1 } = . . . = r _ { m } \cdot \ell _ { m }$ . This is achieved by using the EPO update direction for each output of the Pareto hypernetwork. The recent EPO algorithm can guarantee convergence to a local Pareto optimal point on a specified ray $\pmb { r }$ , addressing the theoretical limitations of LS. Since the EPO descent direction is a convex combination of the gradients, we use the EPO algorithm to compute this adaptive weighting and backpropagate the reweighted losses. See pseudo-code in Alg. 1.
+
+The EPO however, unlike LS, cannot be seen as optimizing a certain objective. Therefore, we cannot consider the PHN-EPO optimization, which uses the EPO descent direction, as optimizing a specific objective. We also note that since the EPO needs to solve a linear programming step at each iteration, the runtime of EPO and PHN-EPO are longer than the LS and PHN-LS counterparts (see Figure 5).
+
+PHN Scalability: Using a naive implementation of HNs, their size grows linearly with the target network’s size; this may restrict the scalability of PHNs to large target networks. Fortunately, our approach can be easily extended to handle large target networks. First, one can learn only part of the target network parameters using the HN (Luan et al., 2018), or learn preference-specific normalization layers (Huang & Belongie, 2017; Perez et al., 2017) as in Dosovitskiy & Djolonga (2019). Furthermore, one can apply “chunking": generate different parts of the target network, conditioned on a trainable chunk descriptor input (Oswald et al., 2020). Similarly, one can reduce the dimension of the shared preference representation to achieve a similar effect (see Appendix B.5). The experiments in the main paper used the vanilla version of PHN.
+
+# 5 EXPERIMENTS
+
+We evaluate Pareto HyperNetworks (PHNs) on a set of diverse multi-objective problems. The experiments show the superiority of PHN over previous MOO methods. We make our source code publicly available at: https://github.com/AvivNavon/pareto-hypernetworks.
+
+Compared methods: We compare the following approaches: (1) Our proposed Pareto HyperNetworks (PHN-LS and PHN-EPO) algorithm described in Section 4; (2) Linear scalarization (LS); (3) Pareto MTL (PMTL) Lin et al. (2019); (4) Exact Pareto optimal (EPO) Mahapatra & Rajan (2020); (5) Continuous Pareto MTL (CPMTL) Ma et al. (2020). All competing methods train one model per preference vector to generate Pareto solutions. As these models optimize on each ray separately, we do not treat them as a baseline but as “gold standard” in term of performance. Their runtime, on the other hand, scales linearly with the number of rays.
+
+![](images/8487a6cbc1e66bee0e195af6ce8e4bef5747a4864ad36d9e36f86ef5fa23e9f6.jpg)  
+Figure 3: Multi-MNIST: Test losses produced by the competing method on three Multi-MNIST datasets. Colors correspond to different preference vectors, using the same color map as in Figure 1. PHN generates a continuous Pareto front (solid/dashed line for PHN-EPO/PHN-LS) while mapping any given preference to its corresponding solution (circles for PHN-EPO and x’s for PHN-LS).
+
+Genetic algorithms are perhaps the most popular approaches for MOO. Two leading approaches in that area, that also use preference information are NSGA-III (Deb et al., 2002) and MOEA/D (Zhang & Li, 2007). Unfortunately, while these methods can be applied to small networks (like the toy example in Figure 1, which gave sub-par results even after 100,000 generations), they failed to converge to meaningful solutions in our larger-scale experiments. Therefore, we do not compare PHN with GAs on large-scale setups.
+
+Evaluation metrics: A common metric for evaluating sets of MOO solutions is the Hypervolume indicator (HV) (Zitzler & Thiele, 1999). It measures the volume in the loss space of points dominated by a solution in the evaluated set (see Appendix C). As this volume is unbounded, it is restricted to the volume in a rectangle defined by the solutions and a selected reference point. All solutions need to be bounded by the reference point, therefore we select different reference points for each experiment. In addition, we evaluate Uniformity, proposed by Mahapatra & Rajan (2020). This metric is designed to quantify how well the loss vector $\ell ( \theta )$ is aligned with the input ray $\mathbfit { \Delta } \mathbf { r }$ . Formally, for any point $\theta$ in the solution space and reference $\mathbfit { \Delta } \mathbf { r }$ we define the non-uniformity $\mu _ { r } ( \ell ( \theta ) ) = D _ { K L } ( \hat { \ell } \| \mathbf { 1 } / m )$ , where ˆ\`j = P j ji ri\`i . We define the uniformity as $1 - \mu _ { r }$ . The main metric, HV, measures the quality of our solutions, while the uniformity measures how well the output model fits the desired preference ray. We stress that the baselines are trained and evaluated for uniformity using the same rays.
+
+Training Strategies: In all experiments, our target network shares the same architecture as the baseline models. For our hypernetwork, we use an MLP with 2 hidden layers and linear heads. We set the hidden dimension to 100 for the Multi-MNIST and NYU experiment, and 25 for the Fairness and SARCOS datasets. The Dirichlet parameter $\alpha$ in Alg. 1 is set to 0.2 for all experiments, as we empirically found this value to perform well across datasets and tasks. We provide a more detailed analysis on the selection of the HN’s hidden dimension and hyperparameter $\alpha$ in Appendix B.5. Extensive experimental details are provided in Appendix A.
+
+Hyperparameter tuning: For PHN, we select hyperparameters based on the HV computed on a validation set. Selecting hyperparameters for the baselines is challenging because there are no clear criteria that can be computed quickly with many rays. Selecting HPs based on HV requires to train each baseline multiple times on all rays. Therefore, we select hyperparameters based on a single ray and apply the selected hyperparameters to all rays. Specifically, we collect all models trained using all hyperparameter configurations, and remove dominated solutions. Finally, we select the combination of HPs with the highest uniformity.
+
+# 5.1 ILLUSTRATIVE EXAMPLE
+
+We evaluate PHN on an illustrative MOO problem whose Pareto front is known (Fonseca, 1995). Here,
+
+$$
+\ell _ { 1 } ( \theta ) = 1 - \exp \big \{ - \| \theta - { \bf 1 } / \sqrt { d } \| _ { 2 } ^ { 2 } \big \} , \quad \ell _ { 2 } ( \theta ) = 1 - \exp \big \{ - \| \theta + { \bf 1 } / \sqrt { d } \| _ { 2 } ^ { 2 } \big \} ,
+$$
+
+Table 1: Evaluation of the different methods on three Multi-MNIST datasets.   
+
+<table><tr><td></td><td colspan="2">Multi-Fashion+MNIST</td><td colspan="2">Multi-Fashion</td><td colspan="2">Multi-MNIST</td><td></td></tr><tr><td></td><td>HV个</td><td>Unif.介</td><td>HV介</td><td>Unif.介</td><td>HV个</td><td>Unif.介</td><td>Run-time (min.,Tesla V100)</td></tr><tr><td>LS</td><td>2.70</td><td>0.849</td><td>2.14</td><td>0.835</td><td>2.85</td><td>0.846</td><td>9.0×5=45</td></tr><tr><td>CPMTL</td><td>2.76</td><td>1</td><td>2.16</td><td>-</td><td>2.88</td><td>-</td><td>10.2 × 5=51</td></tr><tr><td>PMTL</td><td>2.67</td><td>0.776</td><td>2.13</td><td>0.192</td><td>2.86</td><td>0.793</td><td>17.0 ×5=85</td></tr><tr><td>EPO</td><td>2.67</td><td>0.892</td><td>2.15</td><td>0.906</td><td>2.85</td><td>0.918</td><td>23.6× 5=118</td></tr><tr><td>PHN-LS (ours)</td><td>2.75</td><td>0.894</td><td>2.19</td><td>0.900</td><td>2.90</td><td>0.901</td><td>12</td></tr><tr><td>PHN-EPO (ours)</td><td>2.78</td><td>0.952</td><td>2.19</td><td>0.921</td><td>2.78</td><td>0.920</td><td>27</td></tr></table>
+
+with parameters $\boldsymbol { \theta } \in \mathbb { R } ^ { d }$ and $d = 1 0 0$ . This problem has a non-convex Pareto front in $\mathbb { R } ^ { 2 }$ . We compared PHN with EPO, PMTL, MGDA, and two genetic algorithms (GAs) NSGA-III and MOEA/D. The results are presented in Figure 1. Linear scalarization (LS) fails to generate solutions in the concave part of the Pareto front, as theoretically shown in Boyd et al. (2004). PMTL and EPO generate solutions along the entire PF, but the solutions generated by PMTL are only optimized to be in the vicinity of the preference vector $\pmb { r }$ . Results for MGDA and MOEA/D algorithms are shown in Appendix B.1. Our PHN-EPO generates a continuous cover of the PF in a single run. In addition, PHN-EPO returns exact Pareto optimal solutions, mapping an input preference to its corresponding point on the front. We note that similar to the EPO method, PHN-EPO can learn non-convex Pareto fronts. Additional synthetic experiments with non-convex, known Pareto fronts are presented in Appendix B.1.
+
+# 5.2 IMAGE CLASSIFICATION
+
+We evaluate all methods on three MOO benchmark datasets: (1) Multi-MNIST (Sabour et al., 2017); (2) Multi-Fashion, and (3) Multi-Fashion $^ +$ MNIST. In each dataset, two instances are sampled uniformly at random from the MNIST (LeCun et al., 1998) or Fashion-MNIST (Xiao et al., 2017) datasets. The two images are merged into a new image by placing one at the top-left corner and the other at the bottom-right corner. The sampled images can be shifted up to four pixels in each direction. Each dataset consists of 120,000 training examples and 20,000 test set examples. We allocate $10 \%$ of each training set for constructing validation sets. We train 5 models for each baseline, using evenly spaced rays. The results are presented in Figure 3 and Table 1. The HV was calculated with reference point $( 2 , 2 )$ . PHN outperforms all other methods in terms of HV and uniformity, with significant improvement in run time. To better understand the behavior of PHN while varying the input preference vector, we visualize the per-pixel prediction contribution in Appendix B.3.
+
+# 5.3 FAIRNESS
+
+![](images/57f9a4447d18ad70f85a2aac96e19919d9180628cfe9c78d9e27ebde8ba946fc.jpg)  
+Figure 4: Accuracy-Fairness trade-off : The trade-off between the main classification loss and the two fairness losses, on the test sets of the three fairness datasets. Each line corresponds to a different solution along the Pareto front, generated using PHN-EPO.
+
+Fairness has become a popular topic in machine learning in recent years (Mehrabi et al., 2019), and numerous approaches have been proposed for modeling and incorporating fairness into ML systems (Dwork et al., 2018; Kamishima et al., 2012; Zafar et al., 2019). Here, we adopt the approach of Bechavod & Ligett (2017). They proposed a 3-dimensional optimization problem, with a classification objective and two fairness objectives: False Positive (FP) fairness, and False Negative (FN) fairness. We compare PHN with EPO, PMTL, and LS on three commonly used fairness datasets: Adult (Dua & Graff, 2017), Default (Yeh & Lien, 2009) and Bank (Moro et al., 2014).
+
+Table 2: Comparison on three Fairness datasets. Run-time is evaluated on the Adult dataset.   
+
+<table><tr><td></td><td colspan="2">Adult</td><td colspan="2">Default</td><td colspan="2">Bank</td></tr><tr><td></td><td>HV个 Unif.介</td><td></td><td>HV个Unif.个</td><td></td><td>HV个Unif.介</td><td>Run-time (min., Tesla V100)</td></tr><tr><td>LS</td><td>0.628</td><td>0.109</td><td>0.522</td><td>0.069</td><td>0.704 0.215</td><td>0.9 × 10=9.0</td></tr><tr><td>PMTL</td><td>0.614</td><td>0.458</td><td>0.548</td><td>0.471</td><td>0.638 0.497</td><td>2.7 × 10= 26.6</td></tr><tr><td>EPO</td><td>0.608</td><td>0.734</td><td>0.537</td><td>0.533</td><td>0.713 0.881</td><td>1.6 × 10= 15.5</td></tr><tr><td>PHN-LS (ours)</td><td>0.658</td><td>0.289</td><td>0.551</td><td>0.108</td><td>0.730 0.615</td><td>1.1</td></tr><tr><td>PHN-EPO (ours)</td><td>0.648</td><td>0.701</td><td>0.548</td><td>0.359</td><td>0.748 0.821</td><td>1.8</td></tr></table>
+
+Each dataset is divided into train/validation/test sets of sizes $7 0 \% / 1 0 \% / 2 0 \%$ respectively. All baselines are trained and evaluated on 10 preference rays. We present the results in Table 2. HV calculated with reference point $( 1 , 1 , 1 )$ . PHN achieves the best HV performance across all datasets, with reduced training time compared to baselines. EPO performs best in terms of uniformity, however, recall it is evaluated on its training rays. We visualize the accuracy-fairness trade-off achieved by PHN in Figure 4. We also tested GAs on this task but the runtime was extremely high (two days, compared with a runtime of minutes for gradient-based methods) with poor performance.
+
+# 5.4 PIXEL-WISE CLASSIFICATION AND REGRESSION
+
+Image segmentation and depth estimation are key problems in computer vision with various applications (Minaee et al., 2020; Bhoi, 2019). We compare PHN to the other methods on the NYUv2 dataset (Silberman et al., 2012). NYUv2 is a challenging indoor scene dataset that consists of 1449 RGBD images and dense per-pixel labeling. We use this dataset as a multitask learning benchmark for semantic segmentation and depth estimation tasks. We split the dataset into train/validation/test sets of sizes $7 0 \% / 1 0 \% / 2 0 \%$ respectively, and use an ENet-based architecture for the target network (Paszke et al., 2016). Each baseline method is trained five times using five evenly spaced rays. The results are presented in Table 3. HV is calculated with reference point $( 3 , 3 )$ . PHN-EPO achieves the best HV and uniformity, while also being faster than the baselines.
+
+Table 3: Comparison on NYUv2.   
+
+<table><tr><td></td><td colspan="3">NYUv2</td></tr><tr><td></td><td>HV介</td><td>Unif.介</td><td>Run-time (hours,Tesla V100)</td></tr><tr><td>LS</td><td>3.550</td><td>0.666</td><td>0.58 × 5=2.92</td></tr><tr><td>PMTL</td><td>3.554</td><td>0.679</td><td>0.96× 5=4.79</td></tr><tr><td>CPMTL</td><td>3.570</td><td>-</td><td>0.71 × 5=3.55</td></tr><tr><td>EPO</td><td>3.266</td><td>0.728</td><td>1.02 × 5=5.11</td></tr><tr><td>PHN-LS (ours)</td><td>3.546</td><td>0.798</td><td>0.67</td></tr><tr><td>PHN-EPO (ours)</td><td>3.589</td><td>0.820</td><td>1.04</td></tr></table>
+
+Table 4: Comparison on SARCOS dataset.   
+
+<table><tr><td></td><td colspan="2">SARCOS</td></tr><tr><td></td><td>HV个 Unif.介</td><td>Run-time (hours,Tesla V100)</td></tr><tr><td>LS</td><td>0.688</td><td>-0.008</td><td>0.17 × 20= 3.3</td></tr><tr><td>EPO</td><td>0.637</td><td>0.548</td><td>0.5× 20=10</td></tr><tr><td>PHN-LS (ours)</td><td>0.693</td><td>0.020</td><td>0.2</td></tr><tr><td>PHN-EPO (ours)</td><td>0.728</td><td>0.258</td><td>0.8</td></tr></table>
+
+# 5.5 MULTITASK REGRESSION
+
+When learning with many objectives using preference-specific models, generating a good coverage of the solution space becomes too expensive in computation and storage space, as the number of rays needed grows with the number of dimensions. As a result, PHN provides a major improvement in training time and solution space coverage (measured by HV). To evaluate PHN in such settings, we use the SARCOS dataset (Vijayakumar), a commonly used dataset for multitask regression (Zhang & Yang, 2017). SARCOS presents an inverse dynamics problem with 7 regression tasks and 21 explanatory variables. We omit PMTL from this experiment because its training time is significantly longer compared to other baselines, with relatively low performance. For LS and EPO, we train 20 preference-specific models. The results are presented in Table 4. HV calculated with reference point $( 1 , . . . , 1 )$ . As before, PHN performs best in terms of HV, while reducing the overall runtime.
+
+# 6 THE QUALITY-RUNTIME TRADE-OFF
+
+While PHN learns the entire front in a single model, the competing methods need to train multiple models to cover the Pareto front. As a result, these methods have a clear trade-off between their performance and their run time. To better understand this trade-off, Figure 5 shows the HV and run time (in log-scale) of LS and EPO when training a various number of models. For Fashion-MNIST, we trained 25 models, and at the inference time, we selected subsets of various sizes (1 ray, 2 rays, etc) and computed their HV. The shaded area in Figure 5 reflects the variance over different selections of ray subsets. We similarly trained 30 EPO/LS models for Adult and 40 for SARCOS. PHN-EPO achieves superior or comparable hypervolume while being an order of magnitude faster. PHN-LS is even faster, but this may come at the expense of a lower hypervolume.
+
+To further understand the source of PHN performance gain, we evaluate the HV on the same rays used for the competing methods (see Appendix B.4). In many cases, PHN outperforms models trained on specific rays, when evaluated on those exact rays. We attribute the performance gain to the inductive bias effect that PHN induces on a model by sharing weights across rays.
+
+![](images/cb4aac8bd4569ca4694a08f7a28530ef9762f887c5a1034e4cad1b07c9e185b6.jpg)  
+Figure 5: HV vs runtime (min.) comparing PHN with preference-specific LS and EPO. For LS and EPO, each value was computed by evaluating with a subset of models. PHN achieves higher HV significantly faster ( $\mathbf { \dot { X } }$ -axis is in log scale). Shaded area marks the variance across selected subsets.
+
+# 7 CONCLUSION
+
+We put forward a novel MOO setup which we term Pareto Front Learning (PFL). Learning the Pareto front using a single unified model that can be applied to a specific objective preference at inference time. We propose PHN, a model for this setup based on a hypernetwork, and evaluate it on a large and diverse set of tasks and datasets. Our experiments show significant gains in both runtime and performance on all datasets. This novel approach provides users with the flexibility of selecting the operating point at inference time and opens new possibilities where user preferences can be tuned and changed at deployment.
+
+# ACKNOWLEDGEMENTS
+
+This study was funded by a grant to GC from the Israel Science Foundation (ISF 737/2018), and by an equipment grant to GC and Bar-Ilan University from the Israel Science Foundation (ISF 2332/18). AS and AN were funded by a grant from the Israeli Innovation Authority, through the AVATAR consortium.
+
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+# Appendix: Learning the Pareto Front with Hypernetworks
+
+A EXPERIMENTAL DETAILS
+
+Pixelwise Classification and Regression: For NYUv2 we performed a hyperparameters search over the number of epochs and learning rates $\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \}$ . We train each method for 150 epochs with Adam (Kingma & Ba, 2015) optimizer and select an optimal set of learning rate and number of epochs according to Sec. 5. We evaluate PHN over 25 evenly spaced rays.
+
+Image classification: For the Multi-MNIST, experiments we use a LeNet-based target network (LeCun et al., 1998) with two task-specific heads. We train all methods using an Adam optimizer with learning rate $1 e - 4$ for 150 epochs and batch size 256. We then choose the best number of epochs for each method using the validation set. We evaluate PHN over 25 rays.
+
+Fairness: We use a 3−layer feed-forward target network with hidden dimensions [40, 20]. We use learnable embeddings for all categorical variables. We train all methods for 35 epochs using an Adam optimizer with learning rates $\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \}$ . We choose the best hyperparameter configuration using the validation set. Further details on the datasets are provided in Table 5. We evaluate PHN over a sample of 150 rays.
+
+Multitask regression: For the SARCOS experiment, we use a $4 -$ layer feed-forward target network with 256 hidden dimension. All categorical features are passed through embedding layers. We train all methods for 1000 epochs using an Adam optimizer and learning rates $\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \}$ . We choose the best hyperparameter configuration base on the validation set. We use 40,036 training examples, 4,448 validation examples, and 4,449 test examples. We evaluate PHN over a sample of 100 rays.
+
+Table 5: Fairness datasets.   
+
+<table><tr><td></td><td>Adult Default</td><td>Bank</td></tr><tr><td>Train</td><td>32559</td><td>21600 29655</td></tr><tr><td>Validation</td><td>3618</td><td>2400 3295</td></tr><tr><td>Test</td><td>9045</td><td>6000 8238</td></tr></table>
+
+# B ADDITIONAL EXPERIMENTS
+
+![](images/b10e7f01b26106c431e2665ebdbaed0a37efee267dddb22da0b52ca32df0edf7.jpg)  
+Figure 6: Illustrative example using the popular task of Fonseca (1995): The genetic algorithm MOEA/D fails to reach the exact pareto front after 100, 000 generations. MGDA reaches Pareto optimal solutions, but is not suitable for generating preference-specific solutions and only covers a small fraction of the pareto front.
+
+![](images/f595400e3ec7c4d6599c327849c4551608426b405a028bf07ec814aa8683bbea.jpg)  
+Figure 7: Known Pareto front: Top: Evaluation on Problem 2 from Evtushenko & Posypkin (2013). Bottom: Evaluation on ZDT2 (Zitzler et al., 2000). PHN converges to the entire PF, mapping a ray to its corresponding solution on the front. Each circle in the PHN plots denotes a solution corresponds to the baseline rays.
+
+# B.1 KNOWN PARETO FRONT
+
+We provide two additional experiments over popular MOO benchmarks whose Pareto front is known. In addition in Figure 6, we provide additional results for MGDA (Sener & Koltun, 2018) and MOEA/D (Zhang & Li, 2007) on the illustrative example from Section 5.1.
+
+We evaluate PHN on two MOO problems with known, non-convex Pareto front: ZDT2 (Zitzler et al., 2000) and problem two from Evtushenko & Posypkin (2013), henceforth Problem2. In both problems, the density of solutions across the Pareto-optimal region is non-uniform.
+
+For a solution $\theta = ( \theta _ { 1 } , \theta _ { 2 } ) \in [ 0 , 1 ] ^ { 2 }$ , Problem2 is given by,
+
+$$
+\ell _ { 1 } ( \theta ) = ( ( \theta _ { 1 } - 1 ) \theta _ { 2 } ^ { 2 } + 1 ) / 3 , \quad \ell _ { 2 } ( \theta ) = \theta _ { 1 } ,
+$$
+
+and ZDT2 is given by,
+
+$$
+\ell _ { 1 } ( \theta ) = \theta _ { 1 } , \quad \ell _ { 2 } ( \theta ) = 1 - \left( \frac { \theta _ { 1 } } { 1 + 9 \theta _ { 2 } } \right) ^ { 2 } .
+$$
+
+The results are presented in Figure 7. PHN-EPO converges to the entire PF using a single HN model.   
+in contrast with PHN-EPO, EPO fails to converge to the PF for all rays in ZDT2.
+
+# B.2 GENERALIZATION TO UNSEEN RAYS
+
+When the number of objectives is large, the curse of dimensionality makes it likely that rays encountered in test time are "far" from rays encountered during training. Although we evaluated PHN on unseen rays1 through all experiments, here we would like to better examine the generalization of PHN in that scenario. We conducted controlled experiment on SARCOS dataset with 7 tasks. During training, we sample rays from a predefined grid on the 7−dimensional simplex. We use the structured approach of Das & Dennis (1998) for generating well-spaced points. During inference, we measure HV and Uniformity on two sets of rays: (i) Randomly sampled train rays (ii) Randomly sampled test rays from a shifted grid, so they are most distant from the train grid. In both cases, test samples differ from training samples. We find that testing with unseen test rays only causes a minor decrease in both HV and Uniformity compared with train rays. The HV is reduced from 0.7337 to 0.7336 and uniformity is reduced from 0.163 to 0.158. This experiment indicates that PHN generalizes well to unseen rays, even in problems with many objectives. We note that the HV results presented here are higher than the ones reported in Section 5.5. We attribute this inconsistency to change of preference vectors distribution used during training. While throught the paper we sample preference rays from a Dirichlet distribution with $\alpha = 0 . 2$ , here we sample from a fixed, evenly spaced grid. We provide analysis on the robustness to the choice of the Dirichlet distribution parameter $\alpha$ in Appendix B.5.
+
+# B.3 MNIST PHN INTERPRETATION
+
+![](images/7034159f14a61aae45d8447e637a74d8506c08f8944c8ea8968877eaaf2a5c31.jpg)  
+Figure 8: PHN interpretation: PHN is provided with a sample from the Multi-MNIST dataset, where an image of 3 and an image 7 are overlayed. We show the effect that conditioning on different preference rays (left) has on the decision made by the model. Red pixels denote increased contribution to the prediction. See text for details.
+
+To better understand the behaviour of PHN while varying the input preference vector, Figure 8 shows the per-pixel contribution to prediction. We use the SHAP framework (Lundberg & Lee, 2017) for visualization. Red pixels denote an increased probability for a given class, while blue pixels denote a decreased probability. The top left and bottom right digits are labeled as $\{ 7 , 3 \}$ respectively. In the first row (ray: (0.99, 0.01)), the model is focused on task 1 (with label 7), most of the red pixels come from the top left digit in column 7, as expected. Similarly, in the third row (ray: (0.01, 0.99)) most of the red pixels are focused on the bottom right digit in column 3.
+
+# B.4 SAME-RAY EVALUATION
+
+In Section 5 of the main text we evaluate PHN on multiple evenly spaced rays. For completeness, we report here the HV and Uniformity achieved by PHN, while evaluated on the same rays used for training and evaluating the baseline methods. The results are provided in Table 6. Surprisingly, in many cases PHN achieves better performance. We attribute this to the inductive bias effect of sharing weights across all preference rays.
+
+Table 6: Evaluation of PHN on baselines’ ray.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">Fashion+MNIST</td><td colspan="2">Multi-Fashion</td><td colspan="2">Multi-MNIST</td><td colspan="2">Adult</td><td colspan="2">Bank</td><td colspan="2">NYUv2</td><td colspan="2">SARCOS HV</td></tr><tr><td>HV</td><td>Unif.</td><td>HV</td><td>Unif.</td><td></td><td>HV Unif.</td><td>HV</td><td>Unif.</td><td>HVUnif.</td><td></td><td>HVUnif.</td><td>HV</td><td>Unif.</td><td>Unif.</td></tr><tr><td>PHN-LS 2.740</td><td></td><td>0.852</td><td>2.190</td><td>0.838</td><td>2.900</td><td>0.840</td><td>0.624 0.282</td><td></td><td>0.538 0.106</td><td></td><td>0.695 0.603</td><td>3.494 0.683</td><td></td><td>0.693-0.051</td></tr><tr><td>PHN-EPO 2.760</td><td></td><td>0.908</td><td>2.180</td><td>0.860</td><td>2.760</td><td>0.857</td><td>0.631 0.742</td><td></td><td>0.531 0.395</td><td></td><td>0.717 0.833</td><td>3.565 0.701</td><td></td><td>0.728 0.190</td></tr></table>
+
+# B.5 DESIGN CHOICE ANALYSIS
+
+In Section 5 we used fixed values of the hidden dimension and the Dirichlet parameter $\alpha$ (see Alg. 1). Here we examine the effect of different choices for these hyperparameters. We use the Multi-Fashion $+ \mathrm { M N I S T }$ dataset, and the PHN-EPO variant of our method. We first fix the hidden dimension to 100, and vary $\alpha$ from 0.1 to 1 (uniform sampling). The results are presented in Figure 9(a). It appears that at least for this dataset, PHN is quite robust to changes in $\alpha$ . Next, we set $\alpha = 0 . 2$ and examine the effect of changing the dimension of the hidden layers in our hypernetwork. We present the results in Figure 9(b). Here, the effect on the HV is stronger, with best HV achieved using hidden dimension of 100.
+
+![](images/94100b9d94d1764688f570cb9f61d97dd18fe84c7bce14ead2fd8ab9b6a85f01.jpg)  
+Figure 9: Hyperparameters analysis: Performance analysis on the Multi-Fashion $^ +$ MNIST dataset, for different architecture designs and sampling choices.
+
+# B.6 ADDITIONAL RUNTIME ANALYSIS
+
+![](images/3fe31436c1fbae5104d452d03147d996e5cdf42ade1715b2ae2e4d5500f89346.jpg)  
+Figure 10: Runtime analysis for NYUv2, and two additional Multi-MNIST dataset.
+
+We extend the runtime analysis from Section 5.5. Figure 10 provides additional runtime analysis on the Multi-MNIST, Multi-Fashion, and NYUv2 datasets. For NYUv2 we omit the EPO method, since it was outperformed by LS in both HV and runtime.
+
+# C HYPERVOLUME
+
+![](images/7d7857e5808b5a11cc623429a5700fdbf7904e1be3714526737d64df67dd26e4.jpg)  
+Figure 11: Hypervolume illustration: The colored area represents the hypervolume obtained by Pareto set $\{ P _ { 1 } , P _ { 2 } , P _ { 3 } \}$ with respect to the reference point.
+
+The hypervolume metric is defined as follows: Given a set of points $S \subset \mathbb { R } ^ { n }$ and a reference point $\pmb { \rho } \in \mathbb { R } _ { + } ^ { n }$ , the hypervolume of $S$ is measured by the region of non-dominated points bounded above by $\rho$ .
+
+$$
+H V ( S ) = V O L \big ( \{ q \in \mathbb { R } _ { + } ^ { n } \mid \exists p \in S : ( p \preceq q ) \land ( q \preceq \rho ) \} \big ) ,
+$$
+
+where $V O L$ is the euclidean volume. We can interpret hypervolume as the measure of the union of the boxes created by the non-dominated points.
+
+$$
+H V ( S ) = V O L \left( \bigcup _ { p \in S } \prod _ { i = 1 } ^ { n } [ p _ { i } , \rho _ { i } ] \right)
+$$
+
+As a measure of quality for MOO solutions, it takes into account both the quality of individual solutions, which is the volume they dominate, and also the diversity, measured in the overlap between dominated regions. See example in Fig. 11.

parse/train/NjF772F4ZZR/NjF772F4ZZR_content_list.json

@@ -0,0 +1,2171 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING THE PARETO FRONT WITH HYPERNETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            552,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aviv Navon∗ Bar-Ilan University, Israel aviv.navon@biu.ac.il ",
+        "bbox": [
+            184,
+            170,
+            380,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aviv Shamsian∗ Bar-Ilan University, Israel aviv.shamsian@biu.ac.il ",
+        "bbox": [
+            586,
+            170,
+            812,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ethan Fetaya† Bar-Ilan University, Israel ethan.fetaya@biu.ac.il ",
+        "bbox": [
+            183,
+            233,
+            400,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gal Chechik†   \nBar-Ilan University, Israel   \nNVIDIA, Israel   \ngal.chechik@biu.ac.il ",
+        "bbox": [
+            511,
+            233,
+            717,
+            289
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            325,
+            544,
+            340
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-objective optimization (MOO) problems are prevalent in machine learning. These problems have a set of optimal solutions, called the Pareto front, where each point on the front represents a different trade-off between possibly conflicting objectives. Recent MOO methods can target a specific desired ray in loss space however, most approaches still face two grave limitations: (i) A separate model has to be trained for each point on the front; and (ii) The exact trade-off must be known before the optimization process. Here, we tackle the problem of learning the entire Pareto front, with the capability of selecting a desired operating point on the front after training. We call this new setup Pareto-Front Learning (PFL). ",
+        "bbox": [
+            233,
+            356,
+            764,
+            482
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We describe an approach to PFL implemented using HyperNetworks, which we term Pareto HyperNetworks (PHNs). PHN learns the entire Pareto front simultaneously using a single hypernetwork, which receives as input a desired preference vector and returns a Pareto-optimal model whose loss vector is in the desired ray. The unified model is runtime efficient compared to training multiple models and generalizes to new operating points not used during training. We evaluate our method on a wide set of problems, from multi-task regression and classification to fairness. PHNs learn the entire Pareto front at roughly the same time as learning a single point on the front and at the same time reach a better solution set. PFL opens the door to new applications where models are selected based on preferences that are only available at run time. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            675,
+            336,
+            690
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-objective optimization (MOO) aims to optimize several possibly conflicting objectives. MOO is abundant in machine learning problems, from multi-task learning (MTL), where the goal is to learn several tasks simultaneously, to constrained problems. In such problems, one aims to learn a single task while finding solutions that satisfy properties like fairness or privacy. It is common to optimize the main task while adding loss terms to encourage the learned model to obtain these properties. ",
+        "bbox": [
+            174,
+            705,
+            823,
+            776
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "MOO problems have a set of optimal solutions, the Pareto front, each reflecting a different trade-off between objectives. Points on the Pareto front can be viewed as an intersection of the front with a specific direction in loss space (a ray, Figure 1). We refer to this direction as a preference vector, as it represents a single trade-off between objectives. When a direction is known in advance, it is possible to obtain the corresponding solution on the front (Mahapatra & Rajan, 2020). ",
+        "bbox": [
+            174,
+            784,
+            825,
+            853
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, in many cases, we are interested in more than one predefined direction, either because the trade-off is not known before training, or because there are many possible trade-offs of interest. For example, network routing optimization aims to maximize bandwidth, minimize latency, and obey fairness. However, the cost and trade-off vary from one application running in the network to another, or even continuously change in time. The challenge remains to design a model that can be applied at inference time to any given preference direction, even ones not seen during training. We call this problem Pareto front learning (PFL). ",
+        "bbox": [
+            176,
+            859,
+            823,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb897b1cbdcf566c5c6ce00658fef5b296bc2700b5ae6699089c44bda6cf4a50.jpg",
+        "image_caption": [
+            "Figure 1: Illustrative example using the popular task of Fonseca (1995): demonstrating the relation between Pareto front, preference rays, and solutions. Pareto front (black solid line) for a 2D loss space and several rays (colored dashed lines) which represent various possible preferences. Left: A single PHN-EPO model learns the entire Pareto front, mapping any given preference ray to its corresponding solution on the front. The data and task are detailed in Section 5.1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            97,
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+            223
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            334,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although several recent studies (Yang et al., 2019; Parisi et al., 2016) suggested to learn the trade-off curve in MOO problems, there is no existing scalable and general-purpose MOO approach that provides Pareto-optimal solutions for numerous preferences in objective space. Classical approaches, like genetic algorithms, do not scale to modern high-dimensional problems. It is possible in principle to run a single-direction optimization multiple times, each for a different preference, but this approach faces two major drawbacks: (i) Scalability – the number of models to be trained to cover the objective space grows exponentially with the number of objectives; and (ii) Flexibility – the decision maker cannot switch freely between preferences unless all models are trained and stored in advance. ",
+        "bbox": [
+            174,
+            411,
+            825,
+            523
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Here we put forward a new view of PFL as a problem of learning a conditional model, where the conditioning is over the preference direction. During training, a single unified model is trained to produce Pareto optimal solutions while satisfying the given preferences. During inference, the model covers the Pareto front by varying the input preference vector. We further describe an architecture that implements this idea using HyperNetworks. Specifically, we train a hypernetwork, termed Pareto Hypernetwork (PHN), that given a preference vector as an input, produces a deep network model tuned for that objective preference. Training is applied to preferences sampled from the $m -$ dimensional simplex where $m$ represents the number of objectives. ",
+        "bbox": [
+            173,
+            530,
+            825,
+            641
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate PHN on a wide set of problems, from multi-class classification, through fairness and image segmentation to multi-task regression. We find that PHN can achieve superior overall solutions while being $1 0 \\sim 5 0$ times faster (see Figure 5). PHN addresses both scalability and flexibility. Training a unified model allows using any objective preference at inference time. Finally, as PHN generates a continuous parametrization of the entire Pareto front, it could open new possibilities to analyze Pareto optimal solutions in large-scale neural networks. ",
+        "bbox": [
+            174,
+            647,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our paper has the following contributions: (1) We define the Pareto-front learning problem – learn a model that at inference time can operate on any given preference vector, providing a Pareto-optimal solution for that specified objective trade-off. (2) We describe Pareto Hypernetworks (PHN), a unified architecture based on hypernetworks that addresses PFL and shows it can be effectively trained. (3) Empirical evaluations on various tasks and datasets demonstrate the ability of PHNs to generate better objective space coverage compared to multiple baseline models, with significant improvement in training time. ",
+        "bbox": [
+            174,
+            738,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MULTI-OBJECTIVE OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            862,
+            488,
+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We start by formally defining the MOO problem. An MOO is defined by $m$ losses $\\ell _ { i } : \\mathbb { R } ^ { d } \\to \\mathbb { R } _ { + }$ $i = 1 , \\ldots , m$ , or in vector form $\\ell : \\mathbb { R } ^ { \\bar { d } }  \\mathbb { R } _ { + } ^ { m }$ . We define a partial ordering on the loss space $\\mathbb { R } _ { + } ^ { m }$ by setting $\\ell ( \\theta _ { 1 } ) \\preceq \\ell ( \\theta _ { 2 } )$ if for all $i \\in [ m ]$ , $\\ell _ { i } ( \\theta _ { 1 } ) \\leq \\ell _ { i } ( \\theta _ { 2 } )$ . We define $\\ell ( \\theta _ { 1 } ) \\prec \\ell ( \\theta _ { 2 } )$ if $\\ell ( \\theta _ { 1 } ) \\preceq \\ell ( \\theta _ { 2 } )$ and for some $i \\in [ m ] , \\ell _ { i } ( \\theta _ { 1 } ) < \\ell _ { i } ( \\theta _ { 2 } )$ . We say that a point $\\theta _ { 1 } \\in \\mathbb { R } ^ { d }$ dominates $\\theta _ { 2 } \\in \\mathbb { R } ^ { d }$ if $\\ell ( \\theta _ { 1 } ) \\prec \\ell ( \\theta _ { 2 } )$ . ",
+        "bbox": [
+            173,
+            895,
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+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            102,
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+            148
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If a point $\\theta _ { 1 }$ dominates $\\theta _ { 2 }$ , then $\\theta _ { 1 }$ is clearly preferable, because it improves some objectives and is not worse on any other objective. Otherwise, solutions present a certain trade-off and selecting one specific solution requires additional information about the user preferences. A point that is not dominated by any other point is called Pareto optimal. The set of all Pareto optimal points is called the Pareto front. Since many modern machine learning models, e.g., deep neural networks, use non-convex optimization, one cannot expect global optimality. We call a point local Pareto optimal if it is Pareto optimal in some open neighborhood of the point. ",
+        "bbox": [
+            173,
+            155,
+            825,
+            252
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The most straightforward approach to MOO is linear scalarization (LS), where one defines a new single loss $\\begin{array} { r } { \\ell _ { r } ( \\mathbf { \\bar { \\boldsymbol { \\theta } } } ) = \\sum _ { i } r _ { i } \\ell _ { i } ( \\boldsymbol { \\theta } ) } \\end{array}$ given a vector $\\pmb { r } \\in \\mathbb { R } _ { + } ^ { m }$ of weights. One can then apply standard, single-objective optimization algorithms. LS has two major limitations. First, it can only reach the convex part of the Pareto front (Boyd et al., 2004, Chapter 4.7), as shown empirically in Figure 1. Second, if one wishes to target a specific ray in loss space, specified by a preference vector, it is not clear which linear weights lead to that desired Pareto optimal point. ",
+        "bbox": [
+            174,
+            260,
+            825,
+            343
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the context of the current work, we highlight two properties that an MOO optimization procedure should possess: scale to many objectives and control which solution on the Pareto front is obtained. Lin et al. (2019) described Pareto multi-task learning (PMTL), an algorithm that splits the loss space into separate cones based on the selected reference rays, and returns a solution per cone using a constrained version of Fliege & Svaiter (2000). This approach allows the user to target several points on the Pareto front; However, it scales poorly with the number of cones and does not converge to the exact desired ray on the Pareto front. ",
+        "bbox": [
+            174,
+            349,
+            825,
+            446
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Convergence to the desired ray in loss space can be achieved using Exact Pareto Optimal (EPO) (Mahapatra & Rajan, 2020). To find the intersection of the Pareto front with a given preference ray $\\mathbfit { \\Delta } \\mathbf { r }$ EPO balances two goals: Finding a descent direction towards the Pareto front and approaching the desired ray. EPO searches for a point in the convex hull of the gradients, known by Désidéri (2012) to include descent directions, that has a maximal angle with a vector $d _ { b a l }$ which pulls the point to the desired ray. EPO combines gradient descent and controlled ascent enabling it to reach an exact Pareto optimal solution if one exists, or the closest Pareto optimal solution. ",
+        "bbox": [
+            174,
+            454,
+            825,
+            551
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 RELATWED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            577,
+            359,
+            593
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multitask learning. In multitask learning (MTL) we simultaneously solve several learning problems while sharing information among tasks (Zhang & Yang, 2017; Ruder, 2017). In some cases, MTL-based models outperform their single task counterparts in terms of per-task performance and computational efficiency (Standley et al., 2019). MTL approaches map the loss vector into a single loss term using a fixed or dynamic weighting scheme. The most frequently used approach is Linear Scalarization (LS), in which each loss term’s weight is chosen apriori. A proper set of weights is commonly selected using grid search. Unfortunately, such an approach scales poorly with the number of tasks. Recently, MTL methods propose dynamically balance the loss terms using gradient magnitude (Chen et al., 2018), the rate of change in losses (Liu et al., 2019), task uncertainty (Kendall et al., 2018), or learning non-linear loss combinations by implicit differentiation (Navon et al., 2020). However, those methods seek a balanced solution and are not suitable for modeling task trade-offs. ",
+        "bbox": [
+            174,
+            611,
+            825,
+            765
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-objective optimization. The goal of Multi-Objective Optimization (MOO) is to find Pareto optimal solutions corresponding to different trade-offs between objectives (Ehrgott, 2005). MOO has a wide variety of applications in machine learning, spanning Reinforcement Learning (Van Moffaert & Nowé, 2014; Pirotta & Restelli, 2016; Parisi et al., 2014; Pirotta et al., 2015; Parisi et al., 2016; Yang et al., 2019), neural architecture search (Lu et al., 2019; Hsu et al., 2018), fairness (Martínez et al., 2020; Liu & Vicente, 2020), and Bayesian optimization (Shah & Ghahramani, 2016; Hernández-Lobato et al., 2016). Genetic algorithms (GA) are a popular approach for small-scale multi-objective optimization problems. GAs are designed to maintain a set of solutions during optimization. Therefore, they can be extended in natural ways to maintain solutions for different rays. In this area, leading approaches include NSGA-III (Deb & Jain, 2013) and MOEA/D (Zhang & Li, ",
+        "bbox": [
+            174,
+            784,
+            826,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2007). However, these gradient-free methods scale poorly with the number of parameters and are not suitable for training large-scale neural networks. ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Sener & Koltun (2018) proposed a gradient-based MOO algorithm for MTL, based on MDGA (Désidéri, 2012), suitable for training large-scale neural networks. Other recent works include (Lin et al., 2019; Mahapatra & Rajan, 2020) detailed in Section 2. Another recent approach that aims at a more complete view of the Pareto front is Ma et al. (2020). They extend a given Pareto optimal solution in its local neighborhood. In a concurrent work, Lin et al. (2020) extends Lin et al. (2019) for approximating the entire Pareto front. They train a single hypernetwork by constantly changing the reference directions in the PMTL objective. The proposed method is conceptually similar to our approach, but since it builds on PMTL, it may not produce an exact mapping between the preference and the corresponding solution. Similarly, Dosovitskiy & Djolonga (2019) proposed learning a single model conditioning on the objective weight vector. The method uses feature-wise linear modulation Perez et al. (2017), and dynamically weighted LS loss criterion. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hypernetworks. Ha et al. (2017) introduced the idea of hypernetworks (HNs) inspired by the genotype-phenotype relation in cellular biology. HN presents an approach of using one network (hypernetwork) to generate weights for a second network (target network). In recent years, HN are widely used in various domains such as computer vision (Klocek et al., 2019; Ha et al., 2017), language modeling (Suarez, 2017), sequence decoding (Nachmani & Wolf, 2019), continual learning (Oswald et al., 2020), federated learning (Shamsian et al., 2021) and hyperparameter optimization (Mackay et al., 2019; Lorraine & Duvenaud, 2018). HN dynamically generates models conditioned on a given input, obtaining a set of customized models using a single learnable network. ",
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+    {
+        "type": "text",
+        "text": "4 PARETO HYPERNETWORKS ",
+        "text_level": 1,
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+        "type": "table",
+        "img_path": "images/4ea140deb8210230d4d78fba9e13a599968a3357b99f26e14f306c6de45674ed.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1PHN</td></tr><tr><td>while not converged do r ~ Dir(α)</td></tr><tr><td>0(Φ,r)=h(r;) Sample mini-batch (x1,y1),.,(xB,yB)</td></tr><tr><td>if LS then</td></tr><tr><td>g←B∑i,jriVΦli(xj,yj,0(Φ,r))</td></tr><tr><td>if EPO then</td></tr><tr><td>β=EPO(0(Φ,r),l,r)</td></tr><tr><td>g←B∑i,jβi√li(xj,yj,0(Φ,r))</td></tr><tr><td></td></tr><tr><td>Φ←-ngΦ</td></tr><tr><td></td></tr><tr><td>return $</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/c9b62b1546101cd36883d200a2a1501c1d2b55c7c15ec0201d0ecb509952d38e.jpg",
+        "image_caption": [
+            "Figure 2: A PHN $h ( \\cdot ; \\phi )$ receives an input preference ray $\\mathbfit { \\Delta } \\mathbf { r }$ and outputs the corresponding model weights $\\theta _ { r }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We start by describing the basic of hypernetworks architecture (Figure 2). Hypernetworks are deep models that generate the weights of another deep model as their output, named the target network. The weights produced by the hypernetwork depend on its input, and one can view the training of a hypernetwork as training a family of target networks simultaneously, conditioned on the input. Let $\\bar { h } ( \\cdot ; \\phi )$ denote the hypernetwork with parameters $\\phi \\in \\mathbb { R } ^ { n }$ , and let $f$ denote the target network whose parameters we denote by $\\theta$ . In our implementation, we parametrize $h$ using a feed-forward network with multiple heads. Each head outputs a different weight tensor of the target network. More precisely, the input is first mapped to a higher dimensional space using an MLP, to construct shared features. These features are passed through fully connected layers to create a weight matrix per layer in the target network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Consider an $m$ −dimensional MOO problem with an objective vector $\\ell$ . Let $\\pmb { r } = ( r _ { 1 } , . . . , r _ { m } ) \\in S _ { m }$ denote a preference vector representing a desired trade-off over the objectives. Here $S _ { m } = \\{ r \\in$ $\\begin{array} { r } { \\mathbb { R } ^ { m } \\ | \\ \\sum _ { j } r _ { j } \\ = 1 , r _ { j } \\ \\ge \\ 0 \\} } \\end{array}$ is the $m -$ dimensional simplex. Our PHN model takes as input the preference vector $\\pmb { r }$ to produce the weights $\\theta _ { r } = \\theta ( \\phi , r ) = h ( r ; \\phi )$ of the target network. The target network is then applied to the training data in a usual manner. Throughout the training, we randomly sample $\\mathbfit { \\Delta } \\mathbf { r }$ at each iteration from a Dirichlet distribution with parameter $\\alpha \\in \\mathbb { R } ^ { m }$ . At inference time, a given preference direction $\\mathbfit { \\Delta } \\mathbf { r }$ , is provided as input to the $\\mathrm { H N } h$ , which outputs the corresponding Pareto optimal solution, $\\theta _ { r }$ , namely, a set of model weights for specific given $\\pmb { r }$ . We train $\\theta$ to be Pareto optimal and corresponds to the preference vector. For that purpose, we explore two alternatives. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "PHN-LS: The first, named PHN-LS, uses linear scalarization with the preference vector $\\mathbfit { \\Delta } \\mathbf { r }$ as loss weights, i.e the loss for input $\\pmb { r }$ is $\\textstyle \\sum _ { i } r _ { i } \\ell _ { i }$ . While linear scalarization has some theoretical limitations, it is a straightforward and fast approach that tends to work well in practice. One can think of the PHN-LS approach as optimizing the loss $\\begin{array} { r } { \\ell _ { L S } ( \\phi ) = \\mathbb { E } _ { r , x , y } [ \\sum _ { i } r _ { i } \\ell _ { i } ( \\bar { x , } y , \\theta ( \\phi , \\pmb { r } ) ) ] } \\end{array}$ where we sample a mini-batch and a weighting $\\mathbfit { \\Delta } \\mathbf { r }$ . We note, as a direct result of the implicit function theorem, the following: ",
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+    {
+        "type": "text",
+        "text": "Proposition 1. Let $\\theta ( \\hat { \\pmb r } )$ be a local Pareto optimal point corresponding to weights $\\hat { r }$ , i.e. $\\begin{array} { r } { \\sum _ { i j } \\overline { { \\hat { r } _ { i } } } \\nabla \\ell _ { i } ( x _ { j } , y _ { j } , \\theta ) = \\mathrm { 0 } } \\end{array}$ and assume the Hessian of $\\textstyle \\sum _ { i } { \\bar { r } } _ { i } \\ell _ { i }$ has full rank at $\\theta ( \\hat { \\pmb r } )$ . There exists $a$ neighborhood $U$ of $\\scriptstyle { \\hat { r } }$ and a smooth mapping $\\theta ( r )$ such that $\\begin{array} { r } { \\forall \\pmb { r } \\in \\pmb { U } : \\sum _ { i j } r _ { i } \\nabla \\ell _ { i } ( x _ { j } , y _ { j } , \\theta ( \\pmb { r } ) ) = 0 . } \\end{array}$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This shows that under the full rank assumption the connection between the LS term $\\pmb { r }$ and the optimal model weights $\\theta ( r )$ is smooth and can be learned using universal approximation (Cybenko, 1989). We also observe that if for all $\\mathbfit { \\Delta } \\mathbf { r }$ , $\\theta ( \\phi , \\pmb { r } )$ is a local Pareto optimal point corresponding to $\\mathbfit { \\Delta } \\mathbf { r }$ , i.e., $\\begin{array} { r } { \\sum _ { i j } r _ { i } \\nabla \\ell _ { i } ( x _ { j } , y _ { j } , \\theta ( \\pmb { r } ) ) = 0 } \\end{array}$ , then $\\dot { \\nabla } _ { \\phi } \\ell _ { L S } ( \\phi ) = 0$ and this is a stationary point for $\\ell _ { L S }$ . However, there could exists stationary points and even local minima such that $\\nabla _ { \\phi } \\ell _ { L S } ( \\phi ) = 0$ but in general $\\begin{array} { r } { \\sum _ { i j } r _ { i } \\nabla \\ell _ { i } ( x _ { j } , y _ { j } , \\theta ( \\pmb { r } ) ) \\neq \\mathbf { \\bar { 0 } } } \\end{array}$ and the output of our hypernetwork is not a local Pareto optimal point. While these bad local minima exist, as in standard deep neural networks, we empirically find that PHNs avoid these solutions and achieve good results. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "PHN-EPO: Our second approach treats the preference $\\mathbfit { \\Delta } \\mathbf { r }$ as a ray in loss space and trains $\\theta ( \\phi , \\pmb { r } )$ to reach a Pareto optimal point on the inverse ray $\\scriptstyle \\mathbf { \\hat { r } } ^ { - 1 }$ , namely, $r _ { 1 } \\cdot \\ell _ { 1 } = . . . = r _ { m } \\cdot \\ell _ { m }$ . This is achieved by using the EPO update direction for each output of the Pareto hypernetwork. The recent EPO algorithm can guarantee convergence to a local Pareto optimal point on a specified ray $\\pmb { r }$ , addressing the theoretical limitations of LS. Since the EPO descent direction is a convex combination of the gradients, we use the EPO algorithm to compute this adaptive weighting and backpropagate the reweighted losses. See pseudo-code in Alg. 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The EPO however, unlike LS, cannot be seen as optimizing a certain objective. Therefore, we cannot consider the PHN-EPO optimization, which uses the EPO descent direction, as optimizing a specific objective. We also note that since the EPO needs to solve a linear programming step at each iteration, the runtime of EPO and PHN-EPO are longer than the LS and PHN-LS counterparts (see Figure 5). ",
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+    {
+        "type": "text",
+        "text": "PHN Scalability: Using a naive implementation of HNs, their size grows linearly with the target network’s size; this may restrict the scalability of PHNs to large target networks. Fortunately, our approach can be easily extended to handle large target networks. First, one can learn only part of the target network parameters using the HN (Luan et al., 2018), or learn preference-specific normalization layers (Huang & Belongie, 2017; Perez et al., 2017) as in Dosovitskiy & Djolonga (2019). Furthermore, one can apply “chunking\": generate different parts of the target network, conditioned on a trainable chunk descriptor input (Oswald et al., 2020). Similarly, one can reduce the dimension of the shared preference representation to achieve a similar effect (see Appendix B.5). The experiments in the main paper used the vanilla version of PHN. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We evaluate Pareto HyperNetworks (PHNs) on a set of diverse multi-objective problems. The experiments show the superiority of PHN over previous MOO methods. We make our source code publicly available at: https://github.com/AvivNavon/pareto-hypernetworks. ",
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+        "type": "text",
+        "text": "Compared methods: We compare the following approaches: (1) Our proposed Pareto HyperNetworks (PHN-LS and PHN-EPO) algorithm described in Section 4; (2) Linear scalarization (LS); (3) Pareto MTL (PMTL) Lin et al. (2019); (4) Exact Pareto optimal (EPO) Mahapatra & Rajan (2020); (5) Continuous Pareto MTL (CPMTL) Ma et al. (2020). All competing methods train one model per preference vector to generate Pareto solutions. As these models optimize on each ray separately, we do not treat them as a baseline but as “gold standard” in term of performance. Their runtime, on the other hand, scales linearly with the number of rays. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/8487a6cbc1e66bee0e195af6ce8e4bef5747a4864ad36d9e36f86ef5fa23e9f6.jpg",
+        "image_caption": [
+            "Figure 3: Multi-MNIST: Test losses produced by the competing method on three Multi-MNIST datasets. Colors correspond to different preference vectors, using the same color map as in Figure 1. PHN generates a continuous Pareto front (solid/dashed line for PHN-EPO/PHN-LS) while mapping any given preference to its corresponding solution (circles for PHN-EPO and x’s for PHN-LS). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Genetic algorithms are perhaps the most popular approaches for MOO. Two leading approaches in that area, that also use preference information are NSGA-III (Deb et al., 2002) and MOEA/D (Zhang & Li, 2007). Unfortunately, while these methods can be applied to small networks (like the toy example in Figure 1, which gave sub-par results even after 100,000 generations), they failed to converge to meaningful solutions in our larger-scale experiments. Therefore, we do not compare PHN with GAs on large-scale setups. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics: A common metric for evaluating sets of MOO solutions is the Hypervolume indicator (HV) (Zitzler & Thiele, 1999). It measures the volume in the loss space of points dominated by a solution in the evaluated set (see Appendix C). As this volume is unbounded, it is restricted to the volume in a rectangle defined by the solutions and a selected reference point. All solutions need to be bounded by the reference point, therefore we select different reference points for each experiment. In addition, we evaluate Uniformity, proposed by Mahapatra & Rajan (2020). This metric is designed to quantify how well the loss vector $\\ell ( \\theta )$ is aligned with the input ray $\\mathbfit { \\Delta } \\mathbf { r }$ . Formally, for any point $\\theta$ in the solution space and reference $\\mathbfit { \\Delta } \\mathbf { r }$ we define the non-uniformity $\\mu _ { r } ( \\ell ( \\theta ) ) = D _ { K L } ( \\hat { \\ell } \\| \\mathbf { 1 } / m )$ , where ˆ\\`j = P j ji ri\\`i . We define the uniformity as $1 - \\mu _ { r }$ . The main metric, HV, measures the quality of our solutions, while the uniformity measures how well the output model fits the desired preference ray. We stress that the baselines are trained and evaluated for uniformity using the same rays. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training Strategies: In all experiments, our target network shares the same architecture as the baseline models. For our hypernetwork, we use an MLP with 2 hidden layers and linear heads. We set the hidden dimension to 100 for the Multi-MNIST and NYU experiment, and 25 for the Fairness and SARCOS datasets. The Dirichlet parameter $\\alpha$ in Alg. 1 is set to 0.2 for all experiments, as we empirically found this value to perform well across datasets and tasks. We provide a more detailed analysis on the selection of the HN’s hidden dimension and hyperparameter $\\alpha$ in Appendix B.5. Extensive experimental details are provided in Appendix A. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter tuning: For PHN, we select hyperparameters based on the HV computed on a validation set. Selecting hyperparameters for the baselines is challenging because there are no clear criteria that can be computed quickly with many rays. Selecting HPs based on HV requires to train each baseline multiple times on all rays. Therefore, we select hyperparameters based on a single ray and apply the selected hyperparameters to all rays. Specifically, we collect all models trained using all hyperparameter configurations, and remove dominated solutions. Finally, we select the combination of HPs with the highest uniformity. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 ILLUSTRATIVE EXAMPLE ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate PHN on an illustrative MOO problem whose Pareto front is known (Fonseca, 1995). Here, ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d756dbd012ba94e11ecb00bb5021ad0b37faa1305a4bce87de30e62b3740cc13.jpg",
+        "text": "$$\n\\ell _ { 1 } ( \\theta ) = 1 - \\exp \\big \\{ - \\| \\theta - { \\bf 1 } / \\sqrt { d } \\| _ { 2 } ^ { 2 } \\big \\} , \\quad \\ell _ { 2 } ( \\theta ) = 1 - \\exp \\big \\{ - \\| \\theta + { \\bf 1 } / \\sqrt { d } \\| _ { 2 } ^ { 2 } \\big \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/91c8e0e553d1fd5f3deedf096c4055c7e7fb107e3ad7b45a6a7f3a8053293f5a.jpg",
+        "table_caption": [
+            "Table 1: Evaluation of the different methods on three Multi-MNIST datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Multi-Fashion+MNIST</td><td colspan=\"2\">Multi-Fashion</td><td colspan=\"2\">Multi-MNIST</td><td></td></tr><tr><td></td><td>HV个</td><td>Unif.介</td><td>HV介</td><td>Unif.介</td><td>HV个</td><td>Unif.介</td><td>Run-time (min.,Tesla V100)</td></tr><tr><td>LS</td><td>2.70</td><td>0.849</td><td>2.14</td><td>0.835</td><td>2.85</td><td>0.846</td><td>9.0×5=45</td></tr><tr><td>CPMTL</td><td>2.76</td><td>1</td><td>2.16</td><td>-</td><td>2.88</td><td>-</td><td>10.2 × 5=51</td></tr><tr><td>PMTL</td><td>2.67</td><td>0.776</td><td>2.13</td><td>0.192</td><td>2.86</td><td>0.793</td><td>17.0 ×5=85</td></tr><tr><td>EPO</td><td>2.67</td><td>0.892</td><td>2.15</td><td>0.906</td><td>2.85</td><td>0.918</td><td>23.6× 5=118</td></tr><tr><td>PHN-LS (ours)</td><td>2.75</td><td>0.894</td><td>2.19</td><td>0.900</td><td>2.90</td><td>0.901</td><td>12</td></tr><tr><td>PHN-EPO (ours)</td><td>2.78</td><td>0.952</td><td>2.19</td><td>0.921</td><td>2.78</td><td>0.920</td><td>27</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with parameters $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { d }$ and $d = 1 0 0$ . This problem has a non-convex Pareto front in $\\mathbb { R } ^ { 2 }$ . We compared PHN with EPO, PMTL, MGDA, and two genetic algorithms (GAs) NSGA-III and MOEA/D. The results are presented in Figure 1. Linear scalarization (LS) fails to generate solutions in the concave part of the Pareto front, as theoretically shown in Boyd et al. (2004). PMTL and EPO generate solutions along the entire PF, but the solutions generated by PMTL are only optimized to be in the vicinity of the preference vector $\\pmb { r }$ . Results for MGDA and MOEA/D algorithms are shown in Appendix B.1. Our PHN-EPO generates a continuous cover of the PF in a single run. In addition, PHN-EPO returns exact Pareto optimal solutions, mapping an input preference to its corresponding point on the front. We note that similar to the EPO method, PHN-EPO can learn non-convex Pareto fronts. Additional synthetic experiments with non-convex, known Pareto fronts are presented in Appendix B.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            452,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate all methods on three MOO benchmark datasets: (1) Multi-MNIST (Sabour et al., 2017); (2) Multi-Fashion, and (3) Multi-Fashion $^ +$ MNIST. In each dataset, two instances are sampled uniformly at random from the MNIST (LeCun et al., 1998) or Fashion-MNIST (Xiao et al., 2017) datasets. The two images are merged into a new image by placing one at the top-left corner and the other at the bottom-right corner. The sampled images can be shifted up to four pixels in each direction. Each dataset consists of 120,000 training examples and 20,000 test set examples. We allocate $10 \\%$ of each training set for constructing validation sets. We train 5 models for each baseline, using evenly spaced rays. The results are presented in Figure 3 and Table 1. The HV was calculated with reference point $( 2 , 2 )$ . PHN outperforms all other methods in terms of HV and uniformity, with significant improvement in run time. To better understand the behavior of PHN while varying the input preference vector, we visualize the per-pixel prediction contribution in Appendix B.3. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 FAIRNESS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/57f9a4447d18ad70f85a2aac96e19919d9180628cfe9c78d9e27ebde8ba946fc.jpg",
+        "image_caption": [
+            "Figure 4: Accuracy-Fairness trade-off : The trade-off between the main classification loss and the two fairness losses, on the test sets of the three fairness datasets. Each line corresponds to a different solution along the Pareto front, generated using PHN-EPO. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fairness has become a popular topic in machine learning in recent years (Mehrabi et al., 2019), and numerous approaches have been proposed for modeling and incorporating fairness into ML systems (Dwork et al., 2018; Kamishima et al., 2012; Zafar et al., 2019). Here, we adopt the approach of Bechavod & Ligett (2017). They proposed a 3-dimensional optimization problem, with a classification objective and two fairness objectives: False Positive (FP) fairness, and False Negative (FN) fairness. We compare PHN with EPO, PMTL, and LS on three commonly used fairness datasets: Adult (Dua & Graff, 2017), Default (Yeh & Lien, 2009) and Bank (Moro et al., 2014). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b44941f5f9a3bc089c7162ba8d7968966c7719cc132a583f1679973f2f8223c5.jpg",
+        "table_caption": [
+            "Table 2: Comparison on three Fairness datasets. Run-time is evaluated on the Adult dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Adult</td><td colspan=\"2\">Default</td><td colspan=\"2\">Bank</td></tr><tr><td></td><td>HV个 Unif.介</td><td></td><td>HV个Unif.个</td><td></td><td>HV个Unif.介</td><td>Run-time (min., Tesla V100)</td></tr><tr><td>LS</td><td>0.628</td><td>0.109</td><td>0.522</td><td>0.069</td><td>0.704 0.215</td><td>0.9 × 10=9.0</td></tr><tr><td>PMTL</td><td>0.614</td><td>0.458</td><td>0.548</td><td>0.471</td><td>0.638 0.497</td><td>2.7 × 10= 26.6</td></tr><tr><td>EPO</td><td>0.608</td><td>0.734</td><td>0.537</td><td>0.533</td><td>0.713 0.881</td><td>1.6 × 10= 15.5</td></tr><tr><td>PHN-LS (ours)</td><td>0.658</td><td>0.289</td><td>0.551</td><td>0.108</td><td>0.730 0.615</td><td>1.1</td></tr><tr><td>PHN-EPO (ours)</td><td>0.648</td><td>0.701</td><td>0.548</td><td>0.359</td><td>0.748 0.821</td><td>1.8</td></tr></table>",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "text",
+        "text": "Each dataset is divided into train/validation/test sets of sizes $7 0 \\% / 1 0 \\% / 2 0 \\%$ respectively. All baselines are trained and evaluated on 10 preference rays. We present the results in Table 2. HV calculated with reference point $( 1 , 1 , 1 )$ . PHN achieves the best HV performance across all datasets, with reduced training time compared to baselines. EPO performs best in terms of uniformity, however, recall it is evaluated on its training rays. We visualize the accuracy-fairness trade-off achieved by PHN in Figure 4. We also tested GAs on this task but the runtime was extremely high (two days, compared with a runtime of minutes for gradient-based methods) with poor performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 PIXEL-WISE CLASSIFICATION AND REGRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Image segmentation and depth estimation are key problems in computer vision with various applications (Minaee et al., 2020; Bhoi, 2019). We compare PHN to the other methods on the NYUv2 dataset (Silberman et al., 2012). NYUv2 is a challenging indoor scene dataset that consists of 1449 RGBD images and dense per-pixel labeling. We use this dataset as a multitask learning benchmark for semantic segmentation and depth estimation tasks. We split the dataset into train/validation/test sets of sizes $7 0 \\% / 1 0 \\% / 2 0 \\%$ respectively, and use an ENet-based architecture for the target network (Paszke et al., 2016). Each baseline method is trained five times using five evenly spaced rays. The results are presented in Table 3. HV is calculated with reference point $( 3 , 3 )$ . PHN-EPO achieves the best HV and uniformity, while also being faster than the baselines. ",
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+    {
+        "type": "table",
+        "img_path": "images/065ac94d4ea39d4f795cd9c37e8badf3ddab1ee00fb999469e5f078f9a880d93.jpg",
+        "table_caption": [
+            "Table 3: Comparison on NYUv2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">NYUv2</td></tr><tr><td></td><td>HV介</td><td>Unif.介</td><td>Run-time (hours,Tesla V100)</td></tr><tr><td>LS</td><td>3.550</td><td>0.666</td><td>0.58 × 5=2.92</td></tr><tr><td>PMTL</td><td>3.554</td><td>0.679</td><td>0.96× 5=4.79</td></tr><tr><td>CPMTL</td><td>3.570</td><td>-</td><td>0.71 × 5=3.55</td></tr><tr><td>EPO</td><td>3.266</td><td>0.728</td><td>1.02 × 5=5.11</td></tr><tr><td>PHN-LS (ours)</td><td>3.546</td><td>0.798</td><td>0.67</td></tr><tr><td>PHN-EPO (ours)</td><td>3.589</td><td>0.820</td><td>1.04</td></tr></table>",
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+        "table_caption": [
+            "Table 4: Comparison on SARCOS dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">SARCOS</td></tr><tr><td></td><td>HV个 Unif.介</td><td>Run-time (hours,Tesla V100)</td></tr><tr><td>LS</td><td>0.688</td><td>-0.008</td><td>0.17 × 20= 3.3</td></tr><tr><td>EPO</td><td>0.637</td><td>0.548</td><td>0.5× 20=10</td></tr><tr><td>PHN-LS (ours)</td><td>0.693</td><td>0.020</td><td>0.2</td></tr><tr><td>PHN-EPO (ours)</td><td>0.728</td><td>0.258</td><td>0.8</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5.5 MULTITASK REGRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When learning with many objectives using preference-specific models, generating a good coverage of the solution space becomes too expensive in computation and storage space, as the number of rays needed grows with the number of dimensions. As a result, PHN provides a major improvement in training time and solution space coverage (measured by HV). To evaluate PHN in such settings, we use the SARCOS dataset (Vijayakumar), a commonly used dataset for multitask regression (Zhang & Yang, 2017). SARCOS presents an inverse dynamics problem with 7 regression tasks and 21 explanatory variables. We omit PMTL from this experiment because its training time is significantly longer compared to other baselines, with relatively low performance. For LS and EPO, we train 20 preference-specific models. The results are presented in Table 4. HV calculated with reference point $( 1 , . . . , 1 )$ . As before, PHN performs best in terms of HV, while reducing the overall runtime. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 THE QUALITY-RUNTIME TRADE-OFF ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "While PHN learns the entire front in a single model, the competing methods need to train multiple models to cover the Pareto front. As a result, these methods have a clear trade-off between their performance and their run time. To better understand this trade-off, Figure 5 shows the HV and run time (in log-scale) of LS and EPO when training a various number of models. For Fashion-MNIST, we trained 25 models, and at the inference time, we selected subsets of various sizes (1 ray, 2 rays, etc) and computed their HV. The shaded area in Figure 5 reflects the variance over different selections of ray subsets. We similarly trained 30 EPO/LS models for Adult and 40 for SARCOS. PHN-EPO achieves superior or comparable hypervolume while being an order of magnitude faster. PHN-LS is even faster, but this may come at the expense of a lower hypervolume. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To further understand the source of PHN performance gain, we evaluate the HV on the same rays used for the competing methods (see Appendix B.4). In many cases, PHN outperforms models trained on specific rays, when evaluated on those exact rays. We attribute the performance gain to the inductive bias effect that PHN induces on a model by sharing weights across rays. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb4aac8bd4569ca4694a08f7a28530ef9762f887c5a1034e4cad1b07c9e185b6.jpg",
+        "image_caption": [
+            "Figure 5: HV vs runtime (min.) comparing PHN with preference-specific LS and EPO. For LS and EPO, each value was computed by evaluating with a subset of models. PHN achieves higher HV significantly faster ( $\\mathbf { \\dot { X } }$ -axis is in log scale). Shaded area marks the variance across selected subsets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We put forward a novel MOO setup which we term Pareto Front Learning (PFL). Learning the Pareto front using a single unified model that can be applied to a specific objective preference at inference time. We propose PHN, a model for this setup based on a hypernetwork, and evaluate it on a large and diverse set of tasks and datasets. Our experiments show significant gains in both runtime and performance on all datasets. This novel approach provides users with the flexibility of selecting the operating point at inference time and opens new possibilities where user preferences can be tuned and changed at deployment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
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+        "type": "text",
+        "text": "This study was funded by a grant to GC from the Israel Science Foundation (ISF 737/2018), and by an equipment grant to GC and Bar-Ilan University from the Israel Science Foundation (ISF 2332/18). AS and AN were funded by a grant from the Israeli Innovation Authority, through the AVATAR consortium. ",
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+        "text": "Appendix: Learning the Pareto Front with Hypernetworks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL DETAILS ",
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "text",
+        "text": "Pixelwise Classification and Regression: For NYUv2 we performed a hyperparameters search over the number of epochs and learning rates $\\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \\}$ . We train each method for 150 epochs with Adam (Kingma & Ba, 2015) optimizer and select an optimal set of learning rate and number of epochs according to Sec. 5. We evaluate PHN over 25 evenly spaced rays. ",
+        "bbox": [
+            174,
+            175,
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+            232
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Image classification: For the Multi-MNIST, experiments we use a LeNet-based target network (LeCun et al., 1998) with two task-specific heads. We train all methods using an Adam optimizer with learning rate $1 e - 4$ for 150 epochs and batch size 256. We then choose the best number of epochs for each method using the validation set. We evaluate PHN over 25 rays. ",
+        "bbox": [
+            174,
+            248,
+            825,
+            305
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Fairness: We use a 3−layer feed-forward target network with hidden dimensions [40, 20]. We use learnable embeddings for all categorical variables. We train all methods for 35 epochs using an Adam optimizer with learning rates $\\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \\}$ . We choose the best hyperparameter configuration using the validation set. Further details on the datasets are provided in Table 5. We evaluate PHN over a sample of 150 rays. ",
+        "bbox": [
+            174,
+            321,
+            825,
+            392
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Multitask regression: For the SARCOS experiment, we use a $4 -$ layer feed-forward target network with 256 hidden dimension. All categorical features are passed through embedding layers. We train all methods for 1000 epochs using an Adam optimizer and learning rates $\\{ 1 e - 3 , 5 e - 4 , 1 e - 4 \\}$ . We choose the best hyperparameter configuration base on the validation set. We use 40,036 training examples, 4,448 validation examples, and 4,449 test examples. We evaluate PHN over a sample of 100 rays. ",
+        "bbox": [
+            174,
+            410,
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+            493
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/17b5e6952c43d5289b5e9b280d1a727bca7d8d7e6aafc44e3184637594b6b68d.jpg",
+        "table_caption": [
+            "Table 5: Fairness datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Adult Default</td><td>Bank</td></tr><tr><td>Train</td><td>32559</td><td>21600 29655</td></tr><tr><td>Validation</td><td>3618</td><td>2400 3295</td></tr><tr><td>Test</td><td>9045</td><td>6000 8238</td></tr></table>",
+        "bbox": [
+            387,
+            535,
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+            602
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            633,
+            442,
+            648
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/b10e7f01b26106c431e2665ebdbaed0a37efee267dddb22da0b52ca32df0edf7.jpg",
+        "image_caption": [
+            "Figure 6: Illustrative example using the popular task of Fonseca (1995): The genetic algorithm MOEA/D fails to reach the exact pareto front after 100, 000 generations. MGDA reaches Pareto optimal solutions, but is not suitable for generating preference-specific solutions and only covers a small fraction of the pareto front. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            667,
+            705,
+            843
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/f595400e3ec7c4d6599c327849c4551608426b405a028bf07ec814aa8683bbea.jpg",
+        "image_caption": [
+            "Figure 7: Known Pareto front: Top: Evaluation on Problem 2 from Evtushenko & Posypkin (2013). Bottom: Evaluation on ZDT2 (Zitzler et al., 2000). PHN converges to the entire PF, mapping a ray to its corresponding solution on the front. Each circle in the PHN plots denotes a solution corresponds to the baseline rays. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            95,
+            805,
+            396
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.1 KNOWN PARETO FRONT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            491,
+            385,
+            506
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We provide two additional experiments over popular MOO benchmarks whose Pareto front is known. In addition in Figure 6, we provide additional results for MGDA (Sener & Koltun, 2018) and MOEA/D (Zhang & Li, 2007) on the illustrative example from Section 5.1. ",
+        "bbox": [
+            173,
+            517,
+            826,
+            560
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We evaluate PHN on two MOO problems with known, non-convex Pareto front: ZDT2 (Zitzler et al., 2000) and problem two from Evtushenko & Posypkin (2013), henceforth Problem2. In both problems, the density of solutions across the Pareto-optimal region is non-uniform. ",
+        "bbox": [
+            174,
+            566,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For a solution $\\theta = ( \\theta _ { 1 } , \\theta _ { 2 } ) \\in [ 0 , 1 ] ^ { 2 }$ , Problem2 is given by, ",
+        "bbox": [
+            173,
+            614,
+            560,
+            631
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e770dc7692264f67c4d73e17833bf8c430eea43bd6a63efc5e34fafdb295e1f9.jpg",
+        "text": "$$\n\\ell _ { 1 } ( \\theta ) = ( ( \\theta _ { 1 } - 1 ) \\theta _ { 2 } ^ { 2 } + 1 ) / 3 , \\quad \\ell _ { 2 } ( \\theta ) = \\theta _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            637,
+            642,
+            656
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "and ZDT2 is given by, ",
+        "bbox": [
+            174,
+            662,
+            320,
+            678
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee36d32e90d18ff476ed732656486603c09c41873a5d0daa259487e0a8a8cfee.jpg",
+        "text": "$$\n\\ell _ { 1 } ( \\theta ) = \\theta _ { 1 } , \\quad \\ell _ { 2 } ( \\theta ) = 1 - \\left( \\frac { \\theta _ { 1 } } { 1 + 9 \\theta _ { 2 } } \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            676,
+            637,
+            713
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The results are presented in Figure 7. PHN-EPO converges to the entire PF using a single HN model.   \nin contrast with PHN-EPO, EPO fails to converge to the PF for all rays in ZDT2. ",
+        "bbox": [
+            174,
+            715,
+            825,
+            744
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2 GENERALIZATION TO UNSEEN RAYS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            761,
+            470,
+            776
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "When the number of objectives is large, the curse of dimensionality makes it likely that rays encountered in test time are \"far\" from rays encountered during training. Although we evaluated PHN on unseen rays1 through all experiments, here we would like to better examine the generalization of PHN in that scenario. We conducted controlled experiment on SARCOS dataset with 7 tasks. During training, we sample rays from a predefined grid on the 7−dimensional simplex. We use the structured approach of Das & Dennis (1998) for generating well-spaced points. During inference, we measure HV and Uniformity on two sets of rays: (i) Randomly sampled train rays (ii) Randomly sampled test rays from a shifted grid, so they are most distant from the train grid. In both cases, test samples differ from training samples. We find that testing with unseen test rays only causes a minor decrease in both HV and Uniformity compared with train rays. The HV is reduced from 0.7337 to 0.7336 and uniformity is reduced from 0.163 to 0.158. This experiment indicates that PHN generalizes well to unseen rays, even in problems with many objectives. We note that the HV results presented here are higher than the ones reported in Section 5.5. We attribute this inconsistency to change of preference vectors distribution used during training. While throught the paper we sample preference rays from a Dirichlet distribution with $\\alpha = 0 . 2$ , here we sample from a fixed, evenly spaced grid. We provide analysis on the robustness to the choice of the Dirichlet distribution parameter $\\alpha$ in Appendix B.5. ",
+        "bbox": [
+            173,
+            786,
+            825,
+            900
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.3 MNIST PHN INTERPRETATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            231,
+            436,
+            246
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/7034159f14a61aae45d8447e637a74d8506c08f8944c8ea8968877eaaf2a5c31.jpg",
+        "image_caption": [
+            "Figure 8: PHN interpretation: PHN is provided with a sample from the Multi-MNIST dataset, where an image of 3 and an image 7 are overlayed. We show the effect that conditioning on different preference rays (left) has on the decision made by the model. Red pixels denote increased contribution to the prediction. See text for details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            255,
+            823,
+            372
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To better understand the behaviour of PHN while varying the input preference vector, Figure 8 shows the per-pixel contribution to prediction. We use the SHAP framework (Lundberg & Lee, 2017) for visualization. Red pixels denote an increased probability for a given class, while blue pixels denote a decreased probability. The top left and bottom right digits are labeled as $\\{ 7 , 3 \\}$ respectively. In the first row (ray: (0.99, 0.01)), the model is focused on task 1 (with label 7), most of the red pixels come from the top left digit in column 7, as expected. Similarly, in the third row (ray: (0.01, 0.99)) most of the red pixels are focused on the bottom right digit in column 3. ",
+        "bbox": [
+            173,
+            463,
+            825,
+            561
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.4 SAME-RAY EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            577,
+            387,
+            592
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Section 5 of the main text we evaluate PHN on multiple evenly spaced rays. For completeness, we report here the HV and Uniformity achieved by PHN, while evaluated on the same rays used for training and evaluating the baseline methods. The results are provided in Table 6. Surprisingly, in many cases PHN achieves better performance. We attribute this to the inductive bias effect of sharing weights across all preference rays. ",
+        "bbox": [
+            173,
+            603,
+            825,
+            672
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/e3d2f243ea07742c6fb81102e9d9f5906a23195333a9b5ed6e01e2c06ae45259.jpg",
+        "table_caption": [
+            "Table 6: Evaluation of PHN on baselines’ ray. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Fashion+MNIST</td><td colspan=\"2\">Multi-Fashion</td><td colspan=\"2\">Multi-MNIST</td><td colspan=\"2\">Adult</td><td colspan=\"2\">Bank</td><td colspan=\"2\">NYUv2</td><td colspan=\"2\">SARCOS HV</td></tr><tr><td>HV</td><td>Unif.</td><td>HV</td><td>Unif.</td><td></td><td>HV Unif.</td><td>HV</td><td>Unif.</td><td>HVUnif.</td><td></td><td>HVUnif.</td><td>HV</td><td>Unif.</td><td>Unif.</td></tr><tr><td>PHN-LS 2.740</td><td></td><td>0.852</td><td>2.190</td><td>0.838</td><td>2.900</td><td>0.840</td><td>0.624 0.282</td><td></td><td>0.538 0.106</td><td></td><td>0.695 0.603</td><td>3.494 0.683</td><td></td><td>0.693-0.051</td></tr><tr><td>PHN-EPO 2.760</td><td></td><td>0.908</td><td>2.180</td><td>0.860</td><td>2.760</td><td>0.857</td><td>0.631 0.742</td><td></td><td>0.531 0.395</td><td></td><td>0.717 0.833</td><td>3.565 0.701</td><td></td><td>0.728 0.190</td></tr></table>",
+        "bbox": [
+            174,
+            710,
+            825,
+            765
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.5 DESIGN CHOICE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            786,
+            406,
+            800
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Section 5 we used fixed values of the hidden dimension and the Dirichlet parameter $\\alpha$ (see Alg. 1). Here we examine the effect of different choices for these hyperparameters. We use the Multi-Fashion $+ \\mathrm { M N I S T }$ dataset, and the PHN-EPO variant of our method. We first fix the hidden dimension to 100, and vary $\\alpha$ from 0.1 to 1 (uniform sampling). The results are presented in Figure 9(a). It appears that at least for this dataset, PHN is quite robust to changes in $\\alpha$ . Next, we set $\\alpha = 0 . 2$ and examine the effect of changing the dimension of the hidden layers in our hypernetwork. We present the results in Figure 9(b). Here, the effect on the HV is stronger, with best HV achieved using hidden dimension of 100. ",
+        "bbox": [
+            173,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/94100b9d94d1764688f570cb9f61d97dd18fe84c7bce14ead2fd8ab9b6a85f01.jpg",
+        "image_caption": [
+            "Figure 9: Hyperparameters analysis: Performance analysis on the Multi-Fashion $^ +$ MNIST dataset, for different architecture designs and sampling choices. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            97,
+            709,
+            255
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "B.6 ADDITIONAL RUNTIME ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            323,
+            455,
+            337
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/3fe31436c1fbae5104d452d03147d996e5cdf42ade1715b2ae2e4d5500f89346.jpg",
+        "image_caption": [
+            "Figure 10: Runtime analysis for NYUv2, and two additional Multi-MNIST dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            348,
+            816,
+            502
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We extend the runtime analysis from Section 5.5. Figure 10 provides additional runtime analysis on the Multi-MNIST, Multi-Fashion, and NYUv2 datasets. For NYUv2 we omit the EPO method, since it was outperformed by LS in both HV and runtime. ",
+        "bbox": [
+            173,
+            561,
+            825,
+            604
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C HYPERVOLUME ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            623,
+            339,
+            640
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d7857e5808b5a11cc623429a5700fdbf7904e1be3714526737d64df67dd26e4.jpg",
+        "image_caption": [
+            "Figure 11: Hypervolume illustration: The colored area represents the hypervolume obtained by Pareto set $\\{ P _ { 1 } , P _ { 2 } , P _ { 3 } \\}$ with respect to the reference point. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            398,
+            660,
+            589,
+            810
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The hypervolume metric is defined as follows: Given a set of points $S \\subset \\mathbb { R } ^ { n }$ and a reference point $\\pmb { \\rho } \\in \\mathbb { R } _ { + } ^ { n }$ , the hypervolume of $S$ is measured by the region of non-dominated points bounded above by $\\rho$ . ",
+        "bbox": [
+            173,
+            867,
+            825,
+            907
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/078b502a9dc82b7996a6127b31c757bd6c929d3b88511f634bbdd160028d2f6f.jpg",
+        "text": "$$\nH V ( S ) = V O L \\big ( \\{ q \\in \\mathbb { R } _ { + } ^ { n } \\mid \\exists p \\in S : ( p \\preceq q ) \\land ( q \\preceq \\rho ) \\} \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            906,
+            696,
+            926
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where $V O L$ is the euclidean volume. We can interpret hypervolume as the measure of the union of the boxes created by the non-dominated points. ",
+        "bbox": [
+            169,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ad36e082a9c00be21ef6ba7c338770ebcb6defc15e8b825fa195a9c529158a8.jpg",
+        "text": "$$\nH V ( S ) = V O L \\left( \\bigcup _ { p \\in S } \\prod _ { i = 1 } ^ { n } [ p _ { i } , \\rho _ { i } ] \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            137,
+            619,
+            204
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "As a measure of quality for MOO solutions, it takes into account both the quality of individual solutions, which is the volume they dominate, and also the diversity, measured in the overlap between dominated regions. See example in Fig. 11. ",
+        "bbox": [
+            173,
+            217,
+            826,
+            260
+        ],
+        "page_idx": 17
+    }
+]

parse/train/S1ewjhEFwr/S1ewjhEFwr.md

@@ -0,0 +1,357 @@
+# STORAGE EFFICIENT AND DYNAMIC FLEXIBLE RUNTIME CHANNEL PRUNING VIA DEEP REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this paper, we propose a deep reinforcement learning (DRL) based framework to efficiently perform runtime channel pruning on convolutional neural networks (CNNs). Our DRL-based framework aims to learn a pruning strategy to determine how many and which channels to be pruned in each convolutional layer, depending on each specific input instance at runtime. The learned policy optimizes the performance of the network by restricting the computational resource on layers under an overall computation budget. Furthermore, unlike other runtime pruning methods which require to store all channels parameters for inference, our framework can reduce parameters storage consumption for deployment by introducing a static pruning component. Comparison experimental results with existing runtime and static pruning methods on state-of-the-art CNNs demonstrate that our proposed framework is able to provide a tradeoff between dynamic flexibility and storage efficiency in runtime channel pruning.
+
+# 1 INTRODUCTION
+
+In recent years, convolutional neural networks (CNNs) have been proven to be effective in a wide range of computer vision tasks, such as image classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), objection detection (He et al., 2017; Zhou et al., 2019; Law & Deng, 2018), segmentation (He et al., 2017; Zhu et al., 2019). Therefore, nowadays, many computervision-based systems, such as automatic-driving cars, security surveillance cameras, and robotics, are built on the power of CNNs. However, since most state-of-the-art CNNs require expensive computation power for inference and huge storage space to store large amount of parameters, the limitation of energy, computation and storage on mobile or edge devices has become the major bottleneck on real-world deployments of CNNs. Existing studies have been focused on speeding up the execution of CNNs for inference on edge devices by model compression using matrix decomposition (Denil et al., 2013; Masana et al., 2017), network quantization (Courbariaux et al., 2016), network pruning (Dong et al., 2017), etc. Among these approaches, channel pruning has shown promising performance (He et al., 2017; Luo et al., 2017; Zhuang et al., 2018; Peng et al., 2019). Specifically, channel pruning discards an entire input or output channel and keep the rest of the model with structures.
+
+Most channel pruning approaches can be categorized into two types: runtime approaches and static approaches. Static channel pruning approaches aim to design a measurement to evaluate the importance of each channel over the whole training dataset and remove the least important channels to minimize the loss of performance after pruning. By permanently pruning a number of channels, the computation and storage cost of CNNs can be dramatically reduced when being deployed, and the inference execution can be accelerated consequently. Runtime channel pruning approaches have been recently proposed to achieve dynamic channel pruning on each specific instance (Gao et al., 2019; Luo & Wu, 2018). To be specific, the goal of runtime approaches aims to evaluate the channel importance at runtime, which is assumed to be different on different input instances. By pruning channels dynamically, different pruned structures can be considered as different routing of data stream inside CNNs. This kind of approaches is able to significantly improve the representation capability of CNNs, and thus achieve better performance in terms of prediction accuracy compared with static approaches. However, previous runtime approaches trade storage cost off dynamic flexibility of pruning. To achieve dynamic pruning on different specific instances, all parameters of kernels are required to be stored (or even more parameters are introduced). This makes runtime approaches not applicable on resource-limited edge devices. Moreover, most of previous runtime approaches only evaluate the importance among channels in each single layer independently, without considering the difference in efficiency among layers.
+
+In this paper, to address the aforementioned issues of runtime channel pruning approaches, we propose a deep reinforcement learning (DRL) based pruning framework. Basically, we aim to apply DRL to prune CNNs by maximizing received rewards, which are designed to satisfy the overall budget constraints along side with network’s training accuracy. Note that automatic channel pruning by DRL is a difficult task because the action space is usually very huge. Specifically, the discrete action space for the DRL agent is as large as the number of channels at each layer, and the action spaces may vary among layers since there are different numbers of channels in different layers. To facilitate pruning CNNs by DRL, for each layer, we first design a novel prediction component to estimate the importance of channels, and then develop a DRL-based component to learn the sparsity ratio of the layer, i.e., how many channels should be pruned.
+
+More specifically, different from previous runtime channel pruning approaches, which only learn runtime importance of each channel, we propose to learn both runtime importance and additionally static importance for each channel. While runtime importance maintains the saliency of specific channels for each given specific input, the static importance captures the overall saliency of the corresponding channel among the whole dataset. According to each type of the channel importance, we further design different DRL agents (i.e., a runtime agent and a static agent) to learn a sparsity ratio in a layer-wise manner. The sparsity ratio learned by the runtime agent together with the estimated runtime importance of channels are used to generate runtime pruning structures, while the sparsity ratio learned by the static agent together with the estimated static importance of channels are used to generate static (permanent) pruning structures. By considering both the pruning structures, our framework is able to provide a trade-off between storage efficiency and dynamic flexibility for runtime channel pruning.
+
+In summary, our contributions are 2-fold. First, we propose to prune channels by taking both runtime and static information of the environment into consideration. Runtime information endows pruning with flexibility based on different input instances while static information reduces the number of parameters in deployment, leading to storage reduction, which cannot be achieved by conventional runtime pruning approaches. Second, we propose to use DRL to determine sparsity ratios, which is different from the previous pruning approaches which manually set sparsity ratios. Extensive experiments demonstrate the effectiveness of our method.
+
+# 2 RELATED WORK AND PRELIMINARY
+
+# 2.1 STRUCTURE PRUNING
+
+Wen et al. (2016) pioneered structure pruning in deep neural network by imposing the $L _ { 2 , 1 }$ norm in training. Under the same framework, Liu et al. (2017) regarded parameters in batch normalization as channel selection signal, which is minimized to achieve pruning during training. He et al. (2017) formulated channel pruning into a two-step iterative process including LASSO regression based channel selection and least square reconstruction. Luo et al. (2017) formulated channel pruning as minimization of difference of output features, which is solved by greedy selection. Zhuang et al. (2018) further considered early prediction, reconstruction loss and final loss to select importance channels. Overall, structure pruning methods accelerate inference by producing regular and compact model. However, this brought regularness requires preserving more parameters to ensure performance.
+
+# 2.2 DYNAMIC PRUNING
+
+Dynamic pruning provides different pruning strategies according to input data. Wang et al. (2018) proposed to reduce computation by skipping layers or channels based on the analysis of input features. Luo & Wu (2018) proposed to use layer input to learn channel importance, which is then binarized for pruning. Gao et al. (2019) applied the same framework while extended features selection in both input and output features. Similarly, Liu & Deng (2018) introduced multiple branches for runtime inference according to inputs. A gating module is learnt to guide the flow of feature maps. Bolukbasi et al. (2017) learned to choose the components of a deep network to be evaluated for each input adaptively. Early exit is introduced to accelerate computation. Dynamic pruning adaptively takes different actions for different inputs, which is able to accelerate the overall inference time. However, the original high-precision model needs to be stored, together with extra parameters for making specified pruning actions. Rosenbaum et al. (2018) proposed to learn routers to route layers output to different next layers, in order to adjust a network to multi-task learning.
+
+![](images/bc77b32ffc241caf859273ec99f771bfda79928ed2f13e13b67d0d9181ac8f0e.jpg)  
+Figure 1: Illustration of our proposed DRL-based runtime pruning framework.
+
+# 2.3 DEEP REINFORCEMENT LEARNING IN PRUNING
+
+Channel selection is on trial using deep reinforcement learning. Lin et al. (2017) trained a LSTM model to remember and provide channel pruning strategy for backbone CNN model, which is conducted using reinforcement learning techniques. He et al. (2018) proposed to determine the compression ratio in each layer by training an agent regarding the pruning-retraining process as an environment.
+
+# 2.4 PRELIMINARY
+
+Reinforcement Learning We consider a standard setup of reinforcement learning: an agent sequentially takes actions over a sequence of time steps in an environment, in order to maximize the cumulative reward (Sutton & Barto, 1998). This problem can be formulated as a Markov Decision Process (MDP) of a tuple $( S , \mathcal { A } , \mathcal { P } , R , \gamma )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is transition probabilities, $R : S \times \mathcal { A }  \mathbb { R }$ is the reward function, and $\gamma \in [ 0 , 1 )$ is the discount factor. The goal of reinforcement learning is to learn a policy $\pi ( a | s )$ that maximizes the objective of cumulative rewards over finite time steps,
+
+$$
+\operatorname* { m a x } _ { \pi } \sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) ,
+$$
+
+where $s _ { t } \in S$ and $a _ { t } \in \mathcal A$ are state and taken action at time step $t$ respectively.
+
+# 3 DRL-BASED RUNTIME PRUNING FRAMEWORK
+
+The overview of our proposed framework is presented in Fig. 1. To prune convolutional layer $t$ , we learn two types of learnable channel importance: runtime channel importance $\mathbf { u } _ { r } \in \mathbb { R } ^ { C \times \mathbf { \breve { 1 } } }$ and static channel importance $\mathbf { u } _ { s } \in \mathbb { R } ^ { C \times 1 }$ , where $C$ is the number of channels in layer $t$ . The runtime channel importance $\mathbf { u } _ { r }$ is generated by a subnetwork importance predictor $f ( \cdot )$ , which takes the input feature map $\mathbf { F } _ { i n }$ as input, while the static channel importance $\mathbf { u } _ { s }$ is randomly initialized and updated during training. Both $\mathbf { u } _ { r }$ and $\mathbf { u } _ { s }$ indicate the channel importance of the full precision output feature map $\mathbf { F } _ { o u t }$ through a convolution layer. Channels are selected to be pruned according to the values of each element in $\mathbf { u } _ { r }$ and $\mathbf { u } _ { s }$ , and how many channels to be selected is decided by the sparsity ratios $d _ { r }$ and $d _ { s }$ , respectively. To learn the sparsity ratios $d _ { r }$ and $d _ { s }$ , two DRL agents, the runtime agent and the static agent, are introduced, where actions $a _ { t } ^ { r }$ and $a _ { t } ^ { s }$ are defined to set values of $d _ { r }$ and $d _ { s }$ , respectively. The detail of the two DRL agents are described in Sec. 3.3. Consequently, a trade-off pruner $g ( \cdot )$ is performed to balance the runtime and static pruning results, and output a decision mask M of binary values (1/0) to indicate which channels to be pruned (1: pruned, 0: preserved), as well as a unified channel importance vector $\mathbf { u } \in \mathbb { R } ^ { C \times 1 }$ as follows,
+
+$$
+[ \mathbf { M } , \mathbf { u } ] = g ( \mathbf { u } _ { r } , \mathbf { u } _ { s } , d _ { r } , d _ { s } ) .
+$$
+
+The final output after pruning is constructed by multiplying the full precision output feature map $\mathbf { F } _ { o u t }$ , by ${ \bf 1 } - { \bf M }$ and u as,
+
+$$
+\hat { \mathbf { F } } _ { o u t } = \mathbf { F } _ { o u t } \otimes ( \mathbf { 1 } - \mathbf { M } ) \otimes \mathbf { u } ,
+$$
+
+where $\otimes$ is the broadcast element-wise multiplier, and 1 is the matrix of the same size as $\mathbf { M }$ with all the elements being 1. In the following, we introduce how to learn the runtime channel importance vector $\mathbf { u } _ { r }$ and the static channel importance vector $\mathbf { u } _ { s }$ in Sec. 3.1, how to construct the trade-off pruner $g ( \cdot )$ in Sec. 3.2, and how to design the two DRL agents in Sec. 3.3.
+
+# 3.1 LEARNABLE CHANNEL IMPORTANCE
+
+We consider that a convolutional layer takes input of feature map $\mathbf { F } _ { i n } \in \mathbb { R } ^ { C _ { i n } \times H _ { i n } \times W _ { i n } }$ and generates an output feature map $\mathbf { F } _ { o u t } \doteq \perp \operatorname { \textbf { \em E } } ^ { C _ { o u t } \times H _ { o u t } ^ { \star } \times W _ { o u t } }$ , where $C _ { * }$ , $H _ { * }$ and $W _ { * }$ are the number of channels, width and height of the feature map $\mathbf { F } _ { * }$ , respectively. Each element of the channel importance vectors $\mathbf { u } _ { r } \in \mathbb { R } ^ { C _ { o u t } }$ and $\mathbf { u } _ { s } \in \mathbb { R } ^ { C _ { o u t } }$ represents the importance value of the corresponding channel, respectively. In the following, we drop the subscript $o u t$ for simplicity in presentation.
+
+# 3.1.1 RUNTIME CHANNEL IMPORTANCE
+
+As mentioned above, the runtime channel importance $\mathbf { u } _ { r }$ of output feature $\mathbf { F } _ { o u t }$ is predicted by a importance predictor $f ( \cdot )$ , which takes $\mathbf { F } _ { i n }$ as input. Therefore, $\mathbf { u } _ { r }$ can be considered as a function of $\mathbf { F } _ { i n }$ , whose values vary over different input instances. In this paper, we design a subnetwork to approximate $f ( \cdot )$ , which is expected to be of a small size and computationally efficient. Similar to many existing dynamic network pruning methods (Gao et al., 2019; Hu et al., 2018; Luo & Wu, 2018), we use global pooling layer as the first layer in $f ( \cdot )$ , because global pooling is computationally efficient and it can reduce the dimension of $\mathbf { F } _ { i n }$ dramatically. We then feed the output of global pooling into a fully-connected layer without any activation function. The output of fully-connected layer is the runtime channel importance vector $\mathbf { u } _ { r }$ .
+
+Which channels to be preserved $/$ pruned at runtime are determined according to the values of $\mathbf { u } _ { r }$ . We denote by $\mathbf { M } _ { r } \in \{ 0 , 1 \} ^ { C }$ a mask for pruning, where if the value is 0, then the corresponding channel is preserved, otherwise pruned. For now, suppose a sparsity ratio $d _ { r }$ for runtime pruning has already been generated via the dynamic DRL agent, which will be introduced in Sec. 3.3. We then prune $\left( C - \lceil d _ { r } C \rceil \right)$ channels with the smallest importance values in $\mathbf { u } _ { r }$ . Accordingly, the value of an element in ${ { \bf { M } } _ { r } }$ is set to be 1 if the corresponding channel is pruned, otherwise 0.
+
+# 3.1.2 STATIC CHANNEL IMPORTANCE
+
+The static channel importance vector $\mathbf { u } _ { s }$ is to capture the global information for pruning, and thus is learned from the whole dataset. It is randomly generated and learned through backpropagation. Similar to runtime channel pruning, given a sparsity ratio $d _ { s }$ learned by the static DRL agent, $( C -$ $\lceil d _ { s } C \rceil )$ channels with smallest importance values in $\mathbf { u } _ { s }$ are pruned, and a mask $\mathbf { M } _ { s } \in \mathbf { \bar { \{ 0 , 1 \} } } ^ { C }$ is generated to indicate the static pruning results.
+
+# 3.2 TRADE-OFF PRUNER
+
+With the runtime and the static pruning decisions, ${ { \bf { M } } _ { r } }$ and $\mathbf { M } _ { s }$ , we now propose a trade-off pruner to generate a unified channel pruning decision. The main idea behind the trade-off pruner is to 1) prune those channels which are agreed to be pruned by both decisions, and 2) prune a portion of the rest channels by weighted votes from both decisions.
+
+To be specific, we define the mask representing channels pruned by both decisions as
+
+$$
+\mathbf { M } _ { o } = \mathbf { M } _ { s } \wedge \mathbf { M } _ { r } ,
+$$
+
+where $\wedge$ is element-wise logical AND and $1 / 0$ in mask represents logical true or false. The channels indicated to be pruned by $\mathbf { M } _ { o }$ (i.e., the corresponding values are 1) are pruned in final. The channels which are determined to be pruned by ${ { \bf { M } } _ { r } }$ but not by $\mathbf { M } _ { s }$ can be represented by a new mask $\overline { { \mathbf { M } } } _ { r } =$ ${ \bf M } _ { r } - { \bf M } _ { o }$ . Similarly, the channels which are determined to be pruned by ${ { \bf { M } } _ { r } }$ but not by $\mathbf { M } _ { s }$ can be represented by another new mask $\overline { { \mathbf { M } } } _ { s } = \mathbf { M } _ { s } - \mathbf { M } _ { o }$ .
+
+To control the trade-off between ${ { \bf { M } } _ { r } }$ and $\mathbf { M } _ { s }$ , we define a rate $R _ { r }$ denoting how much we trust the pruning decision made by ${ { \bf { M } } _ { r } }$ , while $1 - R _ { r }$ is for $\mathbf { M } _ { s }$ . That means the channels selected by $\overline { { \mathbf { M } } } _ { r }$ will be finally pruned with the rate $R _ { r }$ . Specifically, the number of channels which are selected by $\overline { { \mathbf { M } } } _ { r }$ and finally will be pruned is
+
+$$
+C _ { r } ^ { ' } = \lfloor R _ { r } ( \mathbf { 1 } ^ { \top } \overline { { \mathbf { M } } } _ { r } ) \rfloor ,
+$$
+
+where $\mathbf { 1 } ^ { \top } \overline { { \mathbf { M } } } _ { r }$ returns the number of channels selected by $\overline { { \mathbf { M } } } _ { r }$ . We then select the first $C _ { r } ^ { ' }$ -smallest important channels which are recommended to be pruned by $\overline { { \mathbf { M } } } _ { r }$ to form a mask $\widehat { \mathbf { M } } _ { r }$ . Similarly, for static pruning, we select the first $C _ { s } ^ { ' }$ -smallest important channels which are recommended to be pruned by $\overline { { \mathbf { M } } } _ { s }$ to form another mask $\widehat { \mathbf { M } } _ { s }$ , where $C _ { s } ^ { ' } = \lfloor ( 1 - R _ { r } ) ( \mathbf { 1 } ^ { \top } \overline { { \mathbf { M } } } _ { s } ) \rfloor$ .
+
+The final trade-off pruning mask is defined as
+
+$$
+\mathbf { M } = \mathbf { M } _ { o } + \widehat { \mathbf { M } } _ { r } + \widehat { \mathbf { M } } _ { s } .
+$$
+
+Moreover, in this work, the unified channel importance is simply defined as follows,
+
+$$
+\mathbf { u } = \mathbf { u } _ { r } \otimes \mathbf { u } _ { s } .
+$$
+
+With the trade-off pruning mask $\mathbf { M }$ and the unified channel importance $\mathbf { u }$ , the pruned output feature $\hat { \mathbf { F } } _ { o u t }$ can be generated by Eq. 2.
+
+# 3.3 DRL BASED PRUNING
+
+In this section, we present how to formulate the problems of learning the ratios $d _ { s }$ and $d _ { t }$ for static pruning and runtime pruning, as a MDP, and solve it via DRL, respectively.
+
+# 3.3.1 DRL FOR RUNTIME PRUNING
+
+In the MDP for runtime pruning, we consider the $t$ -th layer of the network as the $t$ -th timestamp.   
+The details of the MDP are listed as follows.
+
+State Given an input feature map $\mathbf { F } _ { i n }$ of layer $t$ , we pass it to a global pooling layer to reduce its dimension to $\mathbb { R } ^ { C _ { i n } }$ , where $C _ { i n }$ is the number of input channel of layer $t$ . Since $C _ { i n }$ varies among layers, we feed the output of global pooling to a layer-dependent encoder to project it to a fix-length vector $s _ { t } ^ { r }$ , which is considered as as a state representation of DRL in the context of runtime pruning.
+
+Action The action $a _ { t } ^ { r }$ is defined as the sparsity ratio at layer $t$ , alternating $d _ { r }$ in runtime pruning mentioned in Section 3.1.1. Existing DRL-base pruning method RNP (Lin et al., 2017) uses a unified discrete actions space with $k$ actions which are too coarse to achieve high accuracy. However, finegrained discrete action space as large as number of channels suffers from exploration difficulty. Therefore, instead of using discrete action spaces, we propose a continuous action space with action $a _ { t } ^ { r } \in ( 0 , 1 ]$ . To avoid over-pruning the filters and crashing in training, we set a minimum sparsity ratio $+ \alpha$ such that $a _ { t } ^ { r } \in ( + \alpha , 1 ]$ .
+
+Reward The reward function is proposed to consider both network accuracy and computation budget. We define the accuracy relative reward based on the loss of pruned backbone network,
+
+$$
+R _ { a c c } ^ { r } = - \mathcal { L } _ { C N N } ,
+$$
+
+where $\mathcal { L } _ { C N N }$ is the loss in CNN, and it may vary in scale among different training stage, i.e. large at beginning of training and small near convergence. To avoid the instability brought by the reward scale, $R _ { a c c } ^ { r }$ is normalized by a moving average,
+
+$$
+R _ { a c c } ^ { r ^ { \prime } } = R _ { a c c } ^ { r } / \beta _ { b } ,
+$$
+
+$$
+\beta _ { b } = \lambda \beta _ { b - 1 } + ( 1 - \lambda ) R _ { a c c } ^ { r } ,
+$$
+
+where $\beta _ { b }$ is the moving average at the $b$ -th training batch and $\lambda$ is the moving weight.
+
+To force computation of the pruned network under a given computation budget, we define a exponential reward function of budget regarding reward $R _ { b u d } ^ { r }$ :
+
+$$
+R _ { b u d } ^ { r } = \left\{ \begin{array} { l l } { \exp ( \alpha _ { 1 } ( B _ { c o m } - \overline { { B } } _ { c o m } ) ) - 1 , } & { B _ { c o m } > \overline { { B } } _ { c o m } , } \\ { 0 , } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+where $B _ { c o m }$ is the computation consumption, which is calculated based on the current of pruned strategy, and $\overline { { B } } _ { c o m }$ is the given computation budget constraint. Finally we sum up the two rewards to form sparse rewards, with being non-zero at terminated step $T$ and zeros at other time step $t < T$ ,
+
+$$
+R _ { t } ^ { r } = \left\{ \begin{array} { l l } { R _ { a c c } ^ { r ^ { \prime } } + R _ { b u d } ^ { r } , } & { t = T , } \\ { 0 , } & { t < T . } \end{array} \right.
+$$
+
+Actor-Critic Agent To solve the continuous action space problem, we choose a commonly used actor-critic agent with a Gaussian policy. Actor-critic agent consists of two components: 1) actor outputs the mean and variance to form a Gaussian policy where the continuous action are sampled from; 2) critic outputs a scalar predicting the future discounted accumulated reward and assists the policy training. Actor network and Critic network share one-layer RNN which takes state $s _ { t } ^ { r }$ as input. The output of RNN is fed into actor specific network constructed by two branches of fullyconnected layers, leading to the mean and variance of the Gaussian policy. The action is sampled for the Gaussian distribution outputed by the actor:
+
+$$
+a _ { t } ^ { r } \sim \mathcal { N } ( \mu ( s _ { t } ^ { r } ; \theta ^ { r } ) , \sigma ( s _ { t } ^ { r } ; \theta ^ { r } ) ) ,
+$$
+
+where $\mu ( s _ { t } ^ { r } ; \theta ^ { r } )$ and $\sigma ( s _ { t } ^ { r } ; \theta ^ { r } )$ is the mean and variance outputed from actor network. The Critic specific network has one fully-connected layer after the shared RNN, and outputs the predictive value $V ( s _ { t } ^ { r } ; \theta ^ { r } )$ .
+
+To optimize the actor-critic agent, Proximal Policy Optimization (PPO) (Schulman et al., 2017) is used. Note that we relax the action $a _ { t } ^ { r }$ to $( - \infty , + \infty )$ in PPO, and use truncate function to clip $a _ { t } ^ { r }$ in $( + \alpha , 1 ]$ when perform pruning.
+
+Besides, an additional regularizer is introduced to restrict the relaxed $a _ { t } ^ { r }$ staying in range $( + \alpha , 1 ]$
+
+$$
+\mathcal { L } _ { a } = \frac { 1 } { 2 } | | a _ { t } ^ { r } - \operatorname* { m a x } ( \operatorname* { m i n } ( a _ { t } ^ { r } , 1 ) , + \alpha ) | | _ { 2 } ^ { 2 } .
+$$
+
+# 3.3.2 DRL FOR STATIC PRUNING
+
+Similar to runtime pruning, the MDP in static pruning is also formulated layer-by-layer. The difference against runtime pruning is the definition of state and reward.
+
+State The state $s _ { t } ^ { s }$ in static pruning is defined as the full shape of $\mathbf { F } _ { o u t }$ , and does not depend on $\mathbf { F } _ { o u t }$ and the current input data.
+
+Action Action $a _ { t } ^ { s }$ is sampled from actor’s outputed Gaussian policy, and it is to alternate the sparsity $d _ { s }$ in static pruning mentioned in Section 3.1.2.
+
+Reward The reward function takes both network accuracy and parameters budget into consideration. The accuracy relative is defined as the same as that in runtime pruning:
+
+$$
+R _ { a c c } ^ { s } = R _ { a c c } ^ { r ^ { \prime } } .
+$$
+
+To reduce the number of parameters of network to satisfy the parameters storage budget, the parameters relative reward is defined in an exponential form as,
+
+$$
+R _ { p a r a m } ^ { s } = \left\{ \begin{array} { l l } { \exp ( \alpha _ { 2 } ( B _ { p a r a m } - \overline { { B } } _ { p a r a m } ) ) - 1 , } & { B _ { p a r a m } > \overline { { B } } _ { p a r a m } , } \\ { 0 , } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+where $B _ { p a r a m }$ is the number of preserved parameters after static pruning and $\overline { { B } } _ { p a r a m }$ is the parameters storage budget.
+
+Table 1: Comparison to state-of-the-art runtime pruning methods on CIFAR-10 at sparsity 0.5. Speed-up is calculated on MACs.   
+
+<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td><td>Speed-up</td><td>#Params</td></tr><tr><td>FBS (Gao et al., 2019)</td><td>91.37</td><td>89.88</td><td>-1.49</td><td>3.93×</td><td>1.11×</td></tr><tr><td>RNP (Lin et al., 2017)</td><td>92.07</td><td>84.93</td><td>-7.14</td><td>3.56×</td><td>1.00×</td></tr><tr><td>ours (runtime only Rr = 1)</td><td>92.07</td><td>91.333</td><td>-0.737</td><td>3.92×</td><td>1.31×</td></tr><tr><td>ours (Rr = 0.5)</td><td>92.07</td><td>91.066</td><td>-1.004</td><td>3.92×</td><td>0.78×</td></tr></table>
+
+<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td><td>Speed-up</td><td>#Params</td></tr><tr><td>FBS (Gao et al., 2019)</td><td>91.37</td><td>91.23</td><td>-0.14</td><td>2×</td><td>1.11×</td></tr><tr><td>ours (Rr = 1.0)</td><td>92.07</td><td>93.178</td><td>+1.108</td><td>1.99×</td><td>1.31×</td></tr><tr><td>ours (Rr = 0.5)</td><td>92.07</td><td>92.502</td><td>+0.432</td><td>1.99×</td><td>0.97×</td></tr></table>
+
+Table 2: Comparison to state-of-the-art runtime pruning methods on CIFAR-10 at sparsity 0.7.   
+Speed-up is calculated on MACs.
+
+Actor-Critic Agent This agent is similar to the one in runtime pruning. It has the same architecture as runtime pruning but differs in introducing a fully-connected layer as the encoder before RNN. This agent is also optimized by PPO.
+
+# 3.4 INFERENCE
+
+In inference, the static agent is not required any more because the static pruning strategy does not depend on individual input data points but the full shape of $\mathbf { F } _ { i n }$ . Therefore, the output action $a _ { t } ^ { s }$ is fixed to each layer $t$ . With the action $a _ { t } ^ { s }$ and the rate $R _ { r }$ , we can decide which filters can be pruned permanently. Specifically, channels with $( ( 1 - a _ { t } ^ { s } ) ( 1 - R _ { r } ) )$ -smallest static importance values are pruned permanently.
+
+# 4 EXPERIMENT
+
+We evaluate our DRL pruning framework on two popular datasets: CIFAR-10 (Krizhevsky, 2009) and ImageNet ILSVRC2012 (Russakovsky et al., 2015), to show the advantage over other channel pruning methods. We analyze the effect of hyper-parameters and different sparsity settings on CIFAR-10. For CIFAR-10, we use M-CifarNet (Zhao et al., 2018) as the backbone CNN. On ImageNet ILSVRC2012, ResNet-18 is used as the backbone CNN.
+
+# 4.1 IMPLEMENTATION DETAILS
+
+We start with a pretrained backbone CNN. Firstly we finetune the backbone CNN and train runtime importance predictor jointly, with sparsity $d _ { r } = 1$ and fixed all static pruning importance $\mathbf { u } _ { s }$ to 1. Then we remove the restriction on the static pruning importance $\mathbf { u } _ { s }$ , and train static pruning importance, the backbone CNN and the runtime importance predictor, with sparsity $d _ { s } = 1$ and runtime pruning sparsity fixed as $d _ { r } = 1$ . After finetuning, we use the DRL agents to predict the sparsity given computation and storage constraints. The DRL agents and the CNN with runtime/static importance are trained in alternating manner: We first fix the CNN as well as runtime/static importance and train two DRL agents, regarding the CNN as environments. Then we fix two agents and finetune the CNN and runtime/static importance. We repeat these two steps until convergence is achieved. We use Adam optimizer for both DRL agent and CNN, and set learning rate at $\overline { { 1 } } 0 ^ { - 6 }$ for the DRL agents. For CNN finetuning and runtime/static importance training, the learning rate is set to $1 0 ^ { - 3 }$ on CIFAR-10. On ImageNet ILSVRC2012, the learning rate starts from $1 0 ^ { - 3 }$ and is divided by 10 after 15 millions iterations.
+
+![](images/7149d698e379a247989cb4adf19d30fa5182186ecfe11294cd3330e87a35f835.jpg)  
+Figure 2: Trade-off between runtime pruning and static pruning at sparsity 0.45. X-axis is the rate $R _ { r }$
+
+![](images/785f2daf37a85fbdca7f2c6dbb534f6089b0380f10283987a3300f504ce7e5be.jpg)  
+Figure 3: Comparison accuracy drop for MCifarNet on CIFAR-10 with computational budget.
+
+<table><tr><td>Method</td><td>Baseline top-1 acc.</td><td>Top-1 acc.</td><td>△ top-1 acc.</td><td>Baseline top-5 acc.</td><td>Top-5 acc.</td><td>△ top-5 acc.</td><td>Speed-up</td></tr><tr><td>DCP (Zhuang et al., 2018)</td><td>69.64</td><td>67.35</td><td>-2.29</td><td>88.98</td><td>88.86</td><td>-0.12</td><td>1.71×</td></tr><tr><td>FPGM (He et al., 2019)</td><td>70.28</td><td>68.41</td><td>-1.87</td><td>89.63</td><td>88.48</td><td>-1.15</td><td>1.71×</td></tr><tr><td>Dynamic Sparse Graph (Liu et al.,2019)</td><td>69.48</td><td>64.8</td><td>-4.68</td><td></td><td>■</td><td></td><td>1.4×</td></tr><tr><td>CGNN (Hua et al.,2018)</td><td>69.02</td><td>67.95</td><td>-1.07</td><td>88.84</td><td>88.21</td><td>-0.63</td><td>1.63×</td></tr><tr><td>FBS (Gao et al.,2019)</td><td>70.71</td><td>68.17</td><td>-2.54</td><td>89.68</td><td>88.22</td><td>-1.46</td><td>1.98×</td></tr><tr><td>Ours (Rr = 0.5)</td><td>69.758</td><td>68.79</td><td>-0.968</td><td>89.078</td><td>88.534</td><td>-0.544</td><td>1.94×</td></tr></table>
+
+Table 3: Comparison with the state-of-the-art channel pruning ResNet-18 on ImageNet. Speed-up is calculated on MACs.
+
+# 4.2 EXPERIMENTAL RESULTS ON CIFAR-10
+
+We compare our proposed method with the following state-of-the-art runtime pruning methods: FBS (Gao et al., 2019), RNP (Lin et al., 2017) on CIFAR-10. The comparison results at sparsity 0.5 and 0.7 are shown in Table 1 and Table 2 respectively. Note that for fair comparison with other methods, the computation and storage budget constraints in our method is calculated according to the sparsity of other methods. Under these constraints, our method does not necessarily lead to the same sparsity as other methods in each layer. RNP cannot set exact sparsity ratio. Instead, its average sparsity ratio is accessible only during testing, which is 0.537 in Table 1. The result of FBS is reproduced using the released code1. The column #Params represents the number of parameters compared to the backbone CNN.
+
+Table 1 shows that our method outperforms other state-of-the-art methods, achieving highest accuracy at an overall sparsity ratio of 0.5. Our method has very close computation speed-up compared to FBS, but outperforms FBS around $0 . 4 8 \%$ to $0 . 7 6 \%$ . When the runtime pruning strategy is solely considered by setting $R _ { r } = 1$ , our method surpasses other comparison methods, indicating that our DRL-based framework improves the performance of channel runtime pruning. By balancing runtime and static pruning via setting $R _ { r } = 0 . 5$ , our method reduces the number of the overall stored parameters and achieves lower accuracy drop than other methods. Table 2 shows that our method outperforms FBS at sparsity of 0.7. When $R _ { r } = 0 . 5$ , our method achieves better performance than the baseline CNN with $2 \times$ speed-up and contains less parameters.
+
+We also study the relation between $R _ { r }$ and network compactness in our framework. Fig. 2 demonstrates the impact of $R _ { r }$ when sparsity is 0.45. The hyper-parameter $R _ { r }$ determines how much we trust about runtime pruning. With $R _ { r }$ close to 1, the accuracy becomes higher due to the more dynamic network flexibility but the space of the parameters storage also increases. When $R _ { r }$ diminishes, the network accuracy decreases but the parameter storage is reduced.
+
+Fig. 3 shows the performance of various sparsity ratios in our method. Again, our method does not prune with one single sparsity ratio for all layers, but uses the sparsity ratio to calculate computation and storage constraints, with which the sparsity ratio is learned for each layer. Fig. 3 demonstrates that our method holds the accuracy when sparsity is larger than about 0.5, which corresponds to about $4 \times$ computational acceleration.
+
+# 4.3 EXPERIMENTAL RESULTS ON IMAGENET ILSVRC2012
+
+We compare our method with state-of-the-art channel pruning methods on ImageNet ILSVRC2012 as shown in Table 3. In this experiment, we use ResNet-18 as the backbone CNN. Among the stateof-the-art pruning methods for comparison, FBS (Gao et al., 2019) and CGNN (Hua et al., 2018) are runtime pruning methods. The overall sparsity ratio of our method is 0.7, which is under the same setting of FBS. Our method with $R _ { r } = 0 . 5$ achieves the smallest top-1 accuracy drop compared with other methods, and also achieves the highest top-1 accuracy after pruning. Overall, our proposed method achieves comparable or better performance compared to other methods with more acceleration. Our method has very close MACs to FBS, while the number of preserved parameters is reduced to $8 1 . 2 \%$ of the baseline.
+
+# 5 CONCLUSION
+
+In this paper, we present a deep reinforcement learning based framework for deep neural network channel pruning in both runtime and static sheme. Specially, channels are pruned according to input feature as runtime pruning, and based on entire training dataset as static pruning, with 2 reinforcement agents to determine the corresponding sparsity. Our method combines the merits of runtime and static pruning, and provides trade-off between storage and dynamic flexibility. Extensive experiments demonstrate the effectiveness of our proposed method.
+
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+
+# A ADDITIONAL EXPERIMENTAL RESULTS
+
+A.1 COMPARISON TO SEPARATELY STATIC AND RUNTIME PRUNING
+
+In this section, we compare our method with two additional baseline methods. One is a variation of our method by separately training static pruning and the training runtime pruning. In this method, we start from a pretrained backbone CNN, $f ( \cdot )$ and $u _ { s }$ . Then we add the static DRL agent to prune channels statically by learning the static policy and $u _ { s }$ . Finally, we add the runtime DRL agent to prune channels dynamically, by fixing the static DLR agent and $u _ { s }$ , and updating the runtime DLR agent and $f ( \cdot )$ only. Another method is to combine state-of-the-art static and runtime pruning methods. We start from a pretrain backbone CNN, and then prune channels with the static pruning method FPGM (He et al., 2019), and finally prune channels with the runtime method FBS (Gao et al., 2019). The experimental results are shown in Table4.
+
+<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td></tr><tr><td>ours s (runtime only Rr = 1)</td><td>92.07</td><td>91.333</td><td>-0.737</td></tr><tr><td>ours s(Rr=0.5)</td><td>92.07</td><td>91.066</td><td>-1.004</td></tr><tr><td>ours (separate static and runtime)</td><td>92.07</td><td>90.965</td><td>-1.0105</td></tr><tr><td>FPGM+FBS</td><td>92.07</td><td>90.456</td><td>-1.614</td></tr></table>
+
+Table 4: Comparison to methods with separately static and runtime pruning on Cifar-10 at sparsity 0.5.
+
+# A.2 STORAGE/ACCURACY TRADE-OFF
+
+Besides Fig. 2, to further illustrate the trade-off between storage and accuracy, we show additional results in Fig. 4.
+
+![](images/cb61cd2607b8c848b7d5883f4895e07b06c16c86e7d12f6dae1ddac840f5cca1.jpg)  
+Figure 4: Trade-off between runtime pruning and static pruning at sparsity 0.45. X-axis is storage(number of parameters), Y-axis is accuracy.
+
+# B PSEUDOCODE OF TRAINING PROCESS
+
+# Algorithm 1: Training process
+
+INPUT: pretrained backbone CNN, computation budget $\bar { B } _ { c o m }$ , storage budget $\bar { B } _ { p a r a m }$   
+OUTPUT: backbone CNN, importance predictor $f ( \cdot )$ , static pruning importance $u _ { s }$ , runtime and   
+static DRL agents   
+Add runtime importance predictor $f ( \cdot )$ and static pruning importance $u _ { s }$ .;   
+$u _ { s } \gets \mathbf { 1 }$ ;   
+$d _ { s } \gets 1$ ;   
+$d _ { r } \gets 1$ ;   
+while not converge do fix $u _ { s }$ , update $f ( \cdot )$ and backbone CNN;   
+end   
+while not converge do   
+1 update $u _ { s }$ , $f ( \cdot )$ and backbone CNN;   
+end   
+add runtime DRL agent and static DRL agent to predict actions $a _ { t } ^ { r }$ and $a _ { t } ^ { s }$ , and alternate $d _ { r }$ and $d _ { s }$   
+at each layer $t$ ;   
+while not converge do for $i \gets 1$ to $N _ { 1 }$ do Forward entired model; Compute rewards $R _ { t } ^ { r }$ and $R _ { t } ^ { s }$ using budget $\bar { B } _ { c o m }$ and $\bar { B } _ { p a r a m }$ ; F $\mathrm { i } \times \bar { u _ { s } } , f ( \cdot )$ and backbone CNN, update runtime and static DRL agents by PPO loss; end for $i \gets 1$ to $N _ { 2 }$ do Fix runtime and static DRL agents, update $u _ { s }$ , $f ( \cdot )$ and backbone CNN by cross entropy loss end   
+end
+
+![](images/efd43963a47c3a1afcdfc3c167806714afb5b2f802c5a7ab44ba6c03012e0742.jpg)  
+Figure 5: Training curve of our method on CIFAR-10 at sparsity 0.5 with $R _ { r } = 0 . 5$
+
+# C TRAINING CURVE
+
+We show the training curve of our method in Fig. 5. We train our method on CIFAR-10 at sparsity 0.5 with $R _ { r } = 0 . 5$ . The figure includes the accuracy curves evaluated on training set and test set, showing that our methods is stable during training.

parse/train/S1ewjhEFwr/S1ewjhEFwr_content_list.json

@@ -0,0 +1,1794 @@
+[
+    {
+        "type": "text",
+        "text": "STORAGE EFFICIENT AND DYNAMIC FLEXIBLE RUNTIME CHANNEL PRUNING VIA DEEP REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            821,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            195,
+            398,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a deep reinforcement learning (DRL) based framework to efficiently perform runtime channel pruning on convolutional neural networks (CNNs). Our DRL-based framework aims to learn a pruning strategy to determine how many and which channels to be pruned in each convolutional layer, depending on each specific input instance at runtime. The learned policy optimizes the performance of the network by restricting the computational resource on layers under an overall computation budget. Furthermore, unlike other runtime pruning methods which require to store all channels parameters for inference, our framework can reduce parameters storage consumption for deployment by introducing a static pruning component. Comparison experimental results with existing runtime and static pruning methods on state-of-the-art CNNs demonstrate that our proposed framework is able to provide a tradeoff between dynamic flexibility and storage efficiency in runtime channel pruning. ",
+        "bbox": [
+            233,
+            291,
+            764,
+            470
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            497,
+            336,
+            512
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, convolutional neural networks (CNNs) have been proven to be effective in a wide range of computer vision tasks, such as image classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), objection detection (He et al., 2017; Zhou et al., 2019; Law & Deng, 2018), segmentation (He et al., 2017; Zhu et al., 2019). Therefore, nowadays, many computervision-based systems, such as automatic-driving cars, security surveillance cameras, and robotics, are built on the power of CNNs. However, since most state-of-the-art CNNs require expensive computation power for inference and huge storage space to store large amount of parameters, the limitation of energy, computation and storage on mobile or edge devices has become the major bottleneck on real-world deployments of CNNs. Existing studies have been focused on speeding up the execution of CNNs for inference on edge devices by model compression using matrix decomposition (Denil et al., 2013; Masana et al., 2017), network quantization (Courbariaux et al., 2016), network pruning (Dong et al., 2017), etc. Among these approaches, channel pruning has shown promising performance (He et al., 2017; Luo et al., 2017; Zhuang et al., 2018; Peng et al., 2019). Specifically, channel pruning discards an entire input or output channel and keep the rest of the model with structures. ",
+        "bbox": [
+            174,
+            529,
+            825,
+            736
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Most channel pruning approaches can be categorized into two types: runtime approaches and static approaches. Static channel pruning approaches aim to design a measurement to evaluate the importance of each channel over the whole training dataset and remove the least important channels to minimize the loss of performance after pruning. By permanently pruning a number of channels, the computation and storage cost of CNNs can be dramatically reduced when being deployed, and the inference execution can be accelerated consequently. Runtime channel pruning approaches have been recently proposed to achieve dynamic channel pruning on each specific instance (Gao et al., 2019; Luo & Wu, 2018). To be specific, the goal of runtime approaches aims to evaluate the channel importance at runtime, which is assumed to be different on different input instances. By pruning channels dynamically, different pruned structures can be considered as different routing of data stream inside CNNs. This kind of approaches is able to significantly improve the representation capability of CNNs, and thus achieve better performance in terms of prediction accuracy compared with static approaches. However, previous runtime approaches trade storage cost off dynamic flexibility of pruning. To achieve dynamic pruning on different specific instances, all parameters of kernels are required to be stored (or even more parameters are introduced). This makes runtime approaches not applicable on resource-limited edge devices. Moreover, most of previous runtime approaches only evaluate the importance among channels in each single layer independently, without considering the difference in efficiency among layers. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            172
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, to address the aforementioned issues of runtime channel pruning approaches, we propose a deep reinforcement learning (DRL) based pruning framework. Basically, we aim to apply DRL to prune CNNs by maximizing received rewards, which are designed to satisfy the overall budget constraints along side with network’s training accuracy. Note that automatic channel pruning by DRL is a difficult task because the action space is usually very huge. Specifically, the discrete action space for the DRL agent is as large as the number of channels at each layer, and the action spaces may vary among layers since there are different numbers of channels in different layers. To facilitate pruning CNNs by DRL, for each layer, we first design a novel prediction component to estimate the importance of channels, and then develop a DRL-based component to learn the sparsity ratio of the layer, i.e., how many channels should be pruned. ",
+        "bbox": [
+            174,
+            181,
+            825,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "More specifically, different from previous runtime channel pruning approaches, which only learn runtime importance of each channel, we propose to learn both runtime importance and additionally static importance for each channel. While runtime importance maintains the saliency of specific channels for each given specific input, the static importance captures the overall saliency of the corresponding channel among the whole dataset. According to each type of the channel importance, we further design different DRL agents (i.e., a runtime agent and a static agent) to learn a sparsity ratio in a layer-wise manner. The sparsity ratio learned by the runtime agent together with the estimated runtime importance of channels are used to generate runtime pruning structures, while the sparsity ratio learned by the static agent together with the estimated static importance of channels are used to generate static (permanent) pruning structures. By considering both the pruning structures, our framework is able to provide a trade-off between storage efficiency and dynamic flexibility for runtime channel pruning. ",
+        "bbox": [
+            174,
+            327,
+            825,
+            493
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, our contributions are 2-fold. First, we propose to prune channels by taking both runtime and static information of the environment into consideration. Runtime information endows pruning with flexibility based on different input instances while static information reduces the number of parameters in deployment, leading to storage reduction, which cannot be achieved by conventional runtime pruning approaches. Second, we propose to use DRL to determine sparsity ratios, which is different from the previous pruning approaches which manually set sparsity ratios. Extensive experiments demonstrate the effectiveness of our method. ",
+        "bbox": [
+            174,
+            500,
+            825,
+            597
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK AND PRELIMINARY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            616,
+            509,
+            632
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 STRUCTURE PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            647,
+            369,
+            661
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Wen et al. (2016) pioneered structure pruning in deep neural network by imposing the $L _ { 2 , 1 }$ norm in training. Under the same framework, Liu et al. (2017) regarded parameters in batch normalization as channel selection signal, which is minimized to achieve pruning during training. He et al. (2017) formulated channel pruning into a two-step iterative process including LASSO regression based channel selection and least square reconstruction. Luo et al. (2017) formulated channel pruning as minimization of difference of output features, which is solved by greedy selection. Zhuang et al. (2018) further considered early prediction, reconstruction loss and final loss to select importance channels. Overall, structure pruning methods accelerate inference by producing regular and compact model. However, this brought regularness requires preserving more parameters to ensure performance. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            811
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 DYNAMIC PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            828,
+            352,
+            842
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Dynamic pruning provides different pruning strategies according to input data. Wang et al. (2018) proposed to reduce computation by skipping layers or channels based on the analysis of input features. Luo & Wu (2018) proposed to use layer input to learn channel importance, which is then binarized for pruning. Gao et al. (2019) applied the same framework while extended features selection in both input and output features. Similarly, Liu & Deng (2018) introduced multiple branches for runtime inference according to inputs. A gating module is learnt to guide the flow of feature maps. Bolukbasi et al. (2017) learned to choose the components of a deep network to be evaluated for each input adaptively. Early exit is introduced to accelerate computation. Dynamic pruning adaptively takes different actions for different inputs, which is able to accelerate the overall inference time. However, the original high-precision model needs to be stored, together with extra parameters for making specified pruning actions. Rosenbaum et al. (2018) proposed to learn routers to route layers output to different next layers, in order to adjust a network to multi-task learning. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc77b32ffc241caf859273ec99f771bfda79928ed2f13e13b67d0d9181ac8f0e.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of our proposed DRL-based runtime pruning framework. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            103,
+            758,
+            262
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            299,
+            825,
+            397
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 DEEP REINFORCEMENT LEARNING IN PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            414,
+            542,
+            428
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Channel selection is on trial using deep reinforcement learning. Lin et al. (2017) trained a LSTM model to remember and provide channel pruning strategy for backbone CNN model, which is conducted using reinforcement learning techniques. He et al. (2018) proposed to determine the compression ratio in each layer by training an agent regarding the pruning-retraining process as an environment. ",
+        "bbox": [
+            174,
+            439,
+            825,
+            508
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 PRELIMINARY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            526,
+            312,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement Learning We consider a standard setup of reinforcement learning: an agent sequentially takes actions over a sequence of time steps in an environment, in order to maximize the cumulative reward (Sutton & Barto, 1998). This problem can be formulated as a Markov Decision Process (MDP) of a tuple $( S , \\mathcal { A } , \\mathcal { P } , R , \\gamma )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $\\mathcal { P } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ is transition probabilities, $R : S \\times \\mathcal { A }  \\mathbb { R }$ is the reward function, and $\\gamma \\in [ 0 , 1 )$ is the discount factor. The goal of reinforcement learning is to learn a policy $\\pi ( a | s )$ that maximizes the objective of cumulative rewards over finite time steps, ",
+        "bbox": [
+            173,
+            551,
+            825,
+            650
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/474eb1f0c3a7b96714a542f434db63bc9e136bf372e82480d973fee8d6938969.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            656,
+            562,
+            699
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $s _ { t } \\in S$ and $a _ { t } \\in \\mathcal A$ are state and taken action at time step $t$ respectively. ",
+        "bbox": [
+            173,
+            704,
+            686,
+            719
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 DRL-BASED RUNTIME PRUNING FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            738,
+            594,
+            756
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The overview of our proposed framework is presented in Fig. 1. To prune convolutional layer $t$ , we learn two types of learnable channel importance: runtime channel importance $\\mathbf { u } _ { r } \\in \\mathbb { R } ^ { C \\times \\mathbf { \\breve { 1 } } }$ and static channel importance $\\mathbf { u } _ { s } \\in \\mathbb { R } ^ { C \\times 1 }$ , where $C$ is the number of channels in layer $t$ . The runtime channel importance $\\mathbf { u } _ { r }$ is generated by a subnetwork importance predictor $f ( \\cdot )$ , which takes the input feature map $\\mathbf { F } _ { i n }$ as input, while the static channel importance $\\mathbf { u } _ { s }$ is randomly initialized and updated during training. Both $\\mathbf { u } _ { r }$ and $\\mathbf { u } _ { s }$ indicate the channel importance of the full precision output feature map $\\mathbf { F } _ { o u t }$ through a convolution layer. Channels are selected to be pruned according to the values of each element in $\\mathbf { u } _ { r }$ and $\\mathbf { u } _ { s }$ , and how many channels to be selected is decided by the sparsity ratios $d _ { r }$ and $d _ { s }$ , respectively. To learn the sparsity ratios $d _ { r }$ and $d _ { s }$ , two DRL agents, the runtime agent and the static agent, are introduced, where actions $a _ { t } ^ { r }$ and $a _ { t } ^ { s }$ are defined to set values of $d _ { r }$ and $d _ { s }$ , respectively. The detail of the two DRL agents are described in Sec. 3.3. Consequently, a trade-off pruner $g ( \\cdot )$ is performed to balance the runtime and static pruning results, and output a decision mask M of binary values (1/0) to indicate which channels to be pruned (1: pruned, 0: preserved), as well as a unified channel importance vector $\\mathbf { u } \\in \\mathbb { R } ^ { C \\times 1 }$ as follows, ",
+        "bbox": [
+            173,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            826,
+            146
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e61e5291c8dbd0a16a7541cea1852047be655c6cd97790003d6717158beb0a18.jpg",
+        "text": "$$\n[ \\mathbf { M } , \\mathbf { u } ] = g ( \\mathbf { u } _ { r } , \\mathbf { u } _ { s } , d _ { r } , d _ { s } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            150,
+            589,
+            167
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The final output after pruning is constructed by multiplying the full precision output feature map $\\mathbf { F } _ { o u t }$ , by ${ \\bf 1 } - { \\bf M }$ and u as, ",
+        "bbox": [
+            173,
+            171,
+            825,
+            199
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3edc47cd665c641dcbb3659b1aaefc98d0c09e4300faaf2364c86ff51ebaafa1.jpg",
+        "text": "$$\n\\hat { \\mathbf { F } } _ { o u t } = \\mathbf { F } _ { o u t } \\otimes ( \\mathbf { 1 } - \\mathbf { M } ) \\otimes \\mathbf { u } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            198,
+            599,
+            217
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\otimes$ is the broadcast element-wise multiplier, and 1 is the matrix of the same size as $\\mathbf { M }$ with all the elements being 1. In the following, we introduce how to learn the runtime channel importance vector $\\mathbf { u } _ { r }$ and the static channel importance vector $\\mathbf { u } _ { s }$ in Sec. 3.1, how to construct the trade-off pruner $g ( \\cdot )$ in Sec. 3.2, and how to design the two DRL agents in Sec. 3.3. ",
+        "bbox": [
+            174,
+            218,
+            825,
+            275
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 LEARNABLE CHANNEL IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            290,
+            470,
+            305
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider that a convolutional layer takes input of feature map $\\mathbf { F } _ { i n } \\in \\mathbb { R } ^ { C _ { i n } \\times H _ { i n } \\times W _ { i n } }$ and generates an output feature map $\\mathbf { F } _ { o u t } \\doteq \\perp \\operatorname { \\textbf { \\em E } } ^ { C _ { o u t } \\times H _ { o u t } ^ { \\star } \\times W _ { o u t } }$ , where $C _ { * }$ , $H _ { * }$ and $W _ { * }$ are the number of channels, width and height of the feature map $\\mathbf { F } _ { * }$ , respectively. Each element of the channel importance vectors $\\mathbf { u } _ { r } \\in \\mathbb { R } ^ { C _ { o u t } }$ and $\\mathbf { u } _ { s } \\in \\mathbb { R } ^ { C _ { o u t } }$ represents the importance value of the corresponding channel, respectively. In the following, we drop the subscript $o u t$ for simplicity in presentation. ",
+        "bbox": [
+            174,
+            315,
+            825,
+            387
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 RUNTIME CHANNEL IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            401,
+            465,
+            415
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As mentioned above, the runtime channel importance $\\mathbf { u } _ { r }$ of output feature $\\mathbf { F } _ { o u t }$ is predicted by a importance predictor $f ( \\cdot )$ , which takes $\\mathbf { F } _ { i n }$ as input. Therefore, $\\mathbf { u } _ { r }$ can be considered as a function of $\\mathbf { F } _ { i n }$ , whose values vary over different input instances. In this paper, we design a subnetwork to approximate $f ( \\cdot )$ , which is expected to be of a small size and computationally efficient. Similar to many existing dynamic network pruning methods (Gao et al., 2019; Hu et al., 2018; Luo & Wu, 2018), we use global pooling layer as the first layer in $f ( \\cdot )$ , because global pooling is computationally efficient and it can reduce the dimension of $\\mathbf { F } _ { i n }$ dramatically. We then feed the output of global pooling into a fully-connected layer without any activation function. The output of fully-connected layer is the runtime channel importance vector $\\mathbf { u } _ { r }$ . ",
+        "bbox": [
+            174,
+            424,
+            825,
+            551
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Which channels to be preserved $/$ pruned at runtime are determined according to the values of $\\mathbf { u } _ { r }$ . We denote by $\\mathbf { M } _ { r } \\in \\{ 0 , 1 \\} ^ { C }$ a mask for pruning, where if the value is 0, then the corresponding channel is preserved, otherwise pruned. For now, suppose a sparsity ratio $d _ { r }$ for runtime pruning has already been generated via the dynamic DRL agent, which will be introduced in Sec. 3.3. We then prune $\\left( C - \\lceil d _ { r } C \\rceil \\right)$ channels with the smallest importance values in $\\mathbf { u } _ { r }$ . Accordingly, the value of an element in ${ { \\bf { M } } _ { r } }$ is set to be 1 if the corresponding channel is pruned, otherwise 0. ",
+        "bbox": [
+            174,
+            558,
+            825,
+            641
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 STATIC CHANNEL IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            656,
+            447,
+            670
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The static channel importance vector $\\mathbf { u } _ { s }$ is to capture the global information for pruning, and thus is learned from the whole dataset. It is randomly generated and learned through backpropagation. Similar to runtime channel pruning, given a sparsity ratio $d _ { s }$ learned by the static DRL agent, $( C -$ $\\lceil d _ { s } C \\rceil )$ channels with smallest importance values in $\\mathbf { u } _ { s }$ are pruned, and a mask $\\mathbf { M } _ { s } \\in \\mathbf { \\bar { \\{ 0 , 1 \\} } } ^ { C }$ is generated to indicate the static pruning results. ",
+        "bbox": [
+            174,
+            680,
+            825,
+            751
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 TRADE-OFF PRUNER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            766,
+            357,
+            781
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the runtime and the static pruning decisions, ${ { \\bf { M } } _ { r } }$ and $\\mathbf { M } _ { s }$ , we now propose a trade-off pruner to generate a unified channel pruning decision. The main idea behind the trade-off pruner is to 1) prune those channels which are agreed to be pruned by both decisions, and 2) prune a portion of the rest channels by weighted votes from both decisions. ",
+        "bbox": [
+            174,
+            792,
+            825,
+            849
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To be specific, we define the mask representing channels pruned by both decisions as ",
+        "bbox": [
+            174,
+            854,
+            730,
+            871
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f79087d05b40e0a82ce5a05ca380ec88c51b2b35988867d71efc6080ba995e0.jpg",
+        "text": "$$\n\\mathbf { M } _ { o } = \\mathbf { M } _ { s } \\wedge \\mathbf { M } _ { r } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            875,
+            558,
+            890
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\wedge$ is element-wise logical AND and $1 / 0$ in mask represents logical true or false. The channels indicated to be pruned by $\\mathbf { M } _ { o }$ (i.e., the corresponding values are 1) are pruned in final. The channels which are determined to be pruned by ${ { \\bf { M } } _ { r } }$ but not by $\\mathbf { M } _ { s }$ can be represented by a new mask $\\overline { { \\mathbf { M } } } _ { r } =$ ${ \\bf M } _ { r } - { \\bf M } _ { o }$ . Similarly, the channels which are determined to be pruned by ${ { \\bf { M } } _ { r } }$ but not by $\\mathbf { M } _ { s }$ can be represented by another new mask $\\overline { { \\mathbf { M } } } _ { s } = \\mathbf { M } _ { s } - \\mathbf { M } _ { o }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To control the trade-off between ${ { \\bf { M } } _ { r } }$ and $\\mathbf { M } _ { s }$ , we define a rate $R _ { r }$ denoting how much we trust the pruning decision made by ${ { \\bf { M } } _ { r } }$ , while $1 - R _ { r }$ is for $\\mathbf { M } _ { s }$ . That means the channels selected by $\\overline { { \\mathbf { M } } } _ { r }$ will be finally pruned with the rate $R _ { r }$ . Specifically, the number of channels which are selected by $\\overline { { \\mathbf { M } } } _ { r }$ and finally will be pruned is ",
+        "bbox": [
+            174,
+            154,
+            823,
+            212
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e42f21874872a70bdba0ae81aa7080f8df5c8a4789de2e33f49de4d365f70659.jpg",
+        "text": "$$\nC _ { r } ^ { ' } = \\lfloor R _ { r } ( \\mathbf { 1 } ^ { \\top } \\overline { { \\mathbf { M } } } _ { r } ) \\rfloor ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            210,
+            568,
+            231
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { 1 } ^ { \\top } \\overline { { \\mathbf { M } } } _ { r }$ returns the number of channels selected by $\\overline { { \\mathbf { M } } } _ { r }$ . We then select the first $C _ { r } ^ { ' }$ -smallest important channels which are recommended to be pruned by $\\overline { { \\mathbf { M } } } _ { r }$ to form a mask $\\widehat { \\mathbf { M } } _ { r }$ . Similarly, for static pruning, we select the first $C _ { s } ^ { ' }$ -smallest important channels which are recommended to be pruned by $\\overline { { \\mathbf { M } } } _ { s }$ to form another mask $\\widehat { \\mathbf { M } } _ { s }$ , where $C _ { s } ^ { ' } = \\lfloor ( 1 - R _ { r } ) ( \\mathbf { 1 } ^ { \\top } \\overline { { \\mathbf { M } } } _ { s } ) \\rfloor$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The final trade-off pruning mask is defined as ",
+        "bbox": [
+            176,
+            308,
+            473,
+            323
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a8fbe6107f301fd83dc35ecbbeb28112c26bfa31d9a6a8ee8bef4337f7d9dac5.jpg",
+        "text": "$$\n\\mathbf { M } = \\mathbf { M } _ { o } + \\widehat { \\mathbf { M } } _ { r } + \\widehat { \\mathbf { M } } _ { s } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            329,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Moreover, in this work, the unified channel importance is simply defined as follows, ",
+        "bbox": [
+            173,
+            356,
+            725,
+            371
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5a198a1124c5e92f4857eb025f420d00e4ef14196e89e31370461bcc2bf02b5.jpg",
+        "text": "$$\n\\mathbf { u } = \\mathbf { u } _ { r } \\otimes \\mathbf { u } _ { s } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            380,
+            547,
+            393
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With the trade-off pruning mask $\\mathbf { M }$ and the unified channel importance $\\mathbf { u }$ , the pruned output feature $\\hat { \\mathbf { F } } _ { o u t }$ can be generated by Eq. 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 DRL BASED PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            449,
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+            464
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present how to formulate the problems of learning the ratios $d _ { s }$ and $d _ { t }$ for static pruning and runtime pruning, as a MDP, and solve it via DRL, respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 DRL FOR RUNTIME PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            520,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the MDP for runtime pruning, we consider the $t$ -th layer of the network as the $t$ -th timestamp.   \nThe details of the MDP are listed as follows. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "State Given an input feature map $\\mathbf { F } _ { i n }$ of layer $t$ , we pass it to a global pooling layer to reduce its dimension to $\\mathbb { R } ^ { C _ { i n } }$ , where $C _ { i n }$ is the number of input channel of layer $t$ . Since $C _ { i n }$ varies among layers, we feed the output of global pooling to a layer-dependent encoder to project it to a fix-length vector $s _ { t } ^ { r }$ , which is considered as as a state representation of DRL in the context of runtime pruning. ",
+        "bbox": [
+            174,
+            588,
+            825,
+            643
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Action The action $a _ { t } ^ { r }$ is defined as the sparsity ratio at layer $t$ , alternating $d _ { r }$ in runtime pruning mentioned in Section 3.1.1. Existing DRL-base pruning method RNP (Lin et al., 2017) uses a unified discrete actions space with $k$ actions which are too coarse to achieve high accuracy. However, finegrained discrete action space as large as number of channels suffers from exploration difficulty. Therefore, instead of using discrete action spaces, we propose a continuous action space with action $a _ { t } ^ { r } \\in ( 0 , 1 ]$ . To avoid over-pruning the filters and crashing in training, we set a minimum sparsity ratio $+ \\alpha$ such that $a _ { t } ^ { r } \\in ( + \\alpha , 1 ]$ . ",
+        "bbox": [
+            173,
+            659,
+            825,
+            758
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Reward The reward function is proposed to consider both network accuracy and computation budget. We define the accuracy relative reward based on the loss of pruned backbone network, ",
+        "bbox": [
+            173,
+            772,
+            823,
+            801
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c25bdc820299cc2d7c1776d93146fcd59c95d3239858d1dc660619dd1d7c8bee.jpg",
+        "text": "$$\nR _ { a c c } ^ { r } = - \\mathcal { L } _ { C N N } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            809,
+            558,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { C N N }$ is the loss in CNN, and it may vary in scale among different training stage, i.e. large at beginning of training and small near convergence. To avoid the instability brought by the reward scale, $R _ { a c c } ^ { r }$ is normalized by a moving average, ",
+        "bbox": [
+            174,
+            832,
+            825,
+            875
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b2cf5c76491f52ee741e87057f64a8ff38de870bb9c41aa20a6ec7f0a3506bca.jpg",
+        "text": "$$\nR _ { a c c } ^ { r ^ { \\prime } } = R _ { a c c } ^ { r } / \\beta _ { b } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            882,
+            557,
+            902
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2971d95668901d82e88537ae3cdcb66fb0649a6fc497e681c3a2346c75051075.jpg",
+        "text": "$$\n\\beta _ { b } = \\lambda \\beta _ { b - 1 } + ( 1 - \\lambda ) R _ { a c c } ^ { r } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            907,
+            593,
+            925
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta _ { b }$ is the moving average at the $b$ -th training batch and $\\lambda$ is the moving weight. ",
+        "bbox": [
+            173,
+            103,
+            730,
+            119
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To force computation of the pruned network under a given computation budget, we define a exponential reward function of budget regarding reward $R _ { b u d } ^ { r }$ : ",
+        "bbox": [
+            173,
+            125,
+            825,
+            155
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1813615ef8a9899c01dc046e862adb041baf5073d7ae9045b4cabab04c4da5a3.jpg",
+        "text": "$$\nR _ { b u d } ^ { r } = \\left\\{ \\begin{array} { l l } { \\exp ( \\alpha _ { 1 } ( B _ { c o m } - \\overline { { B } } _ { c o m } ) ) - 1 , } & { B _ { c o m } > \\overline { { B } } _ { c o m } , } \\\\ { 0 , } & { \\mathrm { o t h e r w i s e } , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            160,
+            689,
+            196
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $B _ { c o m }$ is the computation consumption, which is calculated based on the current of pruned strategy, and $\\overline { { B } } _ { c o m }$ is the given computation budget constraint. Finally we sum up the two rewards to form sparse rewards, with being non-zero at terminated step $T$ and zeros at other time step $t < T$ , ",
+        "bbox": [
+            173,
+            202,
+            826,
+            244
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/32d35af3dcbde7e6fdeee3a96dee9c445bdb951e77b91bf6f6eecfe608320a61.jpg",
+        "text": "$$\nR _ { t } ^ { r } = \\left\\{ \\begin{array} { l l } { R _ { a c c } ^ { r ^ { \\prime } } + R _ { b u d } ^ { r } , } & { t = T , } \\\\ { 0 , } & { t < T . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            250,
+            602,
+            286
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Actor-Critic Agent To solve the continuous action space problem, we choose a commonly used actor-critic agent with a Gaussian policy. Actor-critic agent consists of two components: 1) actor outputs the mean and variance to form a Gaussian policy where the continuous action are sampled from; 2) critic outputs a scalar predicting the future discounted accumulated reward and assists the policy training. Actor network and Critic network share one-layer RNN which takes state $s _ { t } ^ { r }$ as input. The output of RNN is fed into actor specific network constructed by two branches of fullyconnected layers, leading to the mean and variance of the Gaussian policy. The action is sampled for the Gaussian distribution outputed by the actor: ",
+        "bbox": [
+            173,
+            300,
+            825,
+            412
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab5742beaa58ae82e1e3602efad7851f18f6c82c85dedf2ed8da1512d86de03b.jpg",
+        "text": "$$\na _ { t } ^ { r } \\sim \\mathcal { N } ( \\mu ( s _ { t } ^ { r } ; \\theta ^ { r } ) , \\sigma ( s _ { t } ^ { r } ; \\theta ^ { r } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            417,
+            598,
+            435
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu ( s _ { t } ^ { r } ; \\theta ^ { r } )$ and $\\sigma ( s _ { t } ^ { r } ; \\theta ^ { r } )$ is the mean and variance outputed from actor network. The Critic specific network has one fully-connected layer after the shared RNN, and outputs the predictive value $V ( s _ { t } ^ { r } ; \\theta ^ { r } )$ . ",
+        "bbox": [
+            176,
+            440,
+            825,
+            483
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To optimize the actor-critic agent, Proximal Policy Optimization (PPO) (Schulman et al., 2017) is used. Note that we relax the action $a _ { t } ^ { r }$ to $( - \\infty , + \\infty )$ in PPO, and use truncate function to clip $a _ { t } ^ { r }$ in $( + \\alpha , 1 ]$ when perform pruning. ",
+        "bbox": [
+            174,
+            489,
+            821,
+            532
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Besides, an additional regularizer is introduced to restrict the relaxed $a _ { t } ^ { r }$ staying in range $( + \\alpha , 1 ]$ ",
+        "bbox": [
+            173,
+            539,
+            807,
+            554
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/150af902c09851bc9580e95eb7e645239208b9182b8133b94abc8fe168242610.jpg",
+        "text": "$$\n\\mathcal { L } _ { a } = \\frac { 1 } { 2 } | | a _ { t } ^ { r } - \\operatorname* { m a x } ( \\operatorname* { m i n } ( a _ { t } ^ { r } , 1 ) , + \\alpha ) | | _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            559,
+            630,
+            589
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 DRL FOR STATIC PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            602,
+            415,
+            616
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similar to runtime pruning, the MDP in static pruning is also formulated layer-by-layer. The difference against runtime pruning is the definition of state and reward. ",
+        "bbox": [
+            174,
+            626,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "State The state $s _ { t } ^ { s }$ in static pruning is defined as the full shape of $\\mathbf { F } _ { o u t }$ , and does not depend on $\\mathbf { F } _ { o u t }$ and the current input data. ",
+        "bbox": [
+            173,
+            669,
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+            698
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Action Action $a _ { t } ^ { s }$ is sampled from actor’s outputed Gaussian policy, and it is to alternate the sparsity $d _ { s }$ in static pruning mentioned in Section 3.1.2. ",
+        "bbox": [
+            173,
+            713,
+            821,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Reward The reward function takes both network accuracy and parameters budget into consideration. The accuracy relative is defined as the same as that in runtime pruning: ",
+        "bbox": [
+            173,
+            756,
+            821,
+            785
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/93371f187fc0717ccd4d5db0a56c820397ddaf3caba6815bdb16326df1f72883.jpg",
+        "text": "$$\nR _ { a c c } ^ { s } = R _ { a c c } ^ { r ^ { \\prime } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            452,
+            791,
+            545,
+            810
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To reduce the number of parameters of network to satisfy the parameters storage budget, the parameters relative reward is defined in an exponential form as, ",
+        "bbox": [
+            169,
+            816,
+            823,
+            844
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d926bc05f14bed2feb7750c7f596d5af420d3c056c1acf4de1e65bb7ab782bb5.jpg",
+        "text": "$$\nR _ { p a r a m } ^ { s } = \\left\\{ \\begin{array} { l l } { \\exp ( \\alpha _ { 2 } ( B _ { p a r a m } - \\overline { { B } } _ { p a r a m } ) ) - 1 , } & { B _ { p a r a m } > \\overline { { B } } _ { p a r a m } , } \\\\ { 0 , } & { \\mathrm { o t h e r w i s e } , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            851,
+            727,
+            887
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $B _ { p a r a m }$ is the number of preserved parameters after static pruning and $\\overline { { B } } _ { p a r a m }$ is the parameters storage budget. ",
+        "bbox": [
+            176,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b0342d02e34aa32756cee647ecf68c12da4f63bded41415848b5fe72c47ab80.jpg",
+        "table_caption": [
+            "Table 1: Comparison to state-of-the-art runtime pruning methods on CIFAR-10 at sparsity 0.5. Speed-up is calculated on MACs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td><td>Speed-up</td><td>#Params</td></tr><tr><td>FBS (Gao et al., 2019)</td><td>91.37</td><td>89.88</td><td>-1.49</td><td>3.93×</td><td>1.11×</td></tr><tr><td>RNP (Lin et al., 2017)</td><td>92.07</td><td>84.93</td><td>-7.14</td><td>3.56×</td><td>1.00×</td></tr><tr><td>ours (runtime only Rr = 1)</td><td>92.07</td><td>91.333</td><td>-0.737</td><td>3.92×</td><td>1.31×</td></tr><tr><td>ours (Rr = 0.5)</td><td>92.07</td><td>91.066</td><td>-1.004</td><td>3.92×</td><td>0.78×</td></tr></table>",
+        "bbox": [
+            210,
+            94,
+            782,
+            176
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/977a00a9353f3240ea835b43a0fc49439a4825c5a28ce62592f9e77aa0be4a34.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td><td>Speed-up</td><td>#Params</td></tr><tr><td>FBS (Gao et al., 2019)</td><td>91.37</td><td>91.23</td><td>-0.14</td><td>2×</td><td>1.11×</td></tr><tr><td>ours (Rr = 1.0)</td><td>92.07</td><td>93.178</td><td>+1.108</td><td>1.99×</td><td>1.31×</td></tr><tr><td>ours (Rr = 0.5)</td><td>92.07</td><td>92.502</td><td>+0.432</td><td>1.99×</td><td>0.97×</td></tr></table>",
+        "bbox": [
+            223,
+            237,
+            769,
+            306
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Comparison to state-of-the-art runtime pruning methods on CIFAR-10 at sparsity 0.7.   \nSpeed-up is calculated on MACs. ",
+        "bbox": [
+            174,
+            316,
+            821,
+            344
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Actor-Critic Agent This agent is similar to the one in runtime pruning. It has the same architecture as runtime pruning but differs in introducing a fully-connected layer as the encoder before RNN. This agent is also optimized by PPO. ",
+        "bbox": [
+            176,
+            378,
+            823,
+            421
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            449,
+            295,
+            463
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In inference, the static agent is not required any more because the static pruning strategy does not depend on individual input data points but the full shape of $\\mathbf { F } _ { i n }$ . Therefore, the output action $a _ { t } ^ { s }$ is fixed to each layer $t$ . With the action $a _ { t } ^ { s }$ and the rate $R _ { r }$ , we can decide which filters can be pruned permanently. Specifically, channels with $( ( 1 - a _ { t } ^ { s } ) ( 1 - R _ { r } ) )$ -smallest static importance values are pruned permanently. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
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+        "text": "We evaluate our DRL pruning framework on two popular datasets: CIFAR-10 (Krizhevsky, 2009) and ImageNet ILSVRC2012 (Russakovsky et al., 2015), to show the advantage over other channel pruning methods. We analyze the effect of hyper-parameters and different sparsity settings on CIFAR-10. For CIFAR-10, we use M-CifarNet (Zhao et al., 2018) as the backbone CNN. On ImageNet ILSVRC2012, ResNet-18 is used as the backbone CNN. ",
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+    {
+        "type": "text",
+        "text": "4.1 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We start with a pretrained backbone CNN. Firstly we finetune the backbone CNN and train runtime importance predictor jointly, with sparsity $d _ { r } = 1$ and fixed all static pruning importance $\\mathbf { u } _ { s }$ to 1. Then we remove the restriction on the static pruning importance $\\mathbf { u } _ { s }$ , and train static pruning importance, the backbone CNN and the runtime importance predictor, with sparsity $d _ { s } = 1$ and runtime pruning sparsity fixed as $d _ { r } = 1$ . After finetuning, we use the DRL agents to predict the sparsity given computation and storage constraints. The DRL agents and the CNN with runtime/static importance are trained in alternating manner: We first fix the CNN as well as runtime/static importance and train two DRL agents, regarding the CNN as environments. Then we fix two agents and finetune the CNN and runtime/static importance. We repeat these two steps until convergence is achieved. We use Adam optimizer for both DRL agent and CNN, and set learning rate at $\\overline { { 1 } } 0 ^ { - 6 }$ for the DRL agents. For CNN finetuning and runtime/static importance training, the learning rate is set to $1 0 ^ { - 3 }$ on CIFAR-10. On ImageNet ILSVRC2012, the learning rate starts from $1 0 ^ { - 3 }$ and is divided by 10 after 15 millions iterations. ",
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+        "type": "image",
+        "img_path": "images/7149d698e379a247989cb4adf19d30fa5182186ecfe11294cd3330e87a35f835.jpg",
+        "image_caption": [
+            "Figure 2: Trade-off between runtime pruning and static pruning at sparsity 0.45. X-axis is the rate $R _ { r }$ "
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/785f2daf37a85fbdca7f2c6dbb534f6089b0380f10283987a3300f504ce7e5be.jpg",
+        "image_caption": [
+            "Figure 3: Comparison accuracy drop for MCifarNet on CIFAR-10 with computational budget. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/1100f375ba0e3335a47775ba81f55ac9b18f826ec725e923ef973cdf302c478f.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Comparison with the state-of-the-art channel pruning ResNet-18 on ImageNet. Speed-up is calculated on MACs. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Baseline top-1 acc.</td><td>Top-1 acc.</td><td>△ top-1 acc.</td><td>Baseline top-5 acc.</td><td>Top-5 acc.</td><td>△ top-5 acc.</td><td>Speed-up</td></tr><tr><td>DCP (Zhuang et al., 2018)</td><td>69.64</td><td>67.35</td><td>-2.29</td><td>88.98</td><td>88.86</td><td>-0.12</td><td>1.71×</td></tr><tr><td>FPGM (He et al., 2019)</td><td>70.28</td><td>68.41</td><td>-1.87</td><td>89.63</td><td>88.48</td><td>-1.15</td><td>1.71×</td></tr><tr><td>Dynamic Sparse Graph (Liu et al.,2019)</td><td>69.48</td><td>64.8</td><td>-4.68</td><td></td><td>■</td><td></td><td>1.4×</td></tr><tr><td>CGNN (Hua et al.,2018)</td><td>69.02</td><td>67.95</td><td>-1.07</td><td>88.84</td><td>88.21</td><td>-0.63</td><td>1.63×</td></tr><tr><td>FBS (Gao et al.,2019)</td><td>70.71</td><td>68.17</td><td>-2.54</td><td>89.68</td><td>88.22</td><td>-1.46</td><td>1.98×</td></tr><tr><td>Ours (Rr = 0.5)</td><td>69.758</td><td>68.79</td><td>-0.968</td><td>89.078</td><td>88.534</td><td>-0.544</td><td>1.94×</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL RESULTS ON CIFAR-10 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our proposed method with the following state-of-the-art runtime pruning methods: FBS (Gao et al., 2019), RNP (Lin et al., 2017) on CIFAR-10. The comparison results at sparsity 0.5 and 0.7 are shown in Table 1 and Table 2 respectively. Note that for fair comparison with other methods, the computation and storage budget constraints in our method is calculated according to the sparsity of other methods. Under these constraints, our method does not necessarily lead to the same sparsity as other methods in each layer. RNP cannot set exact sparsity ratio. Instead, its average sparsity ratio is accessible only during testing, which is 0.537 in Table 1. The result of FBS is reproduced using the released code1. The column #Params represents the number of parameters compared to the backbone CNN. ",
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+    {
+        "type": "text",
+        "text": "Table 1 shows that our method outperforms other state-of-the-art methods, achieving highest accuracy at an overall sparsity ratio of 0.5. Our method has very close computation speed-up compared to FBS, but outperforms FBS around $0 . 4 8 \\%$ to $0 . 7 6 \\%$ . When the runtime pruning strategy is solely considered by setting $R _ { r } = 1$ , our method surpasses other comparison methods, indicating that our DRL-based framework improves the performance of channel runtime pruning. By balancing runtime and static pruning via setting $R _ { r } = 0 . 5$ , our method reduces the number of the overall stored parameters and achieves lower accuracy drop than other methods. Table 2 shows that our method outperforms FBS at sparsity of 0.7. When $R _ { r } = 0 . 5$ , our method achieves better performance than the baseline CNN with $2 \\times$ speed-up and contains less parameters. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also study the relation between $R _ { r }$ and network compactness in our framework. Fig. 2 demonstrates the impact of $R _ { r }$ when sparsity is 0.45. The hyper-parameter $R _ { r }$ determines how much we trust about runtime pruning. With $R _ { r }$ close to 1, the accuracy becomes higher due to the more dynamic network flexibility but the space of the parameters storage also increases. When $R _ { r }$ diminishes, the network accuracy decreases but the parameter storage is reduced. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fig. 3 shows the performance of various sparsity ratios in our method. Again, our method does not prune with one single sparsity ratio for all layers, but uses the sparsity ratio to calculate computation and storage constraints, with which the sparsity ratio is learned for each layer. Fig. 3 demonstrates that our method holds the accuracy when sparsity is larger than about 0.5, which corresponds to about $4 \\times$ computational acceleration. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTAL RESULTS ON IMAGENET ILSVRC2012 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare our method with state-of-the-art channel pruning methods on ImageNet ILSVRC2012 as shown in Table 3. In this experiment, we use ResNet-18 as the backbone CNN. Among the stateof-the-art pruning methods for comparison, FBS (Gao et al., 2019) and CGNN (Hua et al., 2018) are runtime pruning methods. The overall sparsity ratio of our method is 0.7, which is under the same setting of FBS. Our method with $R _ { r } = 0 . 5$ achieves the smallest top-1 accuracy drop compared with other methods, and also achieves the highest top-1 accuracy after pruning. Overall, our proposed method achieves comparable or better performance compared to other methods with more acceleration. Our method has very close MACs to FBS, while the number of preserved parameters is reduced to $8 1 . 2 \\%$ of the baseline. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a deep reinforcement learning based framework for deep neural network channel pruning in both runtime and static sheme. Specially, channels are pruned according to input feature as runtime pruning, and based on entire training dataset as static pruning, with 2 reinforcement agents to determine the corresponding sparsity. Our method combines the merits of runtime and static pruning, and provides trade-off between storage and dynamic flexibility. Extensive experiments demonstrate the effectiveness of our proposed method. ",
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+    {
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+        "text": "Yiren Zhao, Xitong Gao, Robert Mullins, and Chengzhong Xu. Mayo: A framework for autogenerating hardware friendly deep neural networks. In Proceedings of the 2Nd International Workshop on Embedded and Mobile Deep Learning, EMDL’18, pp. 25–30, New York, NY, USA, 2018. ACM. ISBN 978-1-4503-5844-6. doi: 10.1145/3212725.3212726. URL http://doi. acm.org/10.1145/3212725.3212726. ",
+        "bbox": [
+            174,
+            347,
+            825,
+            416
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xingyi Zhou, Dequan Wang, and Philipp Krahenb ¨ uhl. Objects as points. In ¨ arXiv preprint arXiv:1904.07850, 2019. ",
+        "bbox": [
+            171,
+            428,
+            823,
+            457
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yi Zhu, Karan Sapra, Fitsum A. Reda, Kevin J. Shih, Shawn Newsam, Andrew Tao, and Bryan Catanzaro. Improving semantic segmentation via video propagation and label relaxation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019. ",
+        "bbox": [
+            176,
+            468,
+            823,
+            512
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhuangwei Zhuang, Mingkui Tan, Bohan Zhuang, Jing Liu, Yong Guo, Qingyao Wu, Junzhou Huang, and Jinhui Zhu. Discrimination-aware channel pruning for deep neural networks. In Advances in Neural Information Processing Systems, pp. 875–886, 2018. ",
+        "bbox": [
+            173,
+            522,
+            826,
+            565
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            594,
+            537,
+            609
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 COMPARISON TO SEPARATELY STATIC AND RUNTIME PRUNING ",
+        "bbox": [
+            176,
+            626,
+            656,
+            641
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare our method with two additional baseline methods. One is a variation of our method by separately training static pruning and the training runtime pruning. In this method, we start from a pretrained backbone CNN, $f ( \\cdot )$ and $u _ { s }$ . Then we add the static DRL agent to prune channels statically by learning the static policy and $u _ { s }$ . Finally, we add the runtime DRL agent to prune channels dynamically, by fixing the static DLR agent and $u _ { s }$ , and updating the runtime DLR agent and $f ( \\cdot )$ only. Another method is to combine state-of-the-art static and runtime pruning methods. We start from a pretrain backbone CNN, and then prune channels with the static pruning method FPGM (He et al., 2019), and finally prune channels with the runtime method FBS (Gao et al., 2019). The experimental results are shown in Table4. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            779
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/695c96d9b725de1672a27b82f749d23e0cb0eee0e521cee6e8e86b6b317127be.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Comparison to methods with separately static and runtime pruning on Cifar-10 at sparsity 0.5. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Baseline acc.</td><td>Acc.</td><td>△acc.</td></tr><tr><td>ours s (runtime only Rr = 1)</td><td>92.07</td><td>91.333</td><td>-0.737</td></tr><tr><td>ours s(Rr=0.5)</td><td>92.07</td><td>91.066</td><td>-1.004</td></tr><tr><td>ours (separate static and runtime)</td><td>92.07</td><td>90.965</td><td>-1.0105</td></tr><tr><td>FPGM+FBS</td><td>92.07</td><td>90.456</td><td>-1.614</td></tr></table>",
+        "bbox": [
+            259,
+            786,
+            732,
+            869
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 STORAGE/ACCURACY TRADE-OFF ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            454,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Besides Fig. 2, to further illustrate the trade-off between storage and accuracy, we show additional results in Fig. 4. ",
+        "bbox": [
+            174,
+            136,
+            825,
+            165
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb61cd2607b8c848b7d5883f4895e07b06c16c86e7d12f6dae1ddac840f5cca1.jpg",
+        "image_caption": [
+            "Figure 4: Trade-off between runtime pruning and static pruning at sparsity 0.45. X-axis is storage(number of parameters), Y-axis is accuracy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            196,
+            651,
+            377
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B PSEUDOCODE OF TRAINING PROCESS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            486,
+            524,
+            502
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Training process ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            531,
+            379,
+            546
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "INPUT: pretrained backbone CNN, computation budget $\\bar { B } _ { c o m }$ , storage budget $\\bar { B } _ { p a r a m }$   \nOUTPUT: backbone CNN, importance predictor $f ( \\cdot )$ , static pruning importance $u _ { s }$ , runtime and   \nstatic DRL agents   \nAdd runtime importance predictor $f ( \\cdot )$ and static pruning importance $u _ { s }$ .;   \n$u _ { s } \\gets \\mathbf { 1 }$ ;   \n$d _ { s } \\gets 1$ ;   \n$d _ { r } \\gets 1$ ;   \nwhile not converge do fix $u _ { s }$ , update $f ( \\cdot )$ and backbone CNN;   \nend   \nwhile not converge do   \n1 update $u _ { s }$ , $f ( \\cdot )$ and backbone CNN;   \nend   \nadd runtime DRL agent and static DRL agent to predict actions $a _ { t } ^ { r }$ and $a _ { t } ^ { s }$ , and alternate $d _ { r }$ and $d _ { s }$   \nat each layer $t$ ;   \nwhile not converge do for $i \\gets 1$ to $N _ { 1 }$ do Forward entired model; Compute rewards $R _ { t } ^ { r }$ and $R _ { t } ^ { s }$ using budget $\\bar { B } _ { c o m }$ and $\\bar { B } _ { p a r a m }$ ; F $\\mathrm { i } \\times \\bar { u _ { s } } , f ( \\cdot )$ and backbone CNN, update runtime and static DRL agents by PPO loss; end for $i \\gets 1$ to $N _ { 2 }$ do Fix runtime and static DRL agents, update $u _ { s }$ , $f ( \\cdot )$ and backbone CNN by cross entropy loss end   \nend ",
+        "bbox": [
+            171,
+            549,
+            818,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/efd43963a47c3a1afcdfc3c167806714afb5b2f802c5a7ab44ba6c03012e0742.jpg",
+        "image_caption": [
+            "Figure 5: Training curve of our method on CIFAR-10 at sparsity 0.5 with $R _ { r } = 0 . 5$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            104,
+            651,
+            285
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C TRAINING CURVE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
+            356,
+            359
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We show the training curve of our method in Fig. 5. We train our method on CIFAR-10 at sparsity 0.5 with $R _ { r } = 0 . 5$ . The figure includes the accuracy curves evaluated on training set and test set, showing that our methods is stable during training. ",
+        "bbox": [
+            174,
+            375,
+            825,
+            417
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rkxt8oC9FQ/rkxt8oC9FQ.md

@@ -0,0 +1,305 @@
+# PERFECT MATCH: A SIMPLE METHOD FOR LEARNING REPRESENTATIONS FOR COUNTERFACTUAL INFERENCE WITH NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning representations for counterfactual inference from observational data is of high practical relevance for many domains, such as healthcare, public policy and economics. Counterfactual inference enables one to answer ”What if...?” questions, such as ”What would be the outcome if we gave this patient treatment $t _ { 1 } ? ^ { \dag }$ . However, current methods for training neural networks for counterfactual inference on observational data are either overly complex, limited to settings with only two available treatment options, or both. Here, we present Perfect Match (PM), a method for training neural networks for counterfactual inference that is easy to implement, compatible with any architecture, does not add computational complexity or hyperparameters, and extends to any number of treatments. PM is based on the idea of augmenting samples within a minibatch with their propensity-matched nearest neighbours. Our experiments demonstrate that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes across several real-world and semi-synthetic datasets.
+
+# 1 INTRODUCTION
+
+Estimating individual treatment effects1 (ITE) from observational data is an important problem in many domains. In medicine, for example, we would be interested in using data of people that have been treated in the past to predict what medications would lead to better outcomes for new patients (Shalit et al. (2017)). Similarly, in economics, we would for example want to determine how effective job programs would be based on results of past job training programs (LaLonde (1986)).
+
+ITE estimation from observational data is difficult for two reasons: Firstly, we never observe all potential outcomes. If a patient is given a treatment to treat her symptoms, we never observe what would have happened if the patient was prescribed a potential alternative treatment in the same situation. Secondly, the assignment of cases to treatments is typically biased such that cases for which a given treatment is more effective are more likely to have received that treatment. The distribution of samples may therefore differ significantly between the treated group and the overall population. A supervised model na¨ıvely trained to minimise the factual error would overfit to the properties of the treated group, and thus not generalise well to the entire population.
+
+To address these problems, we introduce Perfect Match (PM), a simple method for training neural networks for counterfactual inference that extends to any number of treatment options. PM effectively controls for biased assignment of treatments in observational data by augmenting every sample within a minibatch with its closest matches by propensity score from the other treatment options. PM is easy to use with existing neural network architectures, simple to implement, and does not add any hyperparameters or computational complexity. We perform experiments that demonstrate that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes on several real-world and semi-synthetic datasets. Our experiments also show that PM is more robust to a high level of treatment assignment bias than existing methods. We believe that our simple and effective method for ITE estimation that extends to any number of treatment options could potentially make counterfactual inference more accessible, particularly for medical applications, where multiple available treatment options are the norm.
+
+Contributions. This work contains the following contributions:
+
+• We introduce Perfect Match, a simple methodology for learning neural representations for counterfactual inference based on propensity score matching within minibatches. • We perform extensive experiments on semi-synthetic and real-world data in both the binary and multiple treatment setting. The experiments show that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes.
+
+# 2 RELATED WORK
+
+There are five main categories of approaches to learning to estimate ITEs:
+
+Matching-based Methods. Matching methods estimate the counterfactual outcome of a sample $X$ with respect to treatment $t$ using the factual outcomes of its nearest neighbours having received $t$ , with respect to a metric space. $\mathbf { k }$ -Nearest-Neighbour (kNN) (Ho et al. (2007)) operates in the potentially high-dimensional covariate space, and therefore might suffer from the curse of dimensionality. Propensity Score Matching (PSM) (Rosenbaum & Rubin (1983)) matches on the scalar probability $p ( t | X )$ of $t$ given the covariates $X$ . PSM may require under- or oversampling the original data.
+
+Adjusted Regression Methods. Adjusted regression models apply regression models with both treatment and covariates as input. The simplest case is Ordinary Least Squares, which can either be used for building one model, with the treatment as an input feature, or multiple separate models, one for each treatment (Kallus (2017)). More complex regression models, such as Treatment-Agnostic Representation Networks (TARNET) (Shalit et al. (2017)) may be used to capture non-linear relationships. Methods that make use of propensity scores, and a model of the outcomes, such as Propensity Dropout (PD) (Alaa et al. (2017)), are referred to as doubly robust (Funk et al. (2011)).
+
+Tree-based Methods. Tree-based methods train many weak learners to build expressive ensemble models. Examples of tree-based methods are Bayesian Additive Regression Trees (BART) (Chipman et al. (2010); Chipman & McCulloch (2016)) and Causal Forests (CF) (Wager & Athey (2017)).
+
+Representation-balancing Methods. Representation-balancing methods seek to learn a highlevel representation for which the covariate distributions are balanced across treatment groups. Balancing Neural Networks (Johansson et al. (2016)) attempt to find such representations by minimising the discrepancy distance (Mansour et al. (2009)) between treatment groups. Counterfactual Regression Networks (CFRNET, Shalit et al. (2017)) use different metrics such as the Wasserstein distance.
+
+Distribution-modeling Methods. Generative Adversarial Nets for inference of Individualised Treatment Effects (GANITE) (Yoon et al. (2018)) address ITE estimation using counterfactual and ITE generators. GANITE uses a complex architecture with many hyperparameters and submodels that may be difficult to implement and optimise. Causal Effect Variational Autoencoders (CEVAEs) (Louizos et al. (2017)) use a latent variable modeling approach that is robust to hidden confounding, but does not address biased treatment assignment. Causal Multi-task Gaussian Processes (CMGP) (Alaa & van der Schaar (2017)) apply a multi-task Gaussian Process to ITE estimation. The optimisation of CMGPs involves a matrix inversion of $O ( n ^ { 3 } )$ complexity that limits their scalability.
+
+In contrast to existing methods, PM is a simple doubly robust method built on propensity score matching that can be used to train expressive non-linear neural network models for ITE estimation from observational data in settings with any number of treatments (Table 1). While the underlying idea behind PM is simple and effective, it has, to the best of our knowledge, not yet been explored.
+
+Table 1: Comparison of several representative state-of-the-art methods for counterfactual inference showing whether they were designed for $> 2$ treatments and their complexity of implementation.   
+
+<table><tr><td></td><td>PM</td><td>CF</td><td>BART</td><td>BNN</td><td>CFRNET</td><td>PD</td><td>CEVAE</td><td>CMGP</td><td>GANITE</td></tr><tr><td>&gt;2 Treatments</td><td>√</td><td>√</td><td>√</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>√</td></tr><tr><td>Complexity</td><td>Low</td><td>Low</td><td>Low</td><td>Low</td><td>Medium</td><td>Medium</td><td>High</td><td>High</td><td>High</td></tr></table>
+
+# 3 METHODOLOGY
+
+Problem Setting. We consider a setting in which we are given $N$ observed samples $X$ , where each sample consists of $p$ covariates $x _ { i }$ with $i \in [ 0 , p )$ . For each sample, the potential outcomes are represented as a vector $Y$ with $k$ entries $y _ { j }$ where each entry corresponds to the outcome when applying one treatment $t _ { j }$ out of the set of $k$ available treatments $T = \{ \bar { t } _ { 0 } , . . . , t _ { k - 1 } \}$ with $j \in [ 0 , k )$ . As training data, we receive observed samples $X$ and their factual outcomes $y _ { j }$ when applying one treatment $t _ { j }$ . The set of available treatments can contain two or more treatment options. We refer to the special case of two available treatment options as the binary treatment setting. Given the observational training data with factual outcomes, we wish to train a predictive model that is able to accurately produce a predicted potential outcomes vector $\hat { Y }$ with $k$ entries $\hat { y } _ { j }$ . In literature, this setting is known as the Rubin-Neyman potential outcomes framework (Rubin (2005)).
+
+Precision in Estimation of Heterogenous Effect (PEHE). The primary metric that we optimise for when training models to estimate ITE is the PEHE (Hill (2011)). In the binary setting, the PEHE measures the ability of a predictive model to estimate the difference in effect between two treatments $t _ { 0 }$ and $t _ { 1 }$ for samples $X$ . To compute the PEHE, we measure the mean squared error between the true difference in effect $y _ { 1 } ( n ) - y _ { 0 } ( n )$ , drawn from the noiseless underlying outcome distributions $\mu _ { 1 }$ and $\mu _ { 0 }$ , and the predicted difference in effect $\hat { y } _ { 1 } ( n ) - \hat { y } _ { 0 } ( n )$ indexed by $n$ over $N$ samples:
+
+$$
+\epsilon _ { \mathrm { P E H E } } = \frac { 1 } { N } \sum _ { n = 0 } ^ { N } \Big ( \mathbb { E } _ { y _ { j } ( n ) \sim \mu _ { j } ( n ) } [ y _ { 1 } ( n ) - y _ { 0 } ( n ) ] - [ \hat { y } _ { 1 } ( n ) - \hat { y } _ { 0 } ( n ) ] \Big ) ^ { 2 }
+$$
+
+When the underlying noiseless distributions $\mu _ { j }$ are not known, the true difference in effect $y _ { 1 } ( n ) -$ $y _ { 0 } ( n )$ can be estimated using the noisy ground truth outcomes $y _ { i }$ (Appendix A). As a secondary metric, we consider the error $\epsilon _ { \mathrm { A T E } }$ in estimating the average treatment effect (ATE) (Hill (2011)). The ATE measures the average difference in effect across the whole population (Appendix B). The ATE is not as important as PEHE for models optimised for ITE estimation, but can be a useful indicator of how well an ITE estimator performs at comparing two treatments across the entire population. We can neither calculate $\epsilon _ { \mathrm { P E H E } }$ nor $\epsilon _ { \mathrm { A T E } }$ without knowing the outcome generating process.
+
+Multiple Treatments. Both PEHE and $\epsilon _ { \mathrm { A T E } }$ can be trivially extended to multiple treatments by considering the average PEHE and ATE between every possible pair of treatments. Note that we lose the information about the precision in estimating ITE between specific pairs of treatments by averaging over all $\binom { k } { 2 }$ pairs. However, one can inspect the pair-wise PEHE to get the whole picture.
+
+$$
+\hat { \epsilon } _ { \mathrm { m P E H E } } = \frac { 1 } { \binom { k } { 2 } } \sum _ { i = 0 } ^ { k - 1 } \sum _ { j = 0 } ^ { i - 1 } \hat { \epsilon } _ { \mathrm { P E H E } , i , j } \qquad ( 2 ) \qquad \hat { \epsilon } _ { \mathrm { m A T E } } = \frac { 1 } { \binom { k } { 2 } } \sum _ { i = 0 } ^ { k - 1 } \sum _ { j = 0 } ^ { i - 1 } \hat { \epsilon } _ { \mathrm { A T E } , i , j }
+$$
+
+Perfect Match (PM). We consider fully differentiable neural network models optimised via minibatch stochastic gradient descent to predict potential outcomes $\hat { Y }$ given a sample $X$ . To address the treatment selection bias inherent in observational data, we propose the use of a training methodology based on propensity score matching. A propensity score is the conditional probability $p ( t | X )$ of a given sample $X$ receiving a specific treatment $t$ (Rosenbaum & Rubin (1983); Ho et al. (2007)). The propensity score is a balancing score, meaning that if the distribution of propensity scores across treatment groups is the same, then the distribution of their covariates $x _ { i }$ is also guaranteed to be the same across treatment groups (Ho et al. (2007)). By matching samples with their nearest neighbours by propensity score we can, in the optimal case, balance the covariates $x _ { i }$ . This breaks the dependence of treatment assignment on $X$ if all relevant covariates are observed, and therefore effectively removes any potential treatment assignment bias (Ho et al. (2007)). However, in practice, we do not have access to the true treatment assignment probability $p ( t | X )$ . We therefore have to estimate the propensity score $p ( t | X )$ by training a predictive model to predict the likelihood of receiving a treatment $t$ for a given sample $X$ . Note that propensity scores are not the only balancing score (Ho et al. (2007)). For example, $X$ itself is another balancing score. However, matching on the onedimensional propensity score is preferable because it avoids the curse of dimensionality that would be associated with matching on the potentially high-dimensional $X$ directly. In PM, we match every sample within a minibatch with its nearest neighbours by propensity score from all other treatments (Algorithm 1, more details in Appendix C), taken from the training set. Every minibatch the model is trained on therefore contains the same number of samples for each treatment group, and the covariates $x _ { i }$ of each treatment group are approximately balanced on average. The intuition behind matching on the minibatch level, rather than the dataset level (Ho et al. (2011)), is that it reduces the variance during training which in turn leads to better expected performance for counterfactual inference (Appendix D). In this sense, PM can be seen as a minibatch sampling strategy (Csiba & Richtarik (2018)) specifically designed to improve learning for counterfactual inference. ´
+
+Algorithm 1 Batch Augmentation using Perfect Match (PM). After augmentation, each batch contains an equal number of samples from each treatment group and the covariates $x _ { i }$ across all treatment groups are approximately balanced.
+
+Input: Batch of $B$ random samples $X _ { \mathrm { b a t c h } }$ with assigned treatments $t$ , training set $X _ { \mathrm { t r a i n } }$ of $N$ samples, number of treatment options $k$ , propensity score estimator $E _ { \mathrm { P S } }$ to calculate the probability $\bar { p } ( t | X )$ of treatment assigned given a sample $X$
+
+Output: Batch $X _ { \mathrm { o u t } }$ consisting of $B \times k$ matched samples
+
+<table><tr><td colspan="2">1: procedurePERFECT_MATCH:</td></tr><tr><td>2:</td><td>Xout ←Empty</td></tr><tr><td>3:</td><td>for sample X with treatment t in Xbatch do</td></tr><tr><td>4:</td><td>p(t|X)←Eps(X)</td></tr><tr><td>5:</td><td>fori=O tok-1do</td></tr><tr><td>6:</td><td>ifi≠t then</td></tr><tr><td>7:</td><td>psi←p(t|X)i</td></tr><tr><td>8:</td><td>Xmatched ← get closest match to propensity score psi with treatment i from Xtrain</td></tr><tr><td>9:</td><td>Add sample Xmatched to Xout</td></tr><tr><td>10:</td><td>Add X to Xout</td></tr><tr><td></td><td></td></tr></table>
+
+Model Selection. Besides accounting for the treatment assignment bias, the other major issue in learning for counterfactual inference from observational data is that, given multiple models, it is not trivial to decide which one to select. The root problem is that we do not have direct access to the true error in estimating counterfactual outcomes, only the error in estimating the observed factual outcomes. This makes it difficult to perform parameter and hyperparameter optimisation, as we do not know which models are better than others for counterfactual inference on a given dataset. To rectify this problem, we use a nearest neighbour approximation $\hat { \epsilon } _ { \mathrm { N N - P E H E } }$ of the $\boldsymbol { \hat { \epsilon } } _ { \mathrm { P E H E } }$ metric for the binary (Shalit et al. (2017)) and multiple treatment settings for model selection. The ˆNN-PEHE estimates the treatment effect of a given sample by substituting the true counterfactual outcome with the outcome $y _ { j }$ from a respective nearest neighbour NN matched on $X$ using the Euclidean distance.
+
+$$
+\hat { \epsilon } _ { \mathrm { N N - P E H E } } = \frac { 1 } { N } \sum _ { n = 0 } ^ { N } \Big ( [ y _ { 1 } ( \mathbf { N N } ( n ) ) - y _ { 0 } ( \mathbf { N N } ( n ) ) ] - [ \hat { y } _ { 1 } ( n ) - \hat { y } _ { 0 } ( n ) ] \Big ) ^ { 2 }
+$$
+
+Analogously to Equations (2) and (3), the ˆNN-PEHE metric can be extended to the multiple treatment setting by considering the mean $\hat { \epsilon } _ { \mathrm { N N - P E H E } }$ between all $\binom { k } { 2 }$ possible pairs of treatments (Appendix E).
+
+Model Architecture. No different from other applications, the chosen architecture plays a key role in the performance of neural networks when attempting to learn representations for counterfactual inference. Shalit et al. (2017) claimed that the na¨ıve approach of appending the treatment index $t _ { j }$ may perform poorly if $X$ is high-dimensional, because the influence of $t _ { j }$ on the hidden layers may be lost during training. Shalit et al. (2017) subsequently introduced the TARNET architecture to rectify this issue. Since the original TARNET was limited to the binary treatment setting, we extended the TARNET architecture to the multiple treatment setting (Figure 1). We did so by using $k$ head networks, one for each treatment option over a set of shared base layers, each with $L$ layers. In TARNET, the $j$ th head network is only trained on samples from treatment $t _ { j }$ . The shared layers are trained on all samples. By using a head network for each treatment, we ensure $t _ { j }$ maintains an appropriate degree of influence on the network output at all points during training.
+
+![](images/df697ae164c15f61e6243c85e1aee1e920e1c2b644a0d3442482f69f5b1c439e.jpg)  
+Figure 1: The TARNET architecture with $k$ heads for the multiple treatment setting.
+
+Table 2: Comparisons of the datasets used in our experiments. We evaluate on three semi-synthetic datasets with varying numbers of treatments and samples, and a real-world clinical trial dataset.   
+
+<table><tr><td>Dataset</td><td>Type</td><td># Samples</td><td>#Features</td><td># Treatments</td><td>Counterfactuals</td></tr><tr><td>IHDP</td><td>semi-synthetic</td><td>747</td><td>25</td><td>2</td><td>available</td></tr><tr><td>Jobs</td><td>real-world</td><td>3212</td><td>7</td><td>2</td><td>not available</td></tr><tr><td>News</td><td>semi-synthetic</td><td>5000</td><td>2870</td><td>2/4/8/16</td><td>available</td></tr><tr><td>TCGA</td><td>semi-synthetic</td><td>9659</td><td>20531</td><td>4</td><td>available</td></tr></table>
+
+# 4 EXPERIMENTS
+
+Our experiments aimed to answer the following questions:
+
+1 What is the comparative performance of PM in inferring counterfactual outcomes in the   
+binary and multiple treatment setting compared to existing state-of-the-art methods?   
+2 Does model selection by NN-PEHE outperform selection by factual MSE?   
+3 How does the relative number of matched samples within a minibatch affect performance?   
+4 How well does PM cope with an increasing treatment assignment bias in the observed data?   
+5 How does the presence of hidden confounders influence the performance of PM?   
+6 How do the learning dynamics of minibatch matching compare to dataset-level matching?
+
+# 4.1 DATASETS
+
+We performed experiments on four real-world and semi-synthetic datasets (Table 2) with binary and multiple treatment options in order to gain a better understanding of the empirical properties of PM.
+
+Infant Health and Development Program (IHDP). The IHDP dataset (Hill (2011)) contains data from a randomised study on the impact of specialist visits on the cognitive development of children, and consists of 747 children with 25 covariates describing properties of the children and their mothers. Children that did not receive specialist visits were part of a control group. The outcomes were simulated using the NPCI package from Dorie (2016). The IHDP dataset is biased because the treatment groups had a biased subset of the treated population removed (Shalit et al. (2017)). We used the same simulated outcomes2 as Shalit et al. (2017).
+
+Jobs. The Jobs dataset (LaLonde (1986)) is a blend of data from randomised and observational studies on the effect of professional training programs on unemployment, and is a commonly used benchmark dataset for counterfactual inference in the binary setting. Each sample consisted of demographic covariates, such as age, gender and previous income, and the treatment was enrolment in a job training program. All samples that were not enrolled in a training program were considered control cases. The task was to predict the potential unemployment given the covariates and treatment. We used the same feature and sample set as described in Dehejia & Wahba (2002); Smith & Todd (2005); Shalit et al. (2017). A subgroup of the data was randomised and we therefore were able to estimate the ground truth average treatment effect on the treated (ATT) (Shalit et al. (2017)). Treatment assignment in Jobs is biased because parts of the data come from an observational study.
+
+![](images/9918fa2b90357bcf89b69a749b3ec1a315ad45bc7e9f7de7ce33b085ecab1308.jpg)  
+Figure 2: Correlation analysis of the real PEHE (y-axis) with the mean squared error (MSE; left) and the nearest neighbour approximation of the precision in estimation of heterogenous effect (NN-PEHE; right) across over $2 0 ^ { \circ } 0 0 0$ model evaluations on the validation set of IHDP. Scatterplots show a subsample of $1 ^ { \cdot } 4 0 0$ data points. $\rho$ indicates the Pearson correlation. NN-PEHE correlates significantly better with PEHE than MSE.
+
+![](images/dbb418865d647f6219d58af93434042418c17613b1162905fddcc533dee487a3.jpg)  
+Figure 3: Change in error (y-axes) in terms of precision in estimation of heterogenous effect (PEHE) and average treatment effect (ATE) when increasing the percentage of matches in each minibatch ( $\mathbf { \dot { x } }$ -axis). Symbols correspond to the mean value of $\hat { \epsilon } _ { \mathrm { A T E } }$ (red) and $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ (blue) on the test set of News-8 across 50 repeated runs with new outcomes (lower is better). Performance improves with more matches added.
+
+![](images/0e66495e0af1f70fbbb9c30fe17567012caef29fd2f299eb4bb1e2a7bb301f2e.jpg)  
+Figure 4: Comparison of several state-of-the-art methods for counterfactual inference on the test set of the News-8 dataset when varying the treatment assignment imbalance $\kappa$ $\mathbf { \dot { x } }$ -axis), i.e. how much the treatment assignment is biased towards more effective treatments. Symbols correspond to the mean value of $\hat { \epsilon } _ { \mathrm { A T E } }$ (left) and $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ (right) across 50 repeated runs with new outcomes (lower is better). The shaded area indicates the standard deviation. PM handles an increasing treatment assignment bias $\kappa$ better than existing state-of-the-art methods in terms of both $\hat { \epsilon } _ { \mathrm { A T E } }$ and $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ .
+
+News. The News dataset was first proposed as a benchmark for counterfactual inference by Johansson et al. (2016) and consists of 5000 randomly sampled news articles from the NY Times corpus3. The News dataset contains data on the opinion of media consumers on news items. The samples $X$ represent news items consisting of word counts $x _ { i } \in \mathbb { N }$ , the outcome $y _ { j } \in \mathbb { R }$ is the reader’s opinion of the news item, and the $k$ available treatment options represent various devices that could be used for viewing, e.g. smartphone, tablet, desktop, television or others (Johansson et al. (2016)). We extended the original dataset specification in (Johansson et al. (2016)) to enable the simulation of arbitrary numbers of viewing devices. To model that consumers prefer to read certain media items on specific viewing devices, we train a topic model on the whole NY Times corpus and define $z ( X )$ as the topic distribution of news item $X$ . We then randomly pick $k + 1$ centroids in topic space, with $k$ centroids $z _ { j }$ per viewing device and one control centroid $z _ { c }$ . We assigned a random Gaussian outcome distribution with mean $\mu _ { j } \sim \mathcal { N } ( 0 . 4 5 , 0 . 1 5 )$ and standard deviation $\sigma _ { j } \sim \mathcal { N } ( 0 . 1 , 0 . 0 5 )$ to each centroid. For each sample, we drew ideal potential outcomes from that Gaussian outcome distribution $\tilde { y } _ { j } \sim \mathcal { N } ( \mu _ { j } , \sigma _ { j } ) + \epsilon$ with $\epsilon \sim \mathcal { N } ( 0 , 0 . 1 5 )$ . We then defined the unscaled potential outcomes $\bar { y _ { j } } = \tilde { y } _ { j } * \left[ \mathbf { D } ( z ( X ) , z _ { j } ) + \mathbf { D } ( z ( X ) , z _ { c } ) \right]$ as the ideal potential outcomes $\tilde { y } _ { j }$ weighted by the sum of distances to centroids $z _ { j }$ and the control centroid $z _ { c }$ using the Euclidean distance as distance D. We assigned the observed treatment $t$ using $t | x \sim \mathrm { B e r n } ( \operatorname { s o f t m a x } ( \kappa \bar { y _ { j } } ) )$ with a treatment assignment bias coefficient $\kappa$ , and the true potential outcome $y _ { j } = C y _ { j }$ as the unscaled potential outcomes $\bar { y _ { j } }$ scaled by a coefficient $C = 5 0$ . We used four different variants of this dataset with $k = 2$ , 4, 8, and 16 viewing devices, and $\kappa = 1 0$ , 10, 10, and 7, respectively. Higher values of $\kappa$ indicate a higher expected treatment assignment bias depending on $\bar { y _ { j } }$ . $\kappa = 0$ indicates no assignment bias.
+
+The Cancer Genomic Atlas (TCGA). The TCGA project collected gene expression data from various types of cancers in 9659 individuals (Weinstein et al. (2013)). There were four available clinical treatment options: (1) medication, (2) chemotherapy, (3) surgery, or (4) both surgery and chemotherapy. We used a synthetic outcome function that simulated the risk of cancer recurrence after receiving either of the treatment options based on the real-world gene expression data. Before further processing, we standardised the gene expression data using the mean and standard deviations of gene expression at each gene locus for normal tissue in the training set. To produce the potential outcomes $Y$ , we first selected $k + 1$ gene expression phenotypes as representative phenotypes: One phenotype $X _ { j }$ for each treatment option and one as a control phenotype $X _ { c }$ . Analogously to the News dataset, we assigned a random Gaussian outcome distribution with mean $\mu _ { j } \sim \bar { \mathcal { N } } ( 0 . 4 \bar { 5 } , 0 . 1 5 )$ and standard deviation $\sigma _ { t } \sim \mathcal { N } ( 0 . 1 , 0 . 0 5 )$ to each phenotype. For each sample, we drew ideal potential outcomes from that Gaussian outcome distribution $\bar { \tilde { y } } _ { j } \sim \mathcal N ( \mu _ { j } , \sigma _ { j } ) + \bar { \epsilon }$ with $\epsilon \sim \mathcal { N } ( 0 , 0 . \bar { 1 5 } )$ . We then defined the unscaled potential outcomes $\bar { y _ { j } } = \tilde { y } _ { j } * \left[ \mathbf { D } ( X , X _ { t _ { j } } ) + \mathbf { D } ( X , X _ { c } ) \right]$ as the ideal potential outcomes $\tilde { y } _ { j }$ weighted by the sum of distances to phenotype $X _ { t _ { j } }$ and the control phenotype $X _ { c }$ using the cosine similarity as distance metric D. We assigned the observed treatment $t$ using $t | x \sim \mathrm { B e r n } ( \operatorname { s o f t m a x } ( \kappa \bar { y _ { j } } ) )$ with a treatment assignment bias coefficient $\kappa = 1 0$ , and the true potential outcome $y _ { j } = C y _ { j }$ as the unscaled potential outcomes $\bar { y _ { j } }$ scaled by a coefficient $C = 5 0$ .
+
+All datasets with the exception of IHDP were split into a training $( 6 3 \% )$ , validation $( 2 7 \% )$ and test set ( $10 \%$ of samples). For IHDP we used exactly the same splits as previously used by Shalit et al. (2017). We repeated experiments on IHDP, Jobs, News and TCGA 1000, 10, 50, and 5 times, respectively. We reassigned outcomes and treatments with a new random seed for each repetition.
+
+Table 3: Comparison of methods for counterfactual inference with two available treatment options on IHDP, Jobs and News-2. We report the mean value $\pm$ the standard deviation of $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ , ATE, $R _ { \mathrm { P o l } }$ and $\epsilon _ { \mathrm { A T T } }$ on the test sets over a number of repeated runs. Where available we list the numbers reported by the original authors. n.r. $=$ not reported. ${ \dag \mathbf { \alpha } } _ { = }$ significantly different from PM $( \alpha < 0 . 0 5 )$   
+
+<table><tr><td></td><td colspan="2">IHDP (1000 repeats)</td><td colspan="2">Jobs (10 repeats)</td><td colspan="2">News-2 (50 repeats)</td></tr><tr><td>Method</td><td>√EPEHE</td><td>EATE</td><td>RPo1(π）</td><td>EATT</td><td>VePEHE</td><td>CATE</td></tr><tr><td>PM</td><td>0.84 ± 0.61</td><td>0.24±0.20</td><td>0.18 ± 0.10</td><td>0.16 ± 0.07</td><td>16.76 ± 1.26</td><td>3.99 ± 1.01</td></tr><tr><td>+on X</td><td>0.81 ± 0.57</td><td>0.24 ± 0.20</td><td>0.23 ± 0.11</td><td>0.19 ± 0.09</td><td>17.06 ± 1.22</td><td>4.14 ± 1.27</td></tr><tr><td>+MLP</td><td>0.83 ± 0.57</td><td>0.23 ± 0.20</td><td>0.19 ± 0.12</td><td>0.16 ± 0.07</td><td>+ 18.38 ± 1.46</td><td>+ 5.90 ± 2.07</td></tr><tr><td>kNN</td><td>4.1 1±0.2</td><td>0.79± 0.05</td><td>+ 0.26±0.0</td><td>0.13 ± 0.05</td><td>† 18.14 ± 1.64</td><td>+7.83 ± 2.55</td></tr><tr><td>PSMPM</td><td>n.r.</td><td>n.r.</td><td>0.18 ± 0.12</td><td>0.18 ± 0.09</td><td>† 17.49 ± 1.49</td><td>† 5.02 ± 2.34</td></tr><tr><td>PSMMI</td><td>+ 2.70 ± 3.85</td><td>+ 0.49 ± 0.81</td><td>0.20 ± 0.09</td><td>+ 0.42 ± 0.22</td><td>+ 17.40 ± 1.30</td><td>+ 4.89 ± 2.39</td></tr><tr><td>RF</td><td>6.6 ±0.3</td><td>† 0.96 ± 0.06</td><td>† 0.28 ± 0.0</td><td>+ 0.09 ± 0.04</td><td>† 17.39 ± 1.24</td><td>† 5.50 ± 1.20</td></tr><tr><td>CF</td><td>3.8 ±0.2</td><td>+ 0.40 ±0.03</td><td>0.20±0.0</td><td>+ 0.07 ± 0.03</td><td>† 18.36 ± 1.73</td><td>+ 8.48± 2.46</td></tr><tr><td>BART</td><td>2.3 ±0.1</td><td>+ 0.34± 0.02</td><td>+ 0.25 ± 0.0</td><td>+ 0.08±0.03</td><td>18.53 ± 2.02</td><td>† 5.40 ± 1.53</td></tr><tr><td>GANITE</td><td>2.4 ±0.4</td><td>+ 0.49± 0.05</td><td>0.14 ± 0.01</td><td>+ 0.06± 0.03</td><td>+ 18.28 ± 1.66</td><td>+ 4.65 ± 2.12</td></tr><tr><td>BNN</td><td>+ 2.1 ±0.1</td><td>+ 0.42± 0.03</td><td>0.24 ± 0.0</td><td>† 0.09 ± 0.04</td><td>n.r.</td><td>n.r.</td></tr><tr><td>PD</td><td>n.r.</td><td>n.r.</td><td>n.r.</td><td>n.r.</td><td>† 17.52 ± 1.62</td><td>4.69 ± 3.17</td></tr><tr><td>TARNET</td><td>+ 0.95 ± 0.02</td><td>+ 0.28 ± 0.01</td><td>0.21± 0.0</td><td>0.11 ± 0.04</td><td>17.17 ± 1.25</td><td>+ 4.58 ± 1.29</td></tr><tr><td>CFRNETwass</td><td>+ 0.76± 0.02</td><td>t 0.27 ± 0.01</td><td>0.21± 0.0</td><td>+ 0.09 ± 0.03</td><td>16.93 ± 1.12</td><td>† 4.54 ± 1.48</td></tr><tr><td>CEVAE</td><td>+ 2.7 ±0.1</td><td>+ 0.46 ± 0.02</td><td>+ 0.26 ±0.0</td><td>+ 0.03 ± 0.01</td><td>n.r.</td><td>n.r.</td></tr><tr><td>CMGP</td><td>+ 0.77 ± 0.11</td><td>+ 0.13 ± 0.12</td><td>0.24± 0.05</td><td>+ 0.09 ± 0.07</td><td>n.r.</td><td>n.r.</td></tr></table>
+
+# 4.2 EXPERIMENTAL SETUP
+
+Models. We evaluated PM, ablations, baselines, and all relevant state-of-the-art methods: kNN (Ho et al. (2007)), BART (Chipman et al. (2010); Chipman & McCulloch (2016)), Random Forests (RF) (Breiman (2001)), CF (Wager & Athey (2017)), GANITE (Yoon et al. (2018)), Balancing Neural Network (BNN) (Johansson et al. (2016)), TARNET (Shalit et al. (2017)), Counterfactual Regression Network using the Wasserstein regulariser (CFRNETWass) (Shalit et al. (2017)), CEVAE (Louizos et al. (2017)), CMGP (Alaa & van der Schaar (2017)), and PD (Alaa et al. (2017)). We trained a Support Vector Machine (SVM) with probability estimation (Pedregosa et al. (2011)) to estimate $p ( t | X )$ for PM on the training set. We also evaluated preprocessing the entire training set with PSM using the same matching routine as PM $( \mathrm { P S M } _ { \mathrm { P M } } )$ and the ”MatchIt” package $( \mathrm { P S M } _ { \mathrm { M I } }$ , Ho et al. (2011)) before training a TARNET (Appendix F). In addition, we trained an ablation of PM where we matched on the covariates $X$ $\ l ^ { \prime } + \ l \mathrm { o n } X )$ directly, if $X$ was low-dimensional $\left( p < 2 0 0 \right)$ ), and on a 50-dimensional representation of $X$ obtained via principal components analysis (PCA), if $X$ was high-dimensional, instead of on the propensity score. We also evaluated PM with a multi-layer perceptron $( + \mathrm { M L P } )$ that received the treatment index $t _ { j }$ as an input instead of using a TARNET.
+
+![](images/8bff40e945252228b1e3394856192f4b1553bd7e163b4cc04175983d61d536e5.jpg)  
+Figure 5: Comparison of several state-of-the-art methods for counterfactual inference on the test set of the TCGA dataset when varying the percentage of hidden confounding $\mathbf { \bar { X } }$ -axis), i.e. what percentage of the features that were relevant for treatment assignment are visible to the model. Symbols correspond to the mean value of $\hat { \epsilon } _ { \mathrm { A T E } }$ (left) and $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ (right) across 5 repeated runs with new outcomes (lower is better). The shaded area indicates the standard deviation. PM, TARNET and CFRNET are relatively stable up to $60 \%$ of hidden confounding. The methods degrade strongly in counterfactual prediction performance at levels of hidden confounding higher than $60 \%$ .
+
+Table 4: Comparison of methods for counterfactual inference with more than two available treatment options on the News-4, News-8 and News-16. Numbers are mean value $\pm$ the standard deviation of $\sqrt { \hat { \epsilon } _ { \mathrm { m P E H E } } }$ and $\hat { \epsilon } _ { \mathrm { m A T E } }$ over 50 repeated runs. $^ { \dag } =$ significantly different from PM $( \alpha < 0 . 0 5 )$   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">News-4 (κ = 10)</td><td colspan="2">News-8 (κ = 10)</td><td colspan="2"></td><td colspan="2">News-16 (k = 7)</td></tr><tr><td>VemPEHE</td><td>EmATE</td><td></td><td>VemPEHE</td><td>CmATE</td><td></td><td>VemPEHE</td><td>CmATE</td></tr><tr><td>PM</td><td>21.58 ± 2.58</td><td>10.04 ± 2.71</td><td></td><td>20.76 ± 1.86</td><td>6.51 ± 1.66</td><td></td><td>20.24 ± 1.46</td><td>5.76 ±1.33</td></tr><tr><td>+on X</td><td>21.41 ± 1.75 +</td><td>8.91 ± 2.00</td><td></td><td>20.90 ± 2.07</td><td>6.36 ± 1.65</td><td></td><td>20.67 ± 1.42</td><td>5.73 ± 1.16</td></tr><tr><td>+ MLP</td><td>+ 25.05 ± 2.80</td><td>+14.92±3.14</td><td>+</td><td>24.88±1.98 +</td><td>14.00±2.45+</td><td></td><td>27.05± 2.47 † 16.90 ± 2.01</td><td></td></tr><tr><td>kNN</td><td>+27.92 ± 2.44</td><td></td><td></td><td>4 + 19.40 ± 3.12 † 26.20± 2.18 † 15.11 ± 2.34 + 2</td><td></td><td></td><td>27.64 ± 2.40 † 17.27 ± 2.10</td><td></td></tr><tr><td>PSMPM</td><td>+ 22.74 ± 2.58 +</td><td></td><td></td><td>11.62 ± 2.69 † 22.16±1.79 †</td><td>8.81±1.96 + 2</td><td></td><td>23.57 ± 2.48 † 11.04 ± 2.50</td><td></td></tr><tr><td>PSMMI</td><td>† 37.26± 2.28 ↑</td><td></td><td></td><td>30.19 ± 2.47 + 30.50± 1.70 † 22.09±1.98</td><td></td><td>+</td><td>28.17 ± 2.02 † 18.81 ± 1.74</td><td></td></tr><tr><td>RF</td><td>+ 26.59 ± 3.02 +</td><td></td><td></td><td>18.03 ±3.18 † 23.77± 2.14 +</td><td>12.40±2.29</td><td>+</td><td>26.13± 2.48 †15.91± 2.00</td><td></td></tr><tr><td>CF</td><td>t 27.28 ± 2.59 ↑</td><td>19.04± 3.15</td><td>→</td><td>25.76±2.05 +</td><td>15.12±2.20+</td><td></td><td>26.85± 2.29</td><td>5+17.03 ± 1.99</td></tr><tr><td>BART</td><td>+ 26.41 ± 3.10 † 17.14± 3.51</td><td></td><td></td><td>t 25.78± 2.66 +</td><td>14.80±2.56†</td><td></td><td>27.45± 2.84 + 17.50± 2.49</td><td></td></tr><tr><td>GANITE</td><td>+ 24.50 ± 2.27 + 13.84 ± 2.69</td><td></td><td>†</td><td>23.58±2.48 +</td><td>11.20±2.84+2</td><td></td><td>25.12± 3.53 + 13.20±3.28</td><td></td></tr><tr><td>PD</td><td>20.88± 3.24 †</td><td>8.47 ± 4.51</td><td></td><td>21.19 ± 2.29</td><td>7.29 ± 2.97 †</td><td></td><td>22.28± 2.25 + 10.65± 2.22</td><td></td></tr><tr><td>TARNET</td><td>+23.40±2.20</td><td>)†13.63 ± 2.18</td><td>†</td><td>22.39± 2.32 +</td><td>9.38 ± 1.92</td><td>+</td><td>21.19 ± 2.01 †</td><td>8.30 ± 1.66</td></tr><tr><td>CFRNETWass</td><td>t 22.65± 1.97 + 12.96± 1.69</td><td></td><td></td><td>21.64±1.82 +</td><td>8.79 ± 1.68</td><td>+</td><td>20.87±1.46†</td><td>8.05 ±1.40</td></tr></table>
+
+Architectures. To ensure that differences between methods of learning counterfactual representations for neural networks are not due to differences in architecture, we based the neural architectures for TARNET, CFRNETWass, PD and PM on the same, previously described extension of the TARNET architecture (Shalit et al. (2017), Details in Appendix G) to the multiple treatment setting.
+
+Hyperparameters. For the IHDP, Jobs, News and TCGA dataset we respectively used 30, 30, 10, and 5 optimisation runs for each method using randomly selected hyperparameters from predefined ranges (Appendix H). We selected the best model across the runs based on validation set $\hat { \epsilon } _ { \mathrm { N N } }$ -PEHE.
+
+Metrics. We calculated the PEHE and $\epsilon _ { \mathrm { A T E } }$ for datasets for which we know the outcome generating process. For the Jobs dataset, we did not have access to counterfactual outcomes. We therefore calculated the $\begin{array} { r } { \mathrm { A T T } \ = \ \frac { 1 } { | T _ { 1 } \cap E | } \sum _ { n = 0 } ^ { N } y _ { 1 } ( n ) - \frac { 1 } { | T _ { 0 } \cap E | } \sum _ { n = 0 } ^ { N } y _ { 0 } ( n ) } \end{array}$ PNn=0 y0(n) where E was the randomised subset of the dataset, and reported the error in estimating ATT $\begin{array} { r } { \epsilon _ { \mathrm { A T T } } = | \mathrm { A T T } - \frac { 1 } { | T _ { 1 } \cap E | } \sum _ { n = 0 } ^ { N } \hat { y } _ { 1 } ( n ) | , } \end{array}$ and the policy risk $R _ { \mathrm { P o l } } ( \pi )$ (Shalit et al. (2017); Yoon et al. (2018)). The policy risk is the average loss in value when treating according to the policy $\pi$ implied by an ITE estimator (Appendix I).
+
+# 5 RESULTS AND DISCUSSION
+
+Counterfactual Inference. We evaluated the counterfactual inference performance of the listed models in the binary setting (Table 3) and multiple treatment setting (Table 4). On IHDP, PM reached the third best performance in terms of $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ after CFRNET and CMGP, and the second best $\epsilon _ { \mathrm { A T E } }$ after CMGP. On Jobs, PM reached the second best $R _ { \mathrm { P o l } } ( \pi )$ after GANITE. On the binary News-2, PM outperformed all other methods in terms of $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ and $\epsilon _ { \mathrm { A T E } }$ . On the News-4/8/16 datasets with more than two treatments, PM consistently outperformed all other methods - in some cases by a large margin - on both metrics with the exception of the News-4 dataset, where PM came second to PD. The strong performance of PM across a wide range of datasets with varying amounts of treatment options is remarkable considering how simple it is compared to other, highly specialised methods. Notably, PM consistently outperformed both CFRNET, with the exception of the IHDP dataset, which accounted for covariate imbalances between treatments via regularisation rather than matching, and $\mathrm { P S M } _ { \mathrm { M I } }$ , which accounted for covariate imbalances by preprocessing the entire training set with a matching algorithm (Ho et al. (2011)). We also found that matching on the propensity score was, in almost all cases, not significantly different from matching on $X$ directly when $X$ was low-dimensional, or a low-dimensional representation of $X$ when $X$ was high-dimensional $^ +$ o n $X$ ). This indicates that PM is effective with any low-dimensional balancing score. In addition, using PM with the TARNET architecture outperformed the MLP $+ \mathrm { M L P } )$ in almost all cases, with the exception of the low-dimensional IHDP and Jobs. We conclude that matching on the propensity score or a low-dimensional representation of $X$ and using the TARNET architecture are sensible default configurations, particularly when $X$ is high-dimensional.
+
+![](images/781506b3f9930a44bc60e6f40e4a257c591bc7efdf418d1d1b39c427024efb86.jpg)  
+Figure 6: Comparison of the learning dynamics during training (normalised training epochs; from start ${ } = 0$ to end $= 1 0 0$ of training, $\mathbf { X }$ -axis) of several matching-based methods on the validation set of News-8. The coloured lines correspond to the mean value of the factual error $\mathrm { \cdot } \sqrt { \mathrm { M S E } }$ ; red) and the counterfactual error $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ ; blue) across 10 randomised hyperparameter configurations (lower is better). The shaded area indicates the standard deviation. All methods used exactly the same model and hyperparameters and only differed in how they addressed treatment assignment imbalance. The leftmost figure shows the desired behavior of the counterfactual and factual error jointly decreasing until convergence. PM exhibits the desired behavior of aligning the optimisation of the factual and counterfactual error. $\mathrm { P S M } _ { \mathrm { P M } }$ shows a weaker degree of alignment and much higher variance. In contrast, $\mathrm { P S M } _ { \mathrm { M I } }$ shows the undesired behavior of the counterfactual error increasing as the factual error decreases, indicating that the models were overfitting to the properties of the treated group.
+
+Model Selection. To judge whether NN-PEHE is more suitable for model selection for counterfactual inference than MSE, we compared their respective correlations with the PEHE on IHDP. We found that NN-PEHE correlates significantly better with the PEHE than MSE (Figure 2).
+
+Number of Matches per Minibatch. To determine the impact of matching fewer than $100 \%$ of all samples in a batch, we evaluated PM on News-8 trained with varying percentages of matched samples on the range 0 to $100 \%$ in steps of $10 \%$ (Figure 3). We found that including more matches indeed consistently reduces the counterfactual error up to $100 \%$ of samples matched. Interestingly, we found a large improvement over using no matched samples even for relatively small percentages $( < 4 0 \% )$ of matched samples per batch. This shows that propensity score matching within a batch is indeed highly effective at improving the training of neural networks for counterfactual inference.
+
+Treatment Assignment Bias. To assess how the predictive performance of the different methods is influenced by increasing amounts of treatment assignment bias, we evaluated their performances on News-8 while varying the assignment bias coefficient $\kappa$ on the range of 5 to 20 (Figure 4). We found that PM handles high amounts of assignment bias better than existing state-of-the-art methods.
+
+Hidden Confounding. To analyse the sensitivity of the different methods to hidden confounding, we successively retrained the models on a reduced set of covariates $x _ { i }$ in increasing steps of $10 \%$ $( = \ 2 0 5 3$ gene loci) in the range of $10 \%$ to $90 \%$ (Figure 5). We removed the covariates from the model but kept them for computing assignments and outcomes, i.e. they were hidden confounders. The results showed that PM was competitive with TARNET and CFRNET in terms of robustness to hidden confounding. Notably, PM, TARNET and CFRNET were relatively stable up to levels of $60 \%$ of hidden confounding, likely because some gene loci are highly correlated.
+
+Comparing Minibatch and Dataset Matching. As outlined previously, it is possible to optimise the counterfactual error using the observed factual samples if (1) we manage to break the dependence of treatment assignment on $X$ , and (2) we observe all relevant variables (Ho et al. (2007)). If we were successful in balancing the covariates, we would expect that the counterfactual error is implicitly and consistently improved alongside the factual error, despite optimising the neural network for the observed factual loss only. To elucidate to what degree this is the case when using the various evaluated matching-based methods, we evaluated the respective training dynamics of PM, $\mathrm { P S M } _ { \mathrm { P M } }$ and $\mathrm { P S M } _ { \mathrm { M I } }$ (Figure 6). We found that PM better conforms to the desired behavior than $\mathrm { P S M } _ { \mathrm { P M } }$ and $\mathrm { P S M } _ { \mathrm { M I } }$ . $\mathrm { P S M } _ { \mathrm { P M } }$ , which used the same matching strategy as PM but on the dataset level, showed a much higher variance than PM. $\mathrm { P S M } _ { \mathrm { M I } }$ was overfitting to the treated group.
+
+Limitations. A limitation of this work is that the theory of matching to balance the covariates only holds in the strongly ignorable setting, i.e. under the assumption that there are no unobserved confounders. However, our experiments (Figure 5) and related works showed that even the presence of large amounts of hidden confounders may not necessarily decrease the performance of ITE estimators in practice if we observe proxy variables (Montgomery et al. (2000); Louizos et al. (2017)). In future work, we would like to integrate PM with diagnostic tools that address hidden confounding (Kallus & Zhou (2018)), and the ability to learn from multiple related tasks (Schwab et al. (2018)).
+
+# 6 CONCLUSION
+
+We presented PM, a new and simple method for training neural networks for estimating ITEs from observational data that extends to any number of available treatment options. In addition, we extended the PEHE metric to settings with more than two treatments, and introduced a nearest neighbour approximation of PEHE and mPEHE that can be used for model selection without having access to counterfactual outcomes. We performed experiments on several real-world and semisynthetic datasets that showed that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes. We also found that the NN-PEHE correlates significantly better with real PEHE than MSE, that including more matched samples in each minibatch improves the learning of counterfactual representations, and that PM handles an increasing treatment assignment bias better than existing state-of-the-art methods. PM may be used for settings with any amount of treatment options, is compatible with any existing neural network architecture, simple to implement, and does not introduce any additional hyperparameters or computational complexity.
+
+# REFERENCES
+
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+Ahmed M Alaa, Michael Weisz, and Mihaela van der Schaar. Deep counterfactual networks with propensity-dropout. arXiv preprint arXiv:1706.05966, 2017.   
+Susan Athey, Julie Tibshirani, and Stefan Wager. Generalized random forests. arXiv preprint arXiv:1610.01271, 2016.   
+Leo Breiman. Random forests. Machine learning, 45(1):5–32, 2001.   
+Hugh Chipman and Robert McCulloch. BayesTree: Bayesian additive regression trees. R package version 0.3-1.4, 2016.   
+Hugh A Chipman, Edward I George, Robert E McCulloch, et al. BART: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266–298, 2010.   
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+Vincent Dorie. NPCI: Non-parametrics for causal inference, 2016. URL https://github. com/vdorie/npci.   
+Michele Jonsson Funk, Daniel Westreich, Chris Wiesen, Til Strmer, M. Alan Brookhart, and Marie Davidian. Doubly robust estimation of causal effects. American Journal of Epidemiology, 173 (7):761–767, 2011.   
+Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217–240, 2011.   
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+Daniel E Ho, Kosuke Imai, Gary King, Elizabeth A Stuart, et al. MatchIt: nonparametric preprocessing for parametric causal inference. Journal of Statistical Software, 42(8):1–28, 2011.   
+Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual inference. In International Conference on Machine Learning, pp. 3020–3029, 2016.   
+Nathan Kallus. Recursive partitioning for personalization using observational data. In International Conference on Machine Learning, 2017.
+
+Nathan Kallus and Angela Zhou. Confounding-robust policy improvement. Advances in Neural Information Processing Systems, 2018.
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+Adam Kapelner and Justin Bleich. bartMachine: Machine learning with Bayesian additive regression trees. arXiv preprint arXiv:1312.2171, 2013.
+
+Robert J LaLonde. Evaluating the econometric evaluations of training programs with experimental data. The American economic review, pp. 604–620, 1986.
+
+Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. In Advances in Neural Information Processing Systems, pp. 6446–6456, 2017.
+
+Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Domain adaptation: Learning bounds and algorithms. arXiv preprint arXiv:0902.3430, 2009.
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+F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine Learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
+
+Paul R. Rosenbaum and Donald B. Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41–55, 1983.
+
+Donald B Rubin. Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association, 100(469):322–331, 2005.
+
+Patrick Schwab, Emanuela Keller, Carl Muroi, David J Mack, Christian Strassle, and Walter Karlen. ¨ Not to Cry Wolf: Distantly Supervised Multitask Learning in Critical Care. In International Conference on Machine Learning, 2018.
+
+Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: Generalization bounds and algorithms. In International Conference on Machine Learning, 2017.
+
+Jeffrey A Smith and Petra E Todd. Does matching overcome LaLonde’s critique of nonexperimental estimators? Journal of Econometrics, 125(1-2):305–353, 2005.
+
+Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 2017.
+
+John N Weinstein, Eric A Collisson, Gordon B Mills, Kenna R Mills Shaw, Brad A Ozenberger, Kyle Ellrott, Ilya Shmulevich, Chris Sander, Joshua M Stuart, Cancer Genome Atlas Research Network, et al. The cancer genome atlas pan-cancer analysis project. Nature genetics, 45(10): 1113–1120, 2013.
+
+Jinsung Yoon, James Jordon, and Mihaela van der Schaar. GANITE: Estimation of Individualized Treatment Effects using Generative Adversarial Nets. In International Conference on Learning Representations, 2018.
+
+# SUPPLEMENTARY MATERIAL FOR:”PERFECT MATCH: A SIMPLE METHOD FOR LEARN-ING REPRESENTATIONS FOR COUNTERFACTUAL IN-FERENCE WITH NEURAL NETWORKS”
+
+Anonymous authors Paper under double-blind review
+
+# A PRECISION IN ESTIMATION OF HETEROGENOUS EFFECT (PEHE)
+
+When the underlying noiseless distributions $\mu _ { j }$ are not given, PEHE is calculated using:
+
+$$
+{ \hat { \epsilon } } _ { \mathrm { P E H E } } = { \frac { 1 } { N } } \sum _ { n = 0 } ^ { N } { \Big ( } { \big [ } y _ { 1 } ( n ) - y _ { 0 } ( n ) { \big ] } - { \big [ } { \hat { y } } _ { 1 } ( n ) - { \hat { y } } _ { 0 } ( n ) { \big ] } { \Big ) } ^ { 2 }
+$$
+
+# B AVERAGE TREATMENT EFFECT (ATE)
+
+The ATE measures the average difference in effect across the whole population (Hill (2011)). It can be useful indicator of how well an ITE estimator performs at comparing two treatments across the entire population.
+
+$$
+\epsilon _ { \mathrm { A T E } } = | | \frac { 1 } { N } \sum _ { n = 0 } ^ { N } \mathbb { E } _ { y ( n ) \sim \mu ( n ) } [ y ( n ) ] - \frac { 1 } { N } \sum _ { n = 0 } ^ { N } \hat { y } ( n ) | | _ { 2 } ^ { 2 }
+$$
+
+Similar to Equation (A.1), the counterfactual treatment effect can be estimated using the noisy ground truth outcomes if the underlying noiseless distributions $\mu _ { j }$ are not given:
+
+$$
+\hat { \epsilon } _ { \mathrm { A T E } } = | | \frac { 1 } { N } \sum _ { n = 0 } ^ { N } y ( n ) - \frac { 1 } { N } \sum _ { n = 0 } ^ { N } \hat { y } ( n ) | | _ { 2 } ^ { 2 }
+$$
+
+# C DETAILS PERFECT MATCH (PM)
+
+Algorithm 1 outlines the procedure of using PM to augment a batch with propensity score matches for training a neural network with stochastic gradient descent. We trained a Support Vector Machine (SVM) with probability estimation as propensity score estimator $E _ { \mathrm { P S } }$ . To speed up recalling nearest neighbours, we additionally prepared an index per treatment into $X _ { \mathrm { t r a i n } }$ sorted by propensity score. At augmentation time, we used binary search on this index to find the nearest neighbours by propensity score in $\mathrm { O } ( \log \ N _ { t } )$ where $N _ { t }$ is the number of samples $X$ in $X _ { \mathrm { { t r a i n } } }$ assigned to the treatment group $t$ . To avoid overfitting to specific edge samples when propensity scores are not distributed evenly in the training set, we chose at random from one of the $k { = } 6$ closest samples by propensity score. See Table S1 for a comparison of the effects on predictive performance of choosing varying values of $k$ . For the $^ { \circ } +$ on $X ^ { \dag }$ model we matched, using the Euclidean distance, directly on the covariates $X$ on IHDP and Jobs, and using a low dimensional representation of $X$ , obtained using principal component analysis (PCA) with 50 principal components, on News.
+
+Table S1: Comparison of PM with varying numbers of nearest neighbours $k$ considered for randomised matching on the News-2/4/8/16 datasets $k { = } 1$ corresponds to no randomisation; $k { = } 6$ , in cursive, are the results reported in the main body). We report the mean value $\pm$ the standard deviation of $\sqrt { \hat { \epsilon } _ { \mathrm { P E H E } } }$ on the test sets over 50 runs. There was no significant differences between different levels of $k$ and $k { = } 6$ (significance level $\alpha < 0 . 0 5$ ). In particular, there was no significant difference between using randomisation and not using randomisation.   
+
+<table><tr><td>Method</td><td>News-2 VEPEHE</td><td>News-4 VEPEHE</td><td>News-8 VEPEHE</td><td>News-16 VEPEHE</td></tr><tr><td>PM (k=1)</td><td>16.79 ± 1.16</td><td>21.86 ± 2.24</td><td>20.70 ±1.87</td><td>20.06 ±1.43</td></tr><tr><td>PM (k=3)</td><td>16.82 ± 1.17</td><td>21.49 ± 2.43</td><td>20.50 ± 1.48</td><td>20.28 ± 1.52</td></tr><tr><td>PM(k=6)</td><td>16.76 ± 1.26</td><td>21.58 ± 2.58</td><td>20.76 ± 1.86</td><td>20.24 ± 1.46</td></tr><tr><td>PM (k=9)</td><td>16.91 ± 1.18</td><td>21.56 ± 2.58</td><td>20.75 ± 1.53</td><td>20.47 ± 1.94</td></tr><tr><td>PM (k=12)</td><td>17.11 ± 0.94</td><td>21.16 ± 2.20</td><td>21.10 ± 2.14</td><td>20.28 ± 1.64</td></tr><tr><td>PM (k=15)</td><td>16.87 ± 1.14</td><td>20.94 ± 2.05</td><td>20.92 ±1.80</td><td>20.48 ±1.80</td></tr></table>
+
+# D REDUCED VARIANCE IS LINKED TO IMPROVED PERFORMANCE IN ESTIMATING TREATMENT EFFECTS
+
+In the standard supervised setting, the variance added by minibatch SGD leads to slower convergence (Csiba & Richtrik (2018)), but will not typically lead to models that perform worse if they are trained until a convergence criterion has been met, e.g. early stopping. It is therefore perhaps surprising that we observe a difference not just in convergence (Figure 6), but also in counterfactual estimation performance (Tables 3 and 4) when comparing models trained with and without batch matching.
+
+A potential cause of the observed difference in treatment effect estimation performance is that we do not have access to the true counterfactual error in observational data to select the best model encountered during training. We therefore have to resort to using the factual error (e.g. factual MSE) or an estimator based on the factual error (e.g. NN-PEHE) to select the best encountered model. However, when the variance during training is high, chances are that we select a model based on the factual error that at this point during the training happens to be suboptimal in terms of counterfactual error. Through this mechanism, added variance during training for counterfactual inference is directly linked to worse expected performance in treatment effect estimation (Figure S1). We do not observe the same behavior in standard supervised learning tasks because we have direct access to the underlying objective, and are therefore able to select the best encountered model regardless of the variance during optimisation, i.e. models that during training do not perform well momentarily will not be selected because their measured error is high. We confirmed experimentally that gradient steps are more likely to be in opposing directions when not using PM (Figure S2).
+
+![](images/54a44ae4f2da5d207dbfbccbabbcbc6b9a928298616bc3331ca291361b7bc335.jpg)  
+Figure S1: Illustration of the impact of added variance introduced by minibatch SGD during training. In the supervised setting (left), high variance does in general not impact the final performance in terms of expected factual error (red), because we can select (dotted black line) the best encountered model during training using the observed error. In the counterfactual setting (right), we are not able to compute the true counterfactual error (blue) in observational data, and must therefore resort to selecting based on the factual error or a measure derived from the factual error. Even though we encounter models that perform similarly well, we are less likely to select these models under high variance, because we do not have direct access to the counterfactual error and the minimum of the factual error in general does not correspond to the minimum of the counterfactual error, particularly when optimisation of counterfactual and factual error are not well aligned. The dashed blue line indicates the counterfactual error of the model selected based on factual error.
+
+![](images/e482f386c8f24bebeefb344e713b1622f13fd82b8b3e2937f1e3f15453e8780c.jpg)  
+Figure S2: Comparison of the frequencies of individual gradient steps during training of several matching-based methods on the validation set of News-8. The heatmap entries show the frequencies of pairs of differences in factual error (∆MSE; y-axis) and counterfactual error ( $\Delta \epsilon _ { \mathrm { P E H E } }$ ; $\mathbf { X }$ -axis) after individual gradient steps. All methods used the same model and only differed in how they addressed treatment assignment bias. The leftmost heatmap shows the desired behavior of gradient steps changing the counterfactual and factual error jointly during training. PM most closely resembles the desired behavior of jointly optimising counterfactual and factual error. In contrast, $\mathrm { P S M } _ { \mathrm { P M } }$ and $\mathrm { P S M _ { M I } }$ have a much higher relative frequency of gradient steps that do not move into the same magnitude or direction (off-diagonal entries) for both factual and counterfactual error.
+
+# E NEAREST NEIGHBOUR APPROXIMATION OF PEHE FOR MULTIPLE TREATMENTS (NN-MPEHE)
+
+The ˆNN-PEHE metric can be extended to the multiple treatment setting by considering the mean ˆNN-PEHE between all $\binom { k } { 2 }$ possible pairs of treatments:
+
+$$
+\hat { \epsilon } _ { \mathrm { N N - m P E H E } } = \frac { 1 } { \binom { k } { 2 } } \sum _ { i = 0 } ^ { k - 1 } \sum _ { j = 0 } ^ { i - 1 } \hat { \epsilon } _ { \mathrm { N N - P E H E } , i , j }
+$$
+
+# F PSEUDOCODE PROPENSITY SCORE MATCHING (PSM)
+
+Algorithm S2 outlines the procedure of preprocessing a training set using $\mathrm { P S M } _ { \mathrm { M I } }$ . For $\mathrm { P S M } _ { \mathrm { P M } }$ , we used the PM matching algorithm (see Algorithm 1) to find matches between samples from the treatment groups. For $\mathrm { P S M } _ { \mathrm { M I } }$ , we used the MatchIt package (Ho et al. (2011)) on setting ”nearest” to find matches between sets of samples from two treatment groups (function get matched samples).
+
+<table><tr><td>Algorithm S2 Preprocessng a training set using Propensity Score Matching (PSM). After prepro- cessing,the training set contains an equal number of samples from each treatment group and the covariates xi across all treatment groups are approximately balanced.</td></tr><tr><td>Input: N training samples Xtrain with assgned treatments t, Number of treatment options k, func- tion get_mat ched_samples to find nearest matches by propensity score between sets of</td></tr><tr><td>samples Output: Preprocessed training set Xout consisting of matched samples from each treatment group</td></tr><tr><td>1: procedure PREPROCESS_TRAINING_SET: 2: tbase ← index of treatment with smallest number of samples in Xtrain</td></tr><tr><td>Xbase ← all samples in Xtrain With treatment tbase</td></tr><tr><td>fori=O to k-1 do</td></tr><tr><td>if i≠ tbase then</td></tr><tr><td>Xcurrent ← all samples in Xtrain with treatment i</td></tr><tr><td>Xmatched ← get_mat ched_samples(Xbase,Xcurrent) Add all samples in Xmatched to Xout</td></tr><tr><td></td></tr><tr><td>Add all samples in Xbase to Xout</td></tr></table>
+
+# G TARNET, CFRNET AND PD FOR MULTIPLE TREATMENTS
+
+In TARNET models, we used ELU nonlinearities between the $L$ hidden layers with $M$ hidden units. $L$ and $M$ were hyperparameters that we optimised during hyperparameter optimisation (Section H). We did not use batch normalisation (BN). To extend CFRNET to the multiple treatment setting, we defined the first treatment option as the control treatment and regularised all treatment options to have the same activation distribution in the topmost shared layer (Shalit et al. (2017)). For PD, we only used propensity dropout and the propensity estimation network. We did not make use of the alternating training schedule proposed in (Alaa et al. (2017)). The $\ " { + } \mathrm { M L P ^ { \prime } }$ model was a simple MLP with $L$ hidden layers of $M$ hidden units that received the treatment option index $t _ { j }$ as an additional input along with the covariates $X$ , and output a $k$ -dimensional potential outcome vector $\hat { Y }$ . The MLP used ELU nonlinearities between the $L$ hidden layers, and also did not use BN.
+
+# H HYPERPARAMETERS
+
+To ensure a fair comparison, we used a standardised approach to hyperparameter optimisation for those methods for which we did not have previously reported performance numbers. In particular, each method we trained was optimised over the same amount of hyperparameter optimisation runs. For the methods that used neural network models (TARNET, CFRNET, PD, $\mathrm { P S M } _ { \mathrm { P M } }$ , $\mathrm { P S M _ { M I } }$ , including $^ { , , } +$ on $X ^ { \dag }$ and $\ " { + } \mathrm { M L P ^ { \prime } }$ , and PM), we chose hyperparameters at random from predefined ranges (Table S2). For CFRNET, we additionally varied the weight of the imbalance penalty at random between 0.1, 1.0, and 10.0. All methods that used a neural network model used the TARNET architecture to ensure differences in performance are not due to architectural differences. To train PM for the Jobs dataset, we used fixed hyperparameters: A batch size of 50, 60 hidden units per layer, and 3 hidden layers. For optimisation, we used the Adam optimiser with a learning rate of 0.001 for a maximum of 100 (News, TCGA) or 400 (IHDP, Jobs) with an early stopping patience of 30 on the factual MSE. We used the default hyperparameters for BART and CF from the ”bartMachine” (Kapelner & Bleich (2013)) and ”grf” (Athey et al. (2016)) R-packages. For GANITE, we used our own implementation since there was no open source implementation available, and - in addition to the parameters in Table S2 - optimised over the supervised loss weights $\alpha$ and $\beta$ (Yoon et al. (2018)) between 0.1, 1, and 10. For the GANITE generators and discriminators, we used MLP architectures with $L$ hidden layers of $M$ hidden units each.
+
+Table S2: Hyperparameter ranges used in the performed experiments.   
+
+<table><tr><td>Hyperparameter</td><td>IHDP</td><td>News/TCGA</td></tr><tr><td>Batch size B</td><td>4,8,50,100</td><td>50</td></tr><tr><td>Number of units per hidden layer M5</td><td>50,100,200</td><td>40,60,80</td></tr><tr><td>Number of hidden layers L</td><td>1,2,3</td><td>2,3</td></tr></table>
+
+# I POLICY RISK $R _ { \mathrm { P o L } } ( \pi )$
+
+We define the policy risk $R _ { \mathrm { P o l } } ( \pi )$ as the expected loss in value when treating according to the policy $\pi$ implied by an ITE estimator (Shalit et al. (2017)):
+
+$$
+R _ { \mathrm { P o l } } ( \pi ) = 1 - ( \mathbb { E } [ y _ { 1 } | \pi ( X ) = 1 ] \cdot p ( \pi = 1 ) + \mathbb { E } [ y _ { 0 } | \pi ( X ) = 0 ] \cdot p ( \pi = 0 ) )
+$$
+
+where the policy $\pi ( X )$ for a sample $X$ implied by an ITE estimator that produces potential outcomes $\hat { y }$ is defined as:
+
+$$
+\pi ( X ) = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } { \hat { y } } _ { 1 , X } - { \hat { y } } _ { 0 , X } > \lambda } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } \quad { \mathrm { ( t o ~ t r e a t ) } }
+$$
+
+# J ORIGINS OF PERFORMANCE NUMBERS IN TABLE 3
+
+Where available, performance numbers in Table 3 were taken from the original authors for IHDP and Jobs (Table S3). For News-2/4/8/16 and TCGA, performance numbers were produced by us.
+
+Table S3: Origins of the performance results reported in Table 3. Entries show the reference to the original report of the performance results, or ”ours” if we produced the results. n.r. $=$ not reported   
+
+<table><tr><td>Method</td><td>IHDP</td><td>Jobs</td></tr><tr><td>PM</td><td>ours</td><td>ours</td></tr><tr><td>+on X</td><td>ours</td><td>ours</td></tr><tr><td>+ MLP</td><td>ours</td><td>ours</td></tr><tr><td>kNN</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>PSMPM</td><td>n.r.</td><td>ours</td></tr><tr><td>PSMMI</td><td>ours</td><td>ours</td></tr><tr><td>RF</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>CF</td><td>(Shalit et al. 2 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>BART</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>GANITE</td><td>(Yoon et al. 2018)</td><td>(Yoon et al. 2018)</td></tr><tr><td>BNN</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>PD</td><td>n.r.</td><td>n.r.</td></tr><tr><td>TARNET</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>CFRNETwass</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>CEVAE CMGP</td><td>(Louizos et al. 2017) (Yoon et al. 2018)</td><td>(Louizos et al. 2017) (Yoon et al. 2018)</td></tr></table>

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+        "text": "PERFECT MATCH: A SIMPLE METHOD FOR LEARNING REPRESENTATIONS FOR COUNTERFACTUAL INFERENCE WITH NEURAL NETWORKS ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
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+        "text": "Learning representations for counterfactual inference from observational data is of high practical relevance for many domains, such as healthcare, public policy and economics. Counterfactual inference enables one to answer ”What if...?” questions, such as ”What would be the outcome if we gave this patient treatment $t _ { 1 } ? ^ { \\dag }$ . However, current methods for training neural networks for counterfactual inference on observational data are either overly complex, limited to settings with only two available treatment options, or both. Here, we present Perfect Match (PM), a method for training neural networks for counterfactual inference that is easy to implement, compatible with any architecture, does not add computational complexity or hyperparameters, and extends to any number of treatments. PM is based on the idea of augmenting samples within a minibatch with their propensity-matched nearest neighbours. Our experiments demonstrate that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes across several real-world and semi-synthetic datasets. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Estimating individual treatment effects1 (ITE) from observational data is an important problem in many domains. In medicine, for example, we would be interested in using data of people that have been treated in the past to predict what medications would lead to better outcomes for new patients (Shalit et al. (2017)). Similarly, in economics, we would for example want to determine how effective job programs would be based on results of past job training programs (LaLonde (1986)). ",
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+    },
+    {
+        "type": "text",
+        "text": "ITE estimation from observational data is difficult for two reasons: Firstly, we never observe all potential outcomes. If a patient is given a treatment to treat her symptoms, we never observe what would have happened if the patient was prescribed a potential alternative treatment in the same situation. Secondly, the assignment of cases to treatments is typically biased such that cases for which a given treatment is more effective are more likely to have received that treatment. The distribution of samples may therefore differ significantly between the treated group and the overall population. A supervised model na¨ıvely trained to minimise the factual error would overfit to the properties of the treated group, and thus not generalise well to the entire population. ",
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+        "type": "text",
+        "text": "To address these problems, we introduce Perfect Match (PM), a simple method for training neural networks for counterfactual inference that extends to any number of treatment options. PM effectively controls for biased assignment of treatments in observational data by augmenting every sample within a minibatch with its closest matches by propensity score from the other treatment options. PM is easy to use with existing neural network architectures, simple to implement, and does not add any hyperparameters or computational complexity. We perform experiments that demonstrate that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes on several real-world and semi-synthetic datasets. Our experiments also show that PM is more robust to a high level of treatment assignment bias than existing methods. We believe that our simple and effective method for ITE estimation that extends to any number of treatment options could potentially make counterfactual inference more accessible, particularly for medical applications, where multiple available treatment options are the norm. ",
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+        "type": "text",
+        "text": "Contributions. This work contains the following contributions: ",
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+    {
+        "type": "text",
+        "text": "• We introduce Perfect Match, a simple methodology for learning neural representations for counterfactual inference based on propensity score matching within minibatches. • We perform extensive experiments on semi-synthetic and real-world data in both the binary and multiple treatment setting. The experiments show that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "There are five main categories of approaches to learning to estimate ITEs: ",
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+    },
+    {
+        "type": "text",
+        "text": "Matching-based Methods. Matching methods estimate the counterfactual outcome of a sample $X$ with respect to treatment $t$ using the factual outcomes of its nearest neighbours having received $t$ , with respect to a metric space. $\\mathbf { k }$ -Nearest-Neighbour (kNN) (Ho et al. (2007)) operates in the potentially high-dimensional covariate space, and therefore might suffer from the curse of dimensionality. Propensity Score Matching (PSM) (Rosenbaum & Rubin (1983)) matches on the scalar probability $p ( t | X )$ of $t$ given the covariates $X$ . PSM may require under- or oversampling the original data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Adjusted Regression Methods. Adjusted regression models apply regression models with both treatment and covariates as input. The simplest case is Ordinary Least Squares, which can either be used for building one model, with the treatment as an input feature, or multiple separate models, one for each treatment (Kallus (2017)). More complex regression models, such as Treatment-Agnostic Representation Networks (TARNET) (Shalit et al. (2017)) may be used to capture non-linear relationships. Methods that make use of propensity scores, and a model of the outcomes, such as Propensity Dropout (PD) (Alaa et al. (2017)), are referred to as doubly robust (Funk et al. (2011)). ",
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+    },
+    {
+        "type": "text",
+        "text": "Tree-based Methods. Tree-based methods train many weak learners to build expressive ensemble models. Examples of tree-based methods are Bayesian Additive Regression Trees (BART) (Chipman et al. (2010); Chipman & McCulloch (2016)) and Causal Forests (CF) (Wager & Athey (2017)). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Representation-balancing Methods. Representation-balancing methods seek to learn a highlevel representation for which the covariate distributions are balanced across treatment groups. Balancing Neural Networks (Johansson et al. (2016)) attempt to find such representations by minimising the discrepancy distance (Mansour et al. (2009)) between treatment groups. Counterfactual Regression Networks (CFRNET, Shalit et al. (2017)) use different metrics such as the Wasserstein distance. ",
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+    },
+    {
+        "type": "text",
+        "text": "Distribution-modeling Methods. Generative Adversarial Nets for inference of Individualised Treatment Effects (GANITE) (Yoon et al. (2018)) address ITE estimation using counterfactual and ITE generators. GANITE uses a complex architecture with many hyperparameters and submodels that may be difficult to implement and optimise. Causal Effect Variational Autoencoders (CEVAEs) (Louizos et al. (2017)) use a latent variable modeling approach that is robust to hidden confounding, but does not address biased treatment assignment. Causal Multi-task Gaussian Processes (CMGP) (Alaa & van der Schaar (2017)) apply a multi-task Gaussian Process to ITE estimation. The optimisation of CMGPs involves a matrix inversion of $O ( n ^ { 3 } )$ complexity that limits their scalability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast to existing methods, PM is a simple doubly robust method built on propensity score matching that can be used to train expressive non-linear neural network models for ITE estimation from observational data in settings with any number of treatments (Table 1). While the underlying idea behind PM is simple and effective, it has, to the best of our knowledge, not yet been explored. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8f820d01415352bb5b8dab9b8bbb260adf23d349d4bfffbb3646418be11bc96c.jpg",
+        "table_caption": [
+            "Table 1: Comparison of several representative state-of-the-art methods for counterfactual inference showing whether they were designed for $> 2$ treatments and their complexity of implementation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PM</td><td>CF</td><td>BART</td><td>BNN</td><td>CFRNET</td><td>PD</td><td>CEVAE</td><td>CMGP</td><td>GANITE</td></tr><tr><td>&gt;2 Treatments</td><td>√</td><td>√</td><td>√</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>√</td></tr><tr><td>Complexity</td><td>Low</td><td>Low</td><td>Low</td><td>Low</td><td>Medium</td><td>Medium</td><td>High</td><td>High</td><td>High</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Problem Setting. We consider a setting in which we are given $N$ observed samples $X$ , where each sample consists of $p$ covariates $x _ { i }$ with $i \\in [ 0 , p )$ . For each sample, the potential outcomes are represented as a vector $Y$ with $k$ entries $y _ { j }$ where each entry corresponds to the outcome when applying one treatment $t _ { j }$ out of the set of $k$ available treatments $T = \\{ \\bar { t } _ { 0 } , . . . , t _ { k - 1 } \\}$ with $j \\in [ 0 , k )$ . As training data, we receive observed samples $X$ and their factual outcomes $y _ { j }$ when applying one treatment $t _ { j }$ . The set of available treatments can contain two or more treatment options. We refer to the special case of two available treatment options as the binary treatment setting. Given the observational training data with factual outcomes, we wish to train a predictive model that is able to accurately produce a predicted potential outcomes vector $\\hat { Y }$ with $k$ entries $\\hat { y } _ { j }$ . In literature, this setting is known as the Rubin-Neyman potential outcomes framework (Rubin (2005)). ",
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+    {
+        "type": "text",
+        "text": "Precision in Estimation of Heterogenous Effect (PEHE). The primary metric that we optimise for when training models to estimate ITE is the PEHE (Hill (2011)). In the binary setting, the PEHE measures the ability of a predictive model to estimate the difference in effect between two treatments $t _ { 0 }$ and $t _ { 1 }$ for samples $X$ . To compute the PEHE, we measure the mean squared error between the true difference in effect $y _ { 1 } ( n ) - y _ { 0 } ( n )$ , drawn from the noiseless underlying outcome distributions $\\mu _ { 1 }$ and $\\mu _ { 0 }$ , and the predicted difference in effect $\\hat { y } _ { 1 } ( n ) - \\hat { y } _ { 0 } ( n )$ indexed by $n$ over $N$ samples: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a7e5a1fb9d969a958c5a62216beb02bf3c1d595c2a3f50d8330427a58036719.jpg",
+        "text": "$$\n\\epsilon _ { \\mathrm { P E H E } } = \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } \\Big ( \\mathbb { E } _ { y _ { j } ( n ) \\sim \\mu _ { j } ( n ) } [ y _ { 1 } ( n ) - y _ { 0 } ( n ) ] - [ \\hat { y } _ { 1 } ( n ) - \\hat { y } _ { 0 } ( n ) ] \\Big ) ^ { 2 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "When the underlying noiseless distributions $\\mu _ { j }$ are not known, the true difference in effect $y _ { 1 } ( n ) -$ $y _ { 0 } ( n )$ can be estimated using the noisy ground truth outcomes $y _ { i }$ (Appendix A). As a secondary metric, we consider the error $\\epsilon _ { \\mathrm { A T E } }$ in estimating the average treatment effect (ATE) (Hill (2011)). The ATE measures the average difference in effect across the whole population (Appendix B). The ATE is not as important as PEHE for models optimised for ITE estimation, but can be a useful indicator of how well an ITE estimator performs at comparing two treatments across the entire population. We can neither calculate $\\epsilon _ { \\mathrm { P E H E } }$ nor $\\epsilon _ { \\mathrm { A T E } }$ without knowing the outcome generating process. ",
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+    {
+        "type": "text",
+        "text": "Multiple Treatments. Both \u000fPEHE and $\\epsilon _ { \\mathrm { A T E } }$ can be trivially extended to multiple treatments by considering the average PEHE and ATE between every possible pair of treatments. Note that we lose the information about the precision in estimating ITE between specific pairs of treatments by averaging over all $\\binom { k } { 2 }$ pairs. However, one can inspect the pair-wise PEHE to get the whole picture. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/805bf309977a86718f1130d7f01e8e1483ed4a6927c4a9f8215b97994fb408aa.jpg",
+        "text": "$$\n\\hat { \\epsilon } _ { \\mathrm { m P E H E } } = \\frac { 1 } { \\binom { k } { 2 } } \\sum _ { i = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ { i - 1 } \\hat { \\epsilon } _ { \\mathrm { P E H E } , i , j } \\qquad ( 2 ) \\qquad \\hat { \\epsilon } _ { \\mathrm { m A T E } } = \\frac { 1 } { \\binom { k } { 2 } } \\sum _ { i = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ { i - 1 } \\hat { \\epsilon } _ { \\mathrm { A T E } , i , j }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Perfect Match (PM). We consider fully differentiable neural network models optimised via minibatch stochastic gradient descent to predict potential outcomes $\\hat { Y }$ given a sample $X$ . To address the treatment selection bias inherent in observational data, we propose the use of a training methodology based on propensity score matching. A propensity score is the conditional probability $p ( t | X )$ of a given sample $X$ receiving a specific treatment $t$ (Rosenbaum & Rubin (1983); Ho et al. (2007)). The propensity score is a balancing score, meaning that if the distribution of propensity scores across treatment groups is the same, then the distribution of their covariates $x _ { i }$ is also guaranteed to be the same across treatment groups (Ho et al. (2007)). By matching samples with their nearest neighbours by propensity score we can, in the optimal case, balance the covariates $x _ { i }$ . This breaks the dependence of treatment assignment on $X$ if all relevant covariates are observed, and therefore effectively removes any potential treatment assignment bias (Ho et al. (2007)). However, in practice, we do not have access to the true treatment assignment probability $p ( t | X )$ . We therefore have to estimate the propensity score $p ( t | X )$ by training a predictive model to predict the likelihood of receiving a treatment $t$ for a given sample $X$ . Note that propensity scores are not the only balancing score (Ho et al. (2007)). For example, $X$ itself is another balancing score. However, matching on the onedimensional propensity score is preferable because it avoids the curse of dimensionality that would be associated with matching on the potentially high-dimensional $X$ directly. In PM, we match every sample within a minibatch with its nearest neighbours by propensity score from all other treatments (Algorithm 1, more details in Appendix C), taken from the training set. Every minibatch the model is trained on therefore contains the same number of samples for each treatment group, and the covariates $x _ { i }$ of each treatment group are approximately balanced on average. The intuition behind matching on the minibatch level, rather than the dataset level (Ho et al. (2011)), is that it reduces the variance during training which in turn leads to better expected performance for counterfactual inference (Appendix D). In this sense, PM can be seen as a minibatch sampling strategy (Csiba & Richtarik (2018)) specifically designed to improve learning for counterfactual inference. ´ ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Batch Augmentation using Perfect Match (PM). After augmentation, each batch contains an equal number of samples from each treatment group and the covariates $x _ { i }$ across all treatment groups are approximately balanced. ",
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+    {
+        "type": "text",
+        "text": "Input: Batch of $B$ random samples $X _ { \\mathrm { b a t c h } }$ with assigned treatments $t$ , training set $X _ { \\mathrm { t r a i n } }$ of $N$ samples, number of treatment options $k$ , propensity score estimator $E _ { \\mathrm { P S } }$ to calculate the probability $\\bar { p } ( t | X )$ of treatment assigned given a sample $X$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Output: Batch $X _ { \\mathrm { o u t } }$ consisting of $B \\times k$ matched samples ",
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+    {
+        "type": "table",
+        "img_path": "images/f2fff6186537b3065f72f8aeebba61f2acfcb4132605c67721765fd21e94f871.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">1: procedurePERFECT_MATCH:</td></tr><tr><td>2:</td><td>Xout ←Empty</td></tr><tr><td>3:</td><td>for sample X with treatment t in Xbatch do</td></tr><tr><td>4:</td><td>p(t|X)←Eps(X)</td></tr><tr><td>5:</td><td>fori=O tok-1do</td></tr><tr><td>6:</td><td>ifi≠t then</td></tr><tr><td>7:</td><td>psi←p(t|X)i</td></tr><tr><td>8:</td><td>Xmatched ← get closest match to propensity score psi with treatment i from Xtrain</td></tr><tr><td>9:</td><td>Add sample Xmatched to Xout</td></tr><tr><td>10:</td><td>Add X to Xout</td></tr><tr><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Model Selection. Besides accounting for the treatment assignment bias, the other major issue in learning for counterfactual inference from observational data is that, given multiple models, it is not trivial to decide which one to select. The root problem is that we do not have direct access to the true error in estimating counterfactual outcomes, only the error in estimating the observed factual outcomes. This makes it difficult to perform parameter and hyperparameter optimisation, as we do not know which models are better than others for counterfactual inference on a given dataset. To rectify this problem, we use a nearest neighbour approximation $\\hat { \\epsilon } _ { \\mathrm { N N - P E H E } }$ of the $\\boldsymbol { \\hat { \\epsilon } } _ { \\mathrm { P E H E } }$ metric for the binary (Shalit et al. (2017)) and multiple treatment settings for model selection. The \u000fˆNN-PEHE estimates the treatment effect of a given sample by substituting the true counterfactual outcome with the outcome $y _ { j }$ from a respective nearest neighbour NN matched on $X$ using the Euclidean distance. ",
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+    {
+        "type": "equation",
+        "img_path": "images/e435c405fe5514fbf9ef407e0f4fd87c0a1f5c41f3ae7deda0ee9294953906cf.jpg",
+        "text": "$$\n\\hat { \\epsilon } _ { \\mathrm { N N - P E H E } } = \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } \\Big ( [ y _ { 1 } ( \\mathbf { N N } ( n ) ) - y _ { 0 } ( \\mathbf { N N } ( n ) ) ] - [ \\hat { y } _ { 1 } ( n ) - \\hat { y } _ { 0 } ( n ) ] \\Big ) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Analogously to Equations (2) and (3), the \u000fˆNN-PEHE metric can be extended to the multiple treatment setting by considering the mean $\\hat { \\epsilon } _ { \\mathrm { N N - P E H E } }$ between all $\\binom { k } { 2 }$ possible pairs of treatments (Appendix E). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Model Architecture. No different from other applications, the chosen architecture plays a key role in the performance of neural networks when attempting to learn representations for counterfactual inference. Shalit et al. (2017) claimed that the na¨ıve approach of appending the treatment index $t _ { j }$ may perform poorly if $X$ is high-dimensional, because the influence of $t _ { j }$ on the hidden layers may be lost during training. Shalit et al. (2017) subsequently introduced the TARNET architecture to rectify this issue. Since the original TARNET was limited to the binary treatment setting, we extended the TARNET architecture to the multiple treatment setting (Figure 1). We did so by using $k$ head networks, one for each treatment option over a set of shared base layers, each with $L$ layers. In TARNET, the $j$ th head network is only trained on samples from treatment $t _ { j }$ . The shared layers are trained on all samples. By using a head network for each treatment, we ensure $t _ { j }$ maintains an appropriate degree of influence on the network output at all points during training. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/df697ae164c15f61e6243c85e1aee1e920e1c2b644a0d3442482f69f5b1c439e.jpg",
+        "image_caption": [
+            "Figure 1: The TARNET architecture with $k$ heads for the multiple treatment setting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3afda24802c480ee6b50775178355e6a516c54e9f816f83c99cef04f13bcf955.jpg",
+        "table_caption": [
+            "Table 2: Comparisons of the datasets used in our experiments. We evaluate on three semi-synthetic datasets with varying numbers of treatments and samples, and a real-world clinical trial dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Type</td><td># Samples</td><td>#Features</td><td># Treatments</td><td>Counterfactuals</td></tr><tr><td>IHDP</td><td>semi-synthetic</td><td>747</td><td>25</td><td>2</td><td>available</td></tr><tr><td>Jobs</td><td>real-world</td><td>3212</td><td>7</td><td>2</td><td>not available</td></tr><tr><td>News</td><td>semi-synthetic</td><td>5000</td><td>2870</td><td>2/4/8/16</td><td>available</td></tr><tr><td>TCGA</td><td>semi-synthetic</td><td>9659</td><td>20531</td><td>4</td><td>available</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our experiments aimed to answer the following questions: ",
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+    },
+    {
+        "type": "text",
+        "text": "1 What is the comparative performance of PM in inferring counterfactual outcomes in the   \nbinary and multiple treatment setting compared to existing state-of-the-art methods?   \n2 Does model selection by NN-PEHE outperform selection by factual MSE?   \n3 How does the relative number of matched samples within a minibatch affect performance?   \n4 How well does PM cope with an increasing treatment assignment bias in the observed data?   \n5 How does the presence of hidden confounders influence the performance of PM?   \n6 How do the learning dynamics of minibatch matching compare to dataset-level matching? ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We performed experiments on four real-world and semi-synthetic datasets (Table 2) with binary and multiple treatment options in order to gain a better understanding of the empirical properties of PM. ",
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+    },
+    {
+        "type": "text",
+        "text": "Infant Health and Development Program (IHDP). The IHDP dataset (Hill (2011)) contains data from a randomised study on the impact of specialist visits on the cognitive development of children, and consists of 747 children with 25 covariates describing properties of the children and their mothers. Children that did not receive specialist visits were part of a control group. The outcomes were simulated using the NPCI package from Dorie (2016). The IHDP dataset is biased because the treatment groups had a biased subset of the treated population removed (Shalit et al. (2017)). We used the same simulated outcomes2 as Shalit et al. (2017). ",
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+    {
+        "type": "text",
+        "text": "Jobs. The Jobs dataset (LaLonde (1986)) is a blend of data from randomised and observational studies on the effect of professional training programs on unemployment, and is a commonly used benchmark dataset for counterfactual inference in the binary setting. Each sample consisted of demographic covariates, such as age, gender and previous income, and the treatment was enrolment in a job training program. All samples that were not enrolled in a training program were considered control cases. The task was to predict the potential unemployment given the covariates and treatment. We used the same feature and sample set as described in Dehejia & Wahba (2002); Smith & Todd (2005); Shalit et al. (2017). A subgroup of the data was randomised and we therefore were able to estimate the ground truth average treatment effect on the treated (ATT) (Shalit et al. (2017)). Treatment assignment in Jobs is biased because parts of the data come from an observational study. ",
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+    {
+        "type": "image",
+        "img_path": "images/9918fa2b90357bcf89b69a749b3ec1a315ad45bc7e9f7de7ce33b085ecab1308.jpg",
+        "image_caption": [
+            "Figure 2: Correlation analysis of the real PEHE (y-axis) with the mean squared error (MSE; left) and the nearest neighbour approximation of the precision in estimation of heterogenous effect (NN-PEHE; right) across over $2 0 ^ { \\circ } 0 0 0$ model evaluations on the validation set of IHDP. Scatterplots show a subsample of $1 ^ { \\cdot } 4 0 0$ data points. $\\rho$ indicates the Pearson correlation. NN-PEHE correlates significantly better with PEHE than MSE. "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/dbb418865d647f6219d58af93434042418c17613b1162905fddcc533dee487a3.jpg",
+        "image_caption": [
+            "Figure 3: Change in error (y-axes) in terms of precision in estimation of heterogenous effect (PEHE) and average treatment effect (ATE) when increasing the percentage of matches in each minibatch ( $\\mathbf { \\dot { x } }$ -axis). Symbols correspond to the mean value of $\\hat { \\epsilon } _ { \\mathrm { A T E } }$ (red) and $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ (blue) on the test set of News-8 across 50 repeated runs with new outcomes (lower is better). Performance improves with more matches added. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0e66495e0af1f70fbbb9c30fe17567012caef29fd2f299eb4bb1e2a7bb301f2e.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of several state-of-the-art methods for counterfactual inference on the test set of the News-8 dataset when varying the treatment assignment imbalance $\\kappa$ $\\mathbf { \\dot { x } }$ -axis), i.e. how much the treatment assignment is biased towards more effective treatments. Symbols correspond to the mean value of $\\hat { \\epsilon } _ { \\mathrm { A T E } }$ (left) and $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ (right) across 50 repeated runs with new outcomes (lower is better). The shaded area indicates the standard deviation. PM handles an increasing treatment assignment bias $\\kappa$ better than existing state-of-the-art methods in terms of both $\\hat { \\epsilon } _ { \\mathrm { A T E } }$ and $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "News. The News dataset was first proposed as a benchmark for counterfactual inference by Johansson et al. (2016) and consists of 5000 randomly sampled news articles from the NY Times corpus3. The News dataset contains data on the opinion of media consumers on news items. The samples $X$ represent news items consisting of word counts $x _ { i } \\in \\mathbb { N }$ , the outcome $y _ { j } \\in \\mathbb { R }$ is the reader’s opinion of the news item, and the $k$ available treatment options represent various devices that could be used for viewing, e.g. smartphone, tablet, desktop, television or others (Johansson et al. (2016)). We extended the original dataset specification in (Johansson et al. (2016)) to enable the simulation of arbitrary numbers of viewing devices. To model that consumers prefer to read certain media items on specific viewing devices, we train a topic model on the whole NY Times corpus and define $z ( X )$ as the topic distribution of news item $X$ . We then randomly pick $k + 1$ centroids in topic space, with $k$ centroids $z _ { j }$ per viewing device and one control centroid $z _ { c }$ . We assigned a random Gaussian outcome distribution with mean $\\mu _ { j } \\sim \\mathcal { N } ( 0 . 4 5 , 0 . 1 5 )$ and standard deviation $\\sigma _ { j } \\sim \\mathcal { N } ( 0 . 1 , 0 . 0 5 )$ to each centroid. For each sample, we drew ideal potential outcomes from that Gaussian outcome distribution $\\tilde { y } _ { j } \\sim \\mathcal { N } ( \\mu _ { j } , \\sigma _ { j } ) + \\epsilon$ with $\\epsilon \\sim \\mathcal { N } ( 0 , 0 . 1 5 )$ . We then defined the unscaled potential outcomes $\\bar { y _ { j } } = \\tilde { y } _ { j } * \\left[ \\mathbf { D } ( z ( X ) , z _ { j } ) + \\mathbf { D } ( z ( X ) , z _ { c } ) \\right]$ as the ideal potential outcomes $\\tilde { y } _ { j }$ weighted by the sum of distances to centroids $z _ { j }$ and the control centroid $z _ { c }$ using the Euclidean distance as distance D. We assigned the observed treatment $t$ using $t | x \\sim \\mathrm { B e r n } ( \\operatorname { s o f t m a x } ( \\kappa \\bar { y _ { j } } ) )$ with a treatment assignment bias coefficient $\\kappa$ , and the true potential outcome $y _ { j } = C y _ { j }$ as the unscaled potential outcomes $\\bar { y _ { j } }$ scaled by a coefficient $C = 5 0$ . We used four different variants of this dataset with $k = 2$ , 4, 8, and 16 viewing devices, and $\\kappa = 1 0$ , 10, 10, and 7, respectively. Higher values of $\\kappa$ indicate a higher expected treatment assignment bias depending on $\\bar { y _ { j } }$ . $\\kappa = 0$ indicates no assignment bias. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Cancer Genomic Atlas (TCGA). The TCGA project collected gene expression data from various types of cancers in 9659 individuals (Weinstein et al. (2013)). There were four available clinical treatment options: (1) medication, (2) chemotherapy, (3) surgery, or (4) both surgery and chemotherapy. We used a synthetic outcome function that simulated the risk of cancer recurrence after receiving either of the treatment options based on the real-world gene expression data. Before further processing, we standardised the gene expression data using the mean and standard deviations of gene expression at each gene locus for normal tissue in the training set. To produce the potential outcomes $Y$ , we first selected $k + 1$ gene expression phenotypes as representative phenotypes: One phenotype $X _ { j }$ for each treatment option and one as a control phenotype $X _ { c }$ . Analogously to the News dataset, we assigned a random Gaussian outcome distribution with mean $\\mu _ { j } \\sim \\bar { \\mathcal { N } } ( 0 . 4 \\bar { 5 } , 0 . 1 5 )$ and standard deviation $\\sigma _ { t } \\sim \\mathcal { N } ( 0 . 1 , 0 . 0 5 )$ to each phenotype. For each sample, we drew ideal potential outcomes from that Gaussian outcome distribution $\\bar { \\tilde { y } } _ { j } \\sim \\mathcal N ( \\mu _ { j } , \\sigma _ { j } ) + \\bar { \\epsilon }$ with $\\epsilon \\sim \\mathcal { N } ( 0 , 0 . \\bar { 1 5 } )$ . We then defined the unscaled potential outcomes $\\bar { y _ { j } } = \\tilde { y } _ { j } * \\left[ \\mathbf { D } ( X , X _ { t _ { j } } ) + \\mathbf { D } ( X , X _ { c } ) \\right]$ as the ideal potential outcomes $\\tilde { y } _ { j }$ weighted by the sum of distances to phenotype $X _ { t _ { j } }$ and the control phenotype $X _ { c }$ using the cosine similarity as distance metric D. We assigned the observed treatment $t$ using $t | x \\sim \\mathrm { B e r n } ( \\operatorname { s o f t m a x } ( \\kappa \\bar { y _ { j } } ) )$ with a treatment assignment bias coefficient $\\kappa = 1 0$ , and the true potential outcome $y _ { j } = C y _ { j }$ as the unscaled potential outcomes $\\bar { y _ { j } }$ scaled by a coefficient $C = 5 0$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "All datasets with the exception of IHDP were split into a training $( 6 3 \\% )$ , validation $( 2 7 \\% )$ and test set ( $10 \\%$ of samples). For IHDP we used exactly the same splits as previously used by Shalit et al. (2017). We repeated experiments on IHDP, Jobs, News and TCGA 1000, 10, 50, and 5 times, respectively. We reassigned outcomes and treatments with a new random seed for each repetition. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e3f0f5a85d9c5650511c8499d74ee85cea43a8e26a18683376faa7d59ab49cca.jpg",
+        "table_caption": [
+            "Table 3: Comparison of methods for counterfactual inference with two available treatment options on IHDP, Jobs and News-2. We report the mean value $\\pm$ the standard deviation of $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ , \u000fATE, $R _ { \\mathrm { P o l } }$ and $\\epsilon _ { \\mathrm { A T T } }$ on the test sets over a number of repeated runs. Where available we list the numbers reported by the original authors. n.r. $=$ not reported. ${ \\dag \\mathbf { \\alpha } } _ { = }$ significantly different from PM $( \\alpha < 0 . 0 5 )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">IHDP (1000 repeats)</td><td colspan=\"2\">Jobs (10 repeats)</td><td colspan=\"2\">News-2 (50 repeats)</td></tr><tr><td>Method</td><td>√EPEHE</td><td>EATE</td><td>RPo1(π）</td><td>EATT</td><td>VePEHE</td><td>CATE</td></tr><tr><td>PM</td><td>0.84 ± 0.61</td><td>0.24±0.20</td><td>0.18 ± 0.10</td><td>0.16 ± 0.07</td><td>16.76 ± 1.26</td><td>3.99 ± 1.01</td></tr><tr><td>+on X</td><td>0.81 ± 0.57</td><td>0.24 ± 0.20</td><td>0.23 ± 0.11</td><td>0.19 ± 0.09</td><td>17.06 ± 1.22</td><td>4.14 ± 1.27</td></tr><tr><td>+MLP</td><td>0.83 ± 0.57</td><td>0.23 ± 0.20</td><td>0.19 ± 0.12</td><td>0.16 ± 0.07</td><td>+ 18.38 ± 1.46</td><td>+ 5.90 ± 2.07</td></tr><tr><td>kNN</td><td>4.1 1±0.2</td><td>0.79± 0.05</td><td>+ 0.26±0.0</td><td>0.13 ± 0.05</td><td>† 18.14 ± 1.64</td><td>+7.83 ± 2.55</td></tr><tr><td>PSMPM</td><td>n.r.</td><td>n.r.</td><td>0.18 ± 0.12</td><td>0.18 ± 0.09</td><td>† 17.49 ± 1.49</td><td>† 5.02 ± 2.34</td></tr><tr><td>PSMMI</td><td>+ 2.70 ± 3.85</td><td>+ 0.49 ± 0.81</td><td>0.20 ± 0.09</td><td>+ 0.42 ± 0.22</td><td>+ 17.40 ± 1.30</td><td>+ 4.89 ± 2.39</td></tr><tr><td>RF</td><td>6.6 ±0.3</td><td>† 0.96 ± 0.06</td><td>† 0.28 ± 0.0</td><td>+ 0.09 ± 0.04</td><td>† 17.39 ± 1.24</td><td>† 5.50 ± 1.20</td></tr><tr><td>CF</td><td>3.8 ±0.2</td><td>+ 0.40 ±0.03</td><td>0.20±0.0</td><td>+ 0.07 ± 0.03</td><td>† 18.36 ± 1.73</td><td>+ 8.48± 2.46</td></tr><tr><td>BART</td><td>2.3 ±0.1</td><td>+ 0.34± 0.02</td><td>+ 0.25 ± 0.0</td><td>+ 0.08±0.03</td><td>18.53 ± 2.02</td><td>† 5.40 ± 1.53</td></tr><tr><td>GANITE</td><td>2.4 ±0.4</td><td>+ 0.49± 0.05</td><td>0.14 ± 0.01</td><td>+ 0.06± 0.03</td><td>+ 18.28 ± 1.66</td><td>+ 4.65 ± 2.12</td></tr><tr><td>BNN</td><td>+ 2.1 ±0.1</td><td>+ 0.42± 0.03</td><td>0.24 ± 0.0</td><td>† 0.09 ± 0.04</td><td>n.r.</td><td>n.r.</td></tr><tr><td>PD</td><td>n.r.</td><td>n.r.</td><td>n.r.</td><td>n.r.</td><td>† 17.52 ± 1.62</td><td>4.69 ± 3.17</td></tr><tr><td>TARNET</td><td>+ 0.95 ± 0.02</td><td>+ 0.28 ± 0.01</td><td>0.21± 0.0</td><td>0.11 ± 0.04</td><td>17.17 ± 1.25</td><td>+ 4.58 ± 1.29</td></tr><tr><td>CFRNETwass</td><td>+ 0.76± 0.02</td><td>t 0.27 ± 0.01</td><td>0.21± 0.0</td><td>+ 0.09 ± 0.03</td><td>16.93 ± 1.12</td><td>† 4.54 ± 1.48</td></tr><tr><td>CEVAE</td><td>+ 2.7 ±0.1</td><td>+ 0.46 ± 0.02</td><td>+ 0.26 ±0.0</td><td>+ 0.03 ± 0.01</td><td>n.r.</td><td>n.r.</td></tr><tr><td>CMGP</td><td>+ 0.77 ± 0.11</td><td>+ 0.13 ± 0.12</td><td>0.24± 0.05</td><td>+ 0.09 ± 0.07</td><td>n.r.</td><td>n.r.</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Models. We evaluated PM, ablations, baselines, and all relevant state-of-the-art methods: kNN (Ho et al. (2007)), BART (Chipman et al. (2010); Chipman & McCulloch (2016)), Random Forests (RF) (Breiman (2001)), CF (Wager & Athey (2017)), GANITE (Yoon et al. (2018)), Balancing Neural Network (BNN) (Johansson et al. (2016)), TARNET (Shalit et al. (2017)), Counterfactual Regression Network using the Wasserstein regulariser (CFRNETWass) (Shalit et al. (2017)), CEVAE (Louizos et al. (2017)), CMGP (Alaa & van der Schaar (2017)), and PD (Alaa et al. (2017)). We trained a Support Vector Machine (SVM) with probability estimation (Pedregosa et al. (2011)) to estimate $p ( t | X )$ for PM on the training set. We also evaluated preprocessing the entire training set with PSM using the same matching routine as PM $( \\mathrm { P S M } _ { \\mathrm { P M } } )$ and the ”MatchIt” package $( \\mathrm { P S M } _ { \\mathrm { M I } }$ , Ho et al. (2011)) before training a TARNET (Appendix F). In addition, we trained an ablation of PM where we matched on the covariates $X$ $\\ l ^ { \\prime } + \\ l \\mathrm { o n } X )$ directly, if $X$ was low-dimensional $\\left( p < 2 0 0 \\right)$ ), and on a 50-dimensional representation of $X$ obtained via principal components analysis (PCA), if $X$ was high-dimensional, instead of on the propensity score. We also evaluated PM with a multi-layer perceptron $( + \\mathrm { M L P } )$ that received the treatment index $t _ { j }$ as an input instead of using a TARNET. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8bff40e945252228b1e3394856192f4b1553bd7e163b4cc04175983d61d536e5.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of several state-of-the-art methods for counterfactual inference on the test set of the TCGA dataset when varying the percentage of hidden confounding $\\mathbf { \\bar { X } }$ -axis), i.e. what percentage of the features that were relevant for treatment assignment are visible to the model. Symbols correspond to the mean value of $\\hat { \\epsilon } _ { \\mathrm { A T E } }$ (left) and $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ (right) across 5 repeated runs with new outcomes (lower is better). The shaded area indicates the standard deviation. PM, TARNET and CFRNET are relatively stable up to $60 \\%$ of hidden confounding. The methods degrade strongly in counterfactual prediction performance at levels of hidden confounding higher than $60 \\%$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            712,
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/dc62ad5c487d50d5a9c43f630997e93b01c6370c0f2031aa4faa4c99d74c5f29.jpg",
+        "table_caption": [
+            "Table 4: Comparison of methods for counterfactual inference with more than two available treatment options on the News-4, News-8 and News-16. Numbers are mean value $\\pm$ the standard deviation of $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { m P E H E } } }$ and $\\hat { \\epsilon } _ { \\mathrm { m A T E } }$ over 50 repeated runs. $^ { \\dag } =$ significantly different from PM $( \\alpha < 0 . 0 5 )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">News-4 (κ = 10)</td><td colspan=\"2\">News-8 (κ = 10)</td><td colspan=\"2\"></td><td colspan=\"2\">News-16 (k = 7)</td></tr><tr><td>VemPEHE</td><td>EmATE</td><td></td><td>VemPEHE</td><td>CmATE</td><td></td><td>VemPEHE</td><td>CmATE</td></tr><tr><td>PM</td><td>21.58 ± 2.58</td><td>10.04 ± 2.71</td><td></td><td>20.76 ± 1.86</td><td>6.51 ± 1.66</td><td></td><td>20.24 ± 1.46</td><td>5.76 ±1.33</td></tr><tr><td>+on X</td><td>21.41 ± 1.75 +</td><td>8.91 ± 2.00</td><td></td><td>20.90 ± 2.07</td><td>6.36 ± 1.65</td><td></td><td>20.67 ± 1.42</td><td>5.73 ± 1.16</td></tr><tr><td>+ MLP</td><td>+ 25.05 ± 2.80</td><td>+14.92±3.14</td><td>+</td><td>24.88±1.98 +</td><td>14.00±2.45+</td><td></td><td>27.05± 2.47 † 16.90 ± 2.01</td><td></td></tr><tr><td>kNN</td><td>+27.92 ± 2.44</td><td></td><td></td><td>4 + 19.40 ± 3.12 † 26.20± 2.18 † 15.11 ± 2.34 + 2</td><td></td><td></td><td>27.64 ± 2.40 † 17.27 ± 2.10</td><td></td></tr><tr><td>PSMPM</td><td>+ 22.74 ± 2.58 +</td><td></td><td></td><td>11.62 ± 2.69 † 22.16±1.79 †</td><td>8.81±1.96 + 2</td><td></td><td>23.57 ± 2.48 † 11.04 ± 2.50</td><td></td></tr><tr><td>PSMMI</td><td>† 37.26± 2.28 ↑</td><td></td><td></td><td>30.19 ± 2.47 + 30.50± 1.70 † 22.09±1.98</td><td></td><td>+</td><td>28.17 ± 2.02 † 18.81 ± 1.74</td><td></td></tr><tr><td>RF</td><td>+ 26.59 ± 3.02 +</td><td></td><td></td><td>18.03 ±3.18 † 23.77± 2.14 +</td><td>12.40±2.29</td><td>+</td><td>26.13± 2.48 †15.91± 2.00</td><td></td></tr><tr><td>CF</td><td>t 27.28 ± 2.59 ↑</td><td>19.04± 3.15</td><td>→</td><td>25.76±2.05 +</td><td>15.12±2.20+</td><td></td><td>26.85± 2.29</td><td>5+17.03 ± 1.99</td></tr><tr><td>BART</td><td>+ 26.41 ± 3.10 † 17.14± 3.51</td><td></td><td></td><td>t 25.78± 2.66 +</td><td>14.80±2.56†</td><td></td><td>27.45± 2.84 + 17.50± 2.49</td><td></td></tr><tr><td>GANITE</td><td>+ 24.50 ± 2.27 + 13.84 ± 2.69</td><td></td><td>†</td><td>23.58±2.48 +</td><td>11.20±2.84+2</td><td></td><td>25.12± 3.53 + 13.20±3.28</td><td></td></tr><tr><td>PD</td><td>20.88± 3.24 †</td><td>8.47 ± 4.51</td><td></td><td>21.19 ± 2.29</td><td>7.29 ± 2.97 †</td><td></td><td>22.28± 2.25 + 10.65± 2.22</td><td></td></tr><tr><td>TARNET</td><td>+23.40±2.20</td><td>)†13.63 ± 2.18</td><td>†</td><td>22.39± 2.32 +</td><td>9.38 ± 1.92</td><td>+</td><td>21.19 ± 2.01 †</td><td>8.30 ± 1.66</td></tr><tr><td>CFRNETWass</td><td>t 22.65± 1.97 + 12.96± 1.69</td><td></td><td></td><td>21.64±1.82 +</td><td>8.79 ± 1.68</td><td>+</td><td>20.87±1.46†</td><td>8.05 ±1.40</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Architectures. To ensure that differences between methods of learning counterfactual representations for neural networks are not due to differences in architecture, we based the neural architectures for TARNET, CFRNETWass, PD and PM on the same, previously described extension of the TARNET architecture (Shalit et al. (2017), Details in Appendix G) to the multiple treatment setting. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters. For the IHDP, Jobs, News and TCGA dataset we respectively used 30, 30, 10, and 5 optimisation runs for each method using randomly selected hyperparameters from predefined ranges (Appendix H). We selected the best model across the runs based on validation set $\\hat { \\epsilon } _ { \\mathrm { N N } }$ -PEHE. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Metrics. We calculated the \u000fPEHE and $\\epsilon _ { \\mathrm { A T E } }$ for datasets for which we know the outcome generating process. For the Jobs dataset, we did not have access to counterfactual outcomes. We therefore calculated the $\\begin{array} { r } { \\mathrm { A T T } \\ = \\ \\frac { 1 } { | T _ { 1 } \\cap E | } \\sum _ { n = 0 } ^ { N } y _ { 1 } ( n ) - \\frac { 1 } { | T _ { 0 } \\cap E | } \\sum _ { n = 0 } ^ { N } y _ { 0 } ( n ) } \\end{array}$ PNn=0 y0(n) where E was the randomised subset of the dataset, and reported the error in estimating ATT $\\begin{array} { r } { \\epsilon _ { \\mathrm { A T T } } = | \\mathrm { A T T } - \\frac { 1 } { | T _ { 1 } \\cap E | } \\sum _ { n = 0 } ^ { N } \\hat { y } _ { 1 } ( n ) | , } \\end{array}$ and the policy risk $R _ { \\mathrm { P o l } } ( \\pi )$ (Shalit et al. (2017); Yoon et al. (2018)). The policy risk is the average loss in value when treating according to the policy $\\pi$ implied by an ITE estimator (Appendix I). ",
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+    },
+    {
+        "type": "text",
+        "text": "5 RESULTS AND DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Counterfactual Inference. We evaluated the counterfactual inference performance of the listed models in the binary setting (Table 3) and multiple treatment setting (Table 4). On IHDP, PM reached the third best performance in terms of $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ after CFRNET and CMGP, and the second best $\\epsilon _ { \\mathrm { A T E } }$ after CMGP. On Jobs, PM reached the second best $R _ { \\mathrm { P o l } } ( \\pi )$ after GANITE. On the binary News-2, PM outperformed all other methods in terms of $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ and $\\epsilon _ { \\mathrm { A T E } }$ . On the News-4/8/16 datasets with more than two treatments, PM consistently outperformed all other methods - in some cases by a large margin - on both metrics with the exception of the News-4 dataset, where PM came second to PD. The strong performance of PM across a wide range of datasets with varying amounts of treatment options is remarkable considering how simple it is compared to other, highly specialised methods. Notably, PM consistently outperformed both CFRNET, with the exception of the IHDP dataset, which accounted for covariate imbalances between treatments via regularisation rather than matching, and $\\mathrm { P S M } _ { \\mathrm { M I } }$ , which accounted for covariate imbalances by preprocessing the entire training set with a matching algorithm (Ho et al. (2011)). We also found that matching on the propensity score was, in almost all cases, not significantly different from matching on $X$ directly when $X$ was low-dimensional, or a low-dimensional representation of $X$ when $X$ was high-dimensional $^ +$ o n $X$ ). This indicates that PM is effective with any low-dimensional balancing score. In addition, using PM with the TARNET architecture outperformed the MLP $+ \\mathrm { M L P } )$ in almost all cases, with the exception of the low-dimensional IHDP and Jobs. We conclude that matching on the propensity score or a low-dimensional representation of $X$ and using the TARNET architecture are sensible default configurations, particularly when $X$ is high-dimensional. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/781506b3f9930a44bc60e6f40e4a257c591bc7efdf418d1d1b39c427024efb86.jpg",
+        "image_caption": [
+            "Figure 6: Comparison of the learning dynamics during training (normalised training epochs; from start ${ } = 0$ to end $= 1 0 0$ of training, $\\mathbf { X }$ -axis) of several matching-based methods on the validation set of News-8. The coloured lines correspond to the mean value of the factual error $\\mathrm { \\cdot } \\sqrt { \\mathrm { M S E } }$ ; red) and the counterfactual error $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ ; blue) across 10 randomised hyperparameter configurations (lower is better). The shaded area indicates the standard deviation. All methods used exactly the same model and hyperparameters and only differed in how they addressed treatment assignment imbalance. The leftmost figure shows the desired behavior of the counterfactual and factual error jointly decreasing until convergence. PM exhibits the desired behavior of aligning the optimisation of the factual and counterfactual error. $\\mathrm { P S M } _ { \\mathrm { P M } }$ shows a weaker degree of alignment and much higher variance. In contrast, $\\mathrm { P S M } _ { \\mathrm { M I } }$ shows the undesired behavior of the counterfactual error increasing as the factual error decreases, indicating that the models were overfitting to the properties of the treated group. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Model Selection. To judge whether NN-PEHE is more suitable for model selection for counterfactual inference than MSE, we compared their respective correlations with the PEHE on IHDP. We found that NN-PEHE correlates significantly better with the PEHE than MSE (Figure 2). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Number of Matches per Minibatch. To determine the impact of matching fewer than $100 \\%$ of all samples in a batch, we evaluated PM on News-8 trained with varying percentages of matched samples on the range 0 to $100 \\%$ in steps of $10 \\%$ (Figure 3). We found that including more matches indeed consistently reduces the counterfactual error up to $100 \\%$ of samples matched. Interestingly, we found a large improvement over using no matched samples even for relatively small percentages $( < 4 0 \\% )$ of matched samples per batch. This shows that propensity score matching within a batch is indeed highly effective at improving the training of neural networks for counterfactual inference. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Treatment Assignment Bias. To assess how the predictive performance of the different methods is influenced by increasing amounts of treatment assignment bias, we evaluated their performances on News-8 while varying the assignment bias coefficient $\\kappa$ on the range of 5 to 20 (Figure 4). We found that PM handles high amounts of assignment bias better than existing state-of-the-art methods. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hidden Confounding. To analyse the sensitivity of the different methods to hidden confounding, we successively retrained the models on a reduced set of covariates $x _ { i }$ in increasing steps of $10 \\%$ $( = \\ 2 0 5 3$ gene loci) in the range of $10 \\%$ to $90 \\%$ (Figure 5). We removed the covariates from the model but kept them for computing assignments and outcomes, i.e. they were hidden confounders. The results showed that PM was competitive with TARNET and CFRNET in terms of robustness to hidden confounding. Notably, PM, TARNET and CFRNET were relatively stable up to levels of $60 \\%$ of hidden confounding, likely because some gene loci are highly correlated. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Comparing Minibatch and Dataset Matching. As outlined previously, it is possible to optimise the counterfactual error using the observed factual samples if (1) we manage to break the dependence of treatment assignment on $X$ , and (2) we observe all relevant variables (Ho et al. (2007)). If we were successful in balancing the covariates, we would expect that the counterfactual error is implicitly and consistently improved alongside the factual error, despite optimising the neural network for the observed factual loss only. To elucidate to what degree this is the case when using the various evaluated matching-based methods, we evaluated the respective training dynamics of PM, $\\mathrm { P S M } _ { \\mathrm { P M } }$ and $\\mathrm { P S M } _ { \\mathrm { M I } }$ (Figure 6). We found that PM better conforms to the desired behavior than $\\mathrm { P S M } _ { \\mathrm { P M } }$ and $\\mathrm { P S M } _ { \\mathrm { M I } }$ . $\\mathrm { P S M } _ { \\mathrm { P M } }$ , which used the same matching strategy as PM but on the dataset level, showed a much higher variance than PM. $\\mathrm { P S M } _ { \\mathrm { M I } }$ was overfitting to the treated group. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Limitations. A limitation of this work is that the theory of matching to balance the covariates only holds in the strongly ignorable setting, i.e. under the assumption that there are no unobserved confounders. However, our experiments (Figure 5) and related works showed that even the presence of large amounts of hidden confounders may not necessarily decrease the performance of ITE estimators in practice if we observe proxy variables (Montgomery et al. (2000); Louizos et al. (2017)). In future work, we would like to integrate PM with diagnostic tools that address hidden confounding (Kallus & Zhou (2018)), and the ability to learn from multiple related tasks (Schwab et al. (2018)). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We presented PM, a new and simple method for training neural networks for estimating ITEs from observational data that extends to any number of available treatment options. In addition, we extended the PEHE metric to settings with more than two treatments, and introduced a nearest neighbour approximation of PEHE and mPEHE that can be used for model selection without having access to counterfactual outcomes. We performed experiments on several real-world and semisynthetic datasets that showed that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes. We also found that the NN-PEHE correlates significantly better with real PEHE than MSE, that including more matched samples in each minibatch improves the learning of counterfactual representations, and that PM handles an increasing treatment assignment bias better than existing state-of-the-art methods. PM may be used for settings with any amount of treatment options, is compatible with any existing neural network architecture, simple to implement, and does not introduce any additional hyperparameters or computational complexity. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "John N Weinstein, Eric A Collisson, Gordon B Mills, Kenna R Mills Shaw, Brad A Ozenberger, Kyle Ellrott, Ilya Shmulevich, Chris Sander, Joshua M Stuart, Cancer Genome Atlas Research Network, et al. The cancer genome atlas pan-cancer analysis project. Nature genetics, 45(10): 1113–1120, 2013. ",
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+        "text": "Jinsung Yoon, James Jordon, and Mihaela van der Schaar. GANITE: Estimation of Individualized Treatment Effects using Generative Adversarial Nets. In International Conference on Learning Representations, 2018. ",
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+    },
+    {
+        "type": "text",
+        "text": "SUPPLEMENTARY MATERIAL FOR:”PERFECT MATCH: A SIMPLE METHOD FOR LEARN-ING REPRESENTATIONS FOR COUNTERFACTUAL IN-FERENCE WITH NEURAL NETWORKS”",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            220,
+            398,
+            248
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A PRECISION IN ESTIMATION OF HETEROGENOUS EFFECT (PEHE) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            306,
+            748,
+            325
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "When the underlying noiseless distributions $\\mu _ { j }$ are not given, PEHE is calculated using: ",
+        "bbox": [
+            171,
+            343,
+            750,
+            359
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3a0b3941dc4b6db54b577ee0f17af8b509949e35fea32b5898ec7d5f6982fec.jpg",
+        "text": "$$\n{ \\hat { \\epsilon } } _ { \\mathrm { P E H E } } = { \\frac { 1 } { N } } \\sum _ { n = 0 } ^ { N } { \\Big ( } { \\big [ } y _ { 1 } ( n ) - y _ { 0 } ( n ) { \\big ] } - { \\big [ } { \\hat { y } } _ { 1 } ( n ) - { \\hat { y } } _ { 0 } ( n ) { \\big ] } { \\Big ) } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            372,
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+            416
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B AVERAGE TREATMENT EFFECT (ATE) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            438,
+            527,
+            457
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The ATE measures the average difference in effect across the whole population (Hill (2011)). It can be useful indicator of how well an ITE estimator performs at comparing two treatments across the entire population. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/186ab789e07def2c12a9749a49689b02bd5aca848e5d6c92c226bd2ca5613c36.jpg",
+        "text": "$$\n\\epsilon _ { \\mathrm { A T E } } = | | \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } \\mathbb { E } _ { y ( n ) \\sim \\mu ( n ) } [ y ( n ) ] - \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } \\hat { y } ( n ) | | _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Similar to Equation (A.1), the counterfactual treatment effect can be estimated using the noisy ground truth outcomes if the underlying noiseless distributions $\\mu _ { j }$ are not given: ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/26469ea5ed15242f1ffe9d80751418ff35e21fb4a9126b7899b27dad2d03e44f.jpg",
+        "text": "$$\n\\hat { \\epsilon } _ { \\mathrm { A T E } } = | | \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } y ( n ) - \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } \\hat { y } ( n ) | | _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C DETAILS PERFECT MATCH (PM) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 outlines the procedure of using PM to augment a batch with propensity score matches for training a neural network with stochastic gradient descent. We trained a Support Vector Machine (SVM) with probability estimation as propensity score estimator $E _ { \\mathrm { P S } }$ . To speed up recalling nearest neighbours, we additionally prepared an index per treatment into $X _ { \\mathrm { t r a i n } }$ sorted by propensity score. At augmentation time, we used binary search on this index to find the nearest neighbours by propensity score in $\\mathrm { O } ( \\log \\ N _ { t } )$ where $N _ { t }$ is the number of samples $X$ in $X _ { \\mathrm { { t r a i n } } }$ assigned to the treatment group $t$ . To avoid overfitting to specific edge samples when propensity scores are not distributed evenly in the training set, we chose at random from one of the $k { = } 6$ closest samples by propensity score. See Table S1 for a comparison of the effects on predictive performance of choosing varying values of $k$ . For the $^ { \\circ } +$ on $X ^ { \\dag }$ model we matched, using the Euclidean distance, directly on the covariates $X$ on IHDP and Jobs, and using a low dimensional representation of $X$ , obtained using principal component analysis (PCA) with 50 principal components, on News. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/66a58c5268b88dc361b8cb3679243f936e1f0cac2884d6bfac327358d944d3ca.jpg",
+        "table_caption": [
+            "Table S1: Comparison of PM with varying numbers of nearest neighbours $k$ considered for randomised matching on the News-2/4/8/16 datasets $k { = } 1$ corresponds to no randomisation; $k { = } 6$ , in cursive, are the results reported in the main body). We report the mean value $\\pm$ the standard deviation of $\\sqrt { \\hat { \\epsilon } _ { \\mathrm { P E H E } } }$ on the test sets over 50 runs. There was no significant differences between different levels of $k$ and $k { = } 6$ (significance level $\\alpha < 0 . 0 5$ ). In particular, there was no significant difference between using randomisation and not using randomisation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>News-2 VEPEHE</td><td>News-4 VEPEHE</td><td>News-8 VEPEHE</td><td>News-16 VEPEHE</td></tr><tr><td>PM (k=1)</td><td>16.79 ± 1.16</td><td>21.86 ± 2.24</td><td>20.70 ±1.87</td><td>20.06 ±1.43</td></tr><tr><td>PM (k=3)</td><td>16.82 ± 1.17</td><td>21.49 ± 2.43</td><td>20.50 ± 1.48</td><td>20.28 ± 1.52</td></tr><tr><td>PM(k=6)</td><td>16.76 ± 1.26</td><td>21.58 ± 2.58</td><td>20.76 ± 1.86</td><td>20.24 ± 1.46</td></tr><tr><td>PM (k=9)</td><td>16.91 ± 1.18</td><td>21.56 ± 2.58</td><td>20.75 ± 1.53</td><td>20.47 ± 1.94</td></tr><tr><td>PM (k=12)</td><td>17.11 ± 0.94</td><td>21.16 ± 2.20</td><td>21.10 ± 2.14</td><td>20.28 ± 1.64</td></tr><tr><td>PM (k=15)</td><td>16.87 ± 1.14</td><td>20.94 ± 2.05</td><td>20.92 ±1.80</td><td>20.48 ±1.80</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "D REDUCED VARIANCE IS LINKED TO IMPROVED PERFORMANCE IN ESTIMATING TREATMENT EFFECTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            353,
+            756,
+            388
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the standard supervised setting, the variance added by minibatch SGD leads to slower convergence (Csiba & Richtrik (2018)), but will not typically lead to models that perform worse if they are trained until a convergence criterion has been met, e.g. early stopping. It is therefore perhaps surprising that we observe a difference not just in convergence (Figure 6), but also in counterfactual estimation performance (Tables 3 and 4) when comparing models trained with and without batch matching. ",
+        "bbox": [
+            173,
+            406,
+            825,
+            477
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A potential cause of the observed difference in treatment effect estimation performance is that we do not have access to the true counterfactual error in observational data to select the best model encountered during training. We therefore have to resort to using the factual error (e.g. factual MSE) or an estimator based on the factual error (e.g. NN-PEHE) to select the best encountered model. However, when the variance during training is high, chances are that we select a model based on the factual error that at this point during the training happens to be suboptimal in terms of counterfactual error. Through this mechanism, added variance during training for counterfactual inference is directly linked to worse expected performance in treatment effect estimation (Figure S1). We do not observe the same behavior in standard supervised learning tasks because we have direct access to the underlying objective, and are therefore able to select the best encountered model regardless of the variance during optimisation, i.e. models that during training do not perform well momentarily will not be selected because their measured error is high. We confirmed experimentally that gradient steps are more likely to be in opposing directions when not using PM (Figure S2). ",
+        "bbox": [
+            173,
+            484,
+            825,
+            665
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/54a44ae4f2da5d207dbfbccbabbcbc6b9a928298616bc3331ca291361b7bc335.jpg",
+        "image_caption": [
+            "Figure S1: Illustration of the impact of added variance introduced by minibatch SGD during training. In the supervised setting (left), high variance does in general not impact the final performance in terms of expected factual error (red), because we can select (dotted black line) the best encountered model during training using the observed error. In the counterfactual setting (right), we are not able to compute the true counterfactual error (blue) in observational data, and must therefore resort to selecting based on the factual error or a measure derived from the factual error. Even though we encounter models that perform similarly well, we are less likely to select these models under high variance, because we do not have direct access to the counterfactual error and the minimum of the factual error in general does not correspond to the minimum of the counterfactual error, particularly when optimisation of counterfactual and factual error are not well aligned. The dashed blue line indicates the counterfactual error of the model selected based on factual error. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            672,
+            823,
+            785
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e482f386c8f24bebeefb344e713b1622f13fd82b8b3e2937f1e3f15453e8780c.jpg",
+        "image_caption": [
+            "Figure S2: Comparison of the frequencies of individual gradient steps during training of several matching-based methods on the validation set of News-8. The heatmap entries show the frequencies of pairs of differences in factual error (∆MSE; y-axis) and counterfactual error ( $\\Delta \\epsilon _ { \\mathrm { P E H E } }$ ; $\\mathbf { X }$ -axis) after individual gradient steps. All methods used the same model and only differed in how they addressed treatment assignment bias. The leftmost heatmap shows the desired behavior of gradient steps changing the counterfactual and factual error jointly during training. PM most closely resembles the desired behavior of jointly optimising counterfactual and factual error. In contrast, $\\mathrm { P S M } _ { \\mathrm { P M } }$ and $\\mathrm { P S M _ { M I } }$ have a much higher relative frequency of gradient steps that do not move into the same magnitude or direction (off-diagonal entries) for both factual and counterfactual error. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            99,
+            800,
+            205
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E NEAREST NEIGHBOUR APPROXIMATION OF PEHE FOR MULTIPLE TREATMENTS (NN-MPEHE) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            371,
+            756,
+            405
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The \u000fˆNN-PEHE metric can be extended to the multiple treatment setting by considering the mean \u000fˆNN-PEHE between all $\\binom { k } { 2 }$ possible pairs of treatments: ",
+        "bbox": [
+            173,
+            424,
+            825,
+            455
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5aced9e8422c4f2b07f67a1654accc21963dc77ce3f89952e6715215bd20979d.jpg",
+        "text": "$$\n\\hat { \\epsilon } _ { \\mathrm { N N - m P E H E } } = \\frac { 1 } { \\binom { k } { 2 } } \\sum _ { i = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ { i - 1 } \\hat { \\epsilon } _ { \\mathrm { N N - P E H E } , i , j }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            469,
+            622,
+            515
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "F PSEUDOCODE PROPENSITY SCORE MATCHING (PSM) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            537,
+            656,
+            555
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Algorithm S2 outlines the procedure of preprocessing a training set using $\\mathrm { P S M } _ { \\mathrm { M I } }$ . For $\\mathrm { P S M } _ { \\mathrm { P M } }$ , we used the PM matching algorithm (see Algorithm 1) to find matches between samples from the treatment groups. For $\\mathrm { P S M } _ { \\mathrm { M I } }$ , we used the MatchIt package (Ho et al. (2011)) on setting ”nearest” to find matches between sets of samples from two treatment groups (function get matched samples). ",
+        "bbox": [
+            173,
+            573,
+            826,
+            631
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/1f7d7e64ec570c65dcaf91ddfa6caaa622d2a81b0d6aba82be137fb39f20f330.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm S2 Preprocessng a training set using Propensity Score Matching (PSM). After prepro- cessing,the training set contains an equal number of samples from each treatment group and the covariates xi across all treatment groups are approximately balanced.</td></tr><tr><td>Input: N training samples Xtrain with assgned treatments t, Number of treatment options k, func- tion get_mat ched_samples to find nearest matches by propensity score between sets of</td></tr><tr><td>samples Output: Preprocessed training set Xout consisting of matched samples from each treatment group</td></tr><tr><td>1: procedure PREPROCESS_TRAINING_SET: 2: tbase ← index of treatment with smallest number of samples in Xtrain</td></tr><tr><td>Xbase ← all samples in Xtrain With treatment tbase</td></tr><tr><td>fori=O to k-1 do</td></tr><tr><td>if i≠ tbase then</td></tr><tr><td>Xcurrent ← all samples in Xtrain with treatment i</td></tr><tr><td>Xmatched ← get_mat ched_samples(Xbase,Xcurrent) Add all samples in Xmatched to Xout</td></tr><tr><td></td></tr><tr><td>Add all samples in Xbase to Xout</td></tr></table>",
+        "bbox": [
+            173,
+            661,
+            826,
+            900
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "G TARNET, CFRNET AND PD FOR MULTIPLE TREATMENTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            702,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In TARNET models, we used ELU nonlinearities between the $L$ hidden layers with $M$ hidden units. $L$ and $M$ were hyperparameters that we optimised during hyperparameter optimisation (Section H). We did not use batch normalisation (BN). To extend CFRNET to the multiple treatment setting, we defined the first treatment option as the control treatment and regularised all treatment options to have the same activation distribution in the topmost shared layer (Shalit et al. (2017)). For PD, we only used propensity dropout and the propensity estimation network. We did not make use of the alternating training schedule proposed in (Alaa et al. (2017)). The $\\ \" { + } \\mathrm { M L P ^ { \\prime } }$ model was a simple MLP with $L$ hidden layers of $M$ hidden units that received the treatment option index $t _ { j }$ as an additional input along with the covariates $X$ , and output a $k$ -dimensional potential outcome vector $\\hat { Y }$ . The MLP used ELU nonlinearities between the $L$ hidden layers, and also did not use BN. ",
+        "bbox": [
+            173,
+            132,
+            825,
+            275
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "H HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            295,
+            380,
+            310
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To ensure a fair comparison, we used a standardised approach to hyperparameter optimisation for those methods for which we did not have previously reported performance numbers. In particular, each method we trained was optimised over the same amount of hyperparameter optimisation runs. For the methods that used neural network models (TARNET, CFRNET, PD, $\\mathrm { P S M } _ { \\mathrm { P M } }$ , $\\mathrm { P S M _ { M I } }$ , including $^ { , , } +$ on $X ^ { \\dag }$ and $\\ \" { + } \\mathrm { M L P ^ { \\prime } }$ , and PM), we chose hyperparameters at random from predefined ranges (Table S2). For CFRNET, we additionally varied the weight of the imbalance penalty at random between 0.1, 1.0, and 10.0. All methods that used a neural network model used the TARNET architecture to ensure differences in performance are not due to architectural differences. To train PM for the Jobs dataset, we used fixed hyperparameters: A batch size of 50, 60 hidden units per layer, and 3 hidden layers. For optimisation, we used the Adam optimiser with a learning rate of 0.001 for a maximum of 100 (News, TCGA) or 400 (IHDP, Jobs) with an early stopping patience of 30 on the factual MSE. We used the default hyperparameters for BART and CF from the ”bartMachine” (Kapelner & Bleich (2013)) and ”grf” (Athey et al. (2016)) R-packages. For GANITE, we used our own implementation since there was no open source implementation available, and - in addition to the parameters in Table S2 - optimised over the supervised loss weights $\\alpha$ and $\\beta$ (Yoon et al. (2018)) between 0.1, 1, and 10. For the GANITE generators and discriminators, we used MLP architectures with $L$ hidden layers of $M$ hidden units each. ",
+        "bbox": [
+            173,
+            327,
+            825,
+            563
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/a4a66408d557807ead43a7aa22fa8ce64e51aa93159be43c497a0af669b10c99.jpg",
+        "table_caption": [
+            "Table S2: Hyperparameter ranges used in the performed experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>IHDP</td><td>News/TCGA</td></tr><tr><td>Batch size B</td><td>4,8,50,100</td><td>50</td></tr><tr><td>Number of units per hidden layer M5</td><td>50,100,200</td><td>40,60,80</td></tr><tr><td>Number of hidden layers L</td><td>1,2,3</td><td>2,3</td></tr></table>",
+        "bbox": [
+            292,
+            595,
+            709,
+            664
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "I POLICY RISK $R _ { \\mathrm { P o L } } ( \\pi )$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            680,
+            385,
+            698
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We define the policy risk $R _ { \\mathrm { P o l } } ( \\pi )$ as the expected loss in value when treating according to the policy $\\pi$ implied by an ITE estimator (Shalit et al. (2017)): ",
+        "bbox": [
+            171,
+            713,
+            823,
+            741
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64302d1135199d1d275fa0a6e42bc8088c6a4804fad766ab394a6e801ca8997c.jpg",
+        "text": "$$\nR _ { \\mathrm { P o l } } ( \\pi ) = 1 - ( \\mathbb { E } [ y _ { 1 } | \\pi ( X ) = 1 ] \\cdot p ( \\pi = 1 ) + \\mathbb { E } [ y _ { 0 } | \\pi ( X ) = 0 ] \\cdot p ( \\pi = 0 ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            747,
+            746,
+            765
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where the policy $\\pi ( X )$ for a sample $X$ implied by an ITE estimator that produces potential outcomes $\\hat { y }$ is defined as: ",
+        "bbox": [
+            171,
+            771,
+            826,
+            800
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82503d24722df5135ae22c8605cd4a4170a7965ce554cce48eee1be479bcf610.jpg",
+        "text": "$$\n\\pi ( X ) = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } { \\hat { y } } _ { 1 , X } - { \\hat { y } } _ { 0 , X } > \\lambda } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. } \\quad { \\mathrm { ( t o ~ t r e a t ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            808,
+            668,
+            843
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "J ORIGINS OF PERFORMANCE NUMBERS IN TABLE 3 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            627,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Where available, performance numbers in Table 3 were taken from the original authors for IHDP and Jobs (Table S3). For News-2/4/8/16 and TCGA, performance numbers were produced by us. ",
+        "bbox": [
+            174,
+            132,
+            821,
+            161
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/f250afdd14f93b8be427f817094ba42e314ee81e223b7efe471e72e27096f5f9.jpg",
+        "table_caption": [
+            "Table S3: Origins of the performance results reported in Table 3. Entries show the reference to the original report of the performance results, or ”ours” if we produced the results. n.r. $=$ not reported "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>IHDP</td><td>Jobs</td></tr><tr><td>PM</td><td>ours</td><td>ours</td></tr><tr><td>+on X</td><td>ours</td><td>ours</td></tr><tr><td>+ MLP</td><td>ours</td><td>ours</td></tr><tr><td>kNN</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>PSMPM</td><td>n.r.</td><td>ours</td></tr><tr><td>PSMMI</td><td>ours</td><td>ours</td></tr><tr><td>RF</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>CF</td><td>(Shalit et al. 2 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>BART</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>GANITE</td><td>(Yoon et al. 2018)</td><td>(Yoon et al. 2018)</td></tr><tr><td>BNN</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>PD</td><td>n.r.</td><td>n.r.</td></tr><tr><td>TARNET</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td>CFRNETwass</td><td>(Shalit et al. 2017)</td><td>(Shalit et al. 2017)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>CEVAE CMGP</td><td>(Louizos et al. 2017) (Yoon et al. 2018)</td><td>(Louizos et al. 2017) (Yoon et al. 2018)</td></tr></table>",
+        "bbox": [
+            303,
+            210,
+            691,
+            468
+        ],
+        "page_idx": 15
+    }
+]

parse/train/vYVI1CHPaQg/vYVI1CHPaQg.md

@@ -0,0 +1,601 @@
+# A BETTER ALTERNATIVE TO ERROR FEEDBACK FORCOMMUNICATION-EFFICIENT DISTRIBUTED LEARN-ING
+
+Samuel Horváth and Peter Richtárik   
+KAUST   
+Thuwal, Saudi Arabia   
+{samuel.horvath, peter.richtarik}@kaust.edu.sa
+
+# ABSTRACT
+
+Modern large-scale machine learning applications require stochastic optimization algorithms to be implemented on distributed compute systems. A key bottleneck of such systems is the communication overhead for exchanging information (e.g., stochastic gradients) across the workers. Among the many techniques proposed to remedy this issue, one of the most successful is the framework of compressed communication with error feedback (EF). EF remains the only known technique that can deal with the error induced by contractive compressors which are not unbiased, such as Top- $K$ or PowerSGD. In this paper, we propose a new and theoretically and practically better alternative to EF for dealing with contractive compressors. In particular, we propose a construction which can transform any contractive compressor into an induced unbiased compressor. Following this transformation, existing methods able to work with unbiased compressors can be applied. We show that our approach leads to vast improvements over EF, including reduced memory requirements, better communication complexity guarantees and fewer assumptions. We further extend our results to federated learning with partial participation following an arbitrary distribution over the nodes, and demonstrate the benefits thereof. We perform several numerical experiments which validate our theoretical findings.
+
+# 1 INTRODUCTION
+
+We consider distributed optimization problems of the form
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) : = { \textstyle { \frac { 1 } { n } } } \sum _ { i = 1 } ^ { n } f _ { i } ( x ) ,
+$$
+
+where $\boldsymbol { x } \in \mathbb { R } ^ { d }$ represents the weights of a statistical model we wish to train, $n$ is the number of nodes, and $f _ { i } \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ is a smooth differentiable loss function composed of data stored on worker $i$ . In a classical distributed machine learning scenario, $f _ { i } ( x ) : = \mathrm { E } _ { \zeta \sim \mathcal { D } _ { i } } \left[ f _ { \zeta } ( x ) \right]$ is the expected loss of model $x$ with respect to the local data distribution $\mathcal { D } _ { i }$ of the form, and $f _ { \zeta } \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ is the loss on the single data point $\zeta$ . This definition allows for different distributions $\mathcal { D } _ { 1 } , \ldots , \mathcal { D } _ { n }$ on each node, which means that the functions $f _ { 1 } , \ldots , f _ { n }$ can have different minimizers. This framework covers Stochastic Optimization when either $n = 1$ or all $\mathcal { D } _ { i }$ are identical, Empirical Risk Minimization (ERM), when $f _ { i } ( x )$ can be expressed as a finite average, i.e, $\begin{array} { r } { f _ { i } ( x ) = \frac { 1 } { m _ { i } } \dot { \sum _ { i = 1 } ^ { m _ { i } } f _ { i j } ( x ) } } \end{array}$ for some $f _ { i j } : \mathbb { R } ^ { d }  \mathbb { R }$ , and Federated Learning (FL) (Kairouz et al., 2019) where each node represents a client.
+
+Communication Bottleneck. In distributed training, model updates (or gradient vectors) have to be exchanged in each iteration. Due to the size of the communicated messages for commonly considered deep models (Alistarh et al., 2016), this represents significant bottleneck of the whole optimization procedure. To reduce the amount of data that has to be transmitted, several strategies were proposed.
+
+One of the most popular strategies is to incorporate local steps and communicated updates every few iterations only (Stich, 2019a; Lin et al., 2018a; Stich & Karimireddy, 2020; Karimireddy et al.,
+
+2019a; Khaled et al., 2020). Unfortunately, despite their practical success, local methods are poorly understood and their theoretical foundations are currently lacking. Almost all existing error guarantees are dominated by a simple baseline, minibatch SGD (Woodworth et al., 2020).
+
+In this work, we focus on another popular approach: gradient compression. In this approach, instead of transmitting the full dimensional (gradient) vector $g \in \mathbb { R } ^ { d }$ , one transmits a compressed vector $\mathcal C ( g )$ , where $\mathcal { C } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ is a (possibly random) operator chosen such that $\mathcal C ( g )$ can be represented using fewer bits, for instance by using limited bit representation (quantization) or by enforcing sparsity. A particularly popular class of quantization operators is based on random dithering (Goodall, 1951; Roberts, 1962); see (Alistarh et al., 2016; Wen et al., 2017; Zhang et al., 2017; Horváth et al., 2019a; Ramezani-Kebrya et al., 2019). Much sparser vectors can be obtained by random sparsification techniques that randomly mask the input vectors and only preserve a constant number of coordinates (Wangni et al., 2018; Konecný & Richtárik, 2018; Stich et al., 2018; Mishchenko et al., ˇ 2019b; Vogels et al., 2019). There is also a line of work (Horváth et al., $2 0 1 9 \mathrm { a }$ ; Basu et al., 2019) in which a combination of sparsification and quantization was proposed to obtain a more aggressive effect. We will not further distinguish between sparsification and quantization approaches, and refer to all of them as compression operators hereafter.
+
+Considering both practice and theory, compression operators can be split into two groups: biased and unbiased. For the unbiased compressors, $\mathcal { C } ( g )$ is required to be an unbiased estimator of the update $g$ . Once this requirement is lifted, extra tricks are necessary for Distributed Compressed Stochastic Gradient Descent (DCSGD) (Alistarh et al., 2016; 2018; Khirirat et al., 2018) employing such a compressor to work, even if the full gradient is computed by each node. Indeed, the naive approach can lead to exponential divergence (Beznosikov et al., 2020), and Error Feedback (EF) (Seide et al., 2014; Karimireddy et al., 2019b) is the only known mechanism able to remedy the situation.
+
+Contributions. Our contributions can be summarized as follows:
+
+• Induced Compressor. When used within the stabilizing EF framework, biased compressors (e.g., Top- $K$ ) can often achieve superior performance when compared to their unbiased counterparts (e.g., Rand- $K$ ). This is often attributed to their low variance. However, despite ample research in this area, EF remains the only known mechanism that allows the use of these powerful biased compressors. Our key contribution is the development of a simple but remarkably effective alternative—and this is the only alternative we know of—which we argue leads to better and more versatile methods both in theory and practice. In particular, we propose a general construction that can transform any biased compressor, such as Top- $K$ , into an unbiased one for which we coin the name induced compressor (Section 3). Instead of using the desired biased compressor within EF, our proposal is to instead use the induced compressor within an appropriately chosen existing method designed for unbiased compressors, such as distributed compressed SGD (DCSGD) (Khirirat et al., 2018), variance reduced DCSGD (DIANA) (Mishchenko et al., 2019a) or accelerated DIANA (ADIANA) (Li et al., 2020). While EF can bee seen as a version of DCSGD which can work with biased compressors, variance reduced nor accelerated variants of EF were not known at the time of writing this paper.
+
+• Better Theory for DCSGD. As a secondary contribution, we provide a new and tighter theoretical analysis of DCSGD under weaker assumptions. If $f$ is $\mu$ -quasi convex (not necessarily convex) and local functions $f _ { i }$ are $( L , \sigma ^ { 2 } )$ -smooth (weaker version of $L$ -smoothness with strong growth condition), we obtain the rate $\begin{array} { r } { \mathcal { O } \left( \delta _ { n } L r ^ { 0 } \exp \left[ - \frac { \mu T } { 4 \delta _ { n } L } \right] + \frac { ( \delta _ { n } - 1 ) D + { \delta \sigma } ^ { 2 } / n } { \mu T } \right) } \end{array}$ , where $\begin{array} { r } { \delta _ { n } = 1 + \frac { \delta - 1 } { n } } \end{array}$ and $\delta \geq 1$ is the parameter which bounds the second moment of the compression operator, and $T$ is the number of iterations. This rate has linearly decreasing dependence on the number of nodes $n$ , which is strictly better than the best-known rate for DCSGD with EF, whose convergence does not improve as the number of nodes increases, which is one of the main disadvantages of using EF. Moreover, EF requires extra assumptions. In addition, while the best-known rates for EF (Karimireddy et al., 2019b; Beznosikov et al., 2020) are expressed in terms of functional values, our theory guarantees convergence in both iterates and functional values. Another practical implication of our findings is the reduction of the memory requirements by half; this is because in DCSGD one does not need to store the error vector.
+
+• Partial Participation. We further extend our results to obtain the first convergence guarantee for partial participation with arbitrary distributions over nodes, which plays a key role in Federated Learning (FL).
+
+# Algorithm 1 DCSGD
+
+1: Input: $\{ \eta ^ { k } \} _ { k = 0 } ^ { T } > 0 .$ , $x _ { 0 }$   
+2: for $k = 0 , 1 , \hdots T$ do   
+3: Parallel: Worker side   
+4: for $i = 1 , \ldots , n$ do   
+5: obtain $g _ { i } ^ { k }$   
+6: send $\Delta _ { i } ^ { k } = \mathcal { C } ^ { k } ( g _ { i } ^ { k } )$ to master   
+7: [no need to keep track of errors]   
+8: end for   
+9: Master side   
+10: aggregate $\begin{array} { r } { \Delta ^ { k } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \Delta _ { i } ^ { k } } \end{array}$   
+11: broadcast $\Delta ^ { k }$ to each worker   
+12: Parallel: Worker side   
+13: for $i = 1 , \ldots , n$ do   
+14: $x ^ { k + 1 } = x ^ { k } - \eta ^ { k } \Delta ^ { k }$   
+15: end for   
+16: end for
+
+# Algorithm 2 DCSGD with Error Feedback
+
+1: Input: $\{ \eta ^ { k } \} _ { k = 0 } ^ { T } > 0$ , $x _ { 0 }$ , $e _ { i } ^ { 0 } = 0 \forall i \in [ n ]$   
+2: for $k = 0 , 1 , \hdots T$ do   
+3: Parallel: Worker side   
+4: for $i = 1 , \ldots , n$ do   
+5: obtain $g _ { i } ^ { k }$   
+6: send $\Delta _ { i } ^ { k } = \mathcal { C } ^ { k } ( \eta ^ { k } g _ { i } ^ { k } + e _ { i } ^ { k } )$ to master   
+7: $e _ { i } ^ { k + 1 } = \eta ^ { k } g _ { i } ^ { k } + e _ { i } ^ { k } - \Delta _ { i } ^ { k }$   
+8: end for   
+9: 10: Master siaggregate $\begin{array} { r } { \Delta ^ { k } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \Delta _ { i } ^ { k } } \end{array}$   
+11: broadcast $\Delta ^ { k }$ to each worker   
+12: Parallel: Worker side   
+13: for $i = 1 , \ldots , n$ do   
+14: $x ^ { k + 1 } = x ^ { k } - \Delta ^ { k }$   
+15: end for   
+16: end for
+
+• Experimental Validation. Finally, we provide an experimental evaluation on an array of classification tasks with CIFAR10 dataset corroborating our theoretical findings.
+
+# 2 ERROR FEEDBACK IS NOT A GOOD IDEA WHEN USING UNBIASED COMPRESSORS
+
+In this section we first introduce the notions of unbiased and general compression operators, and then compare Distributed Compressed SGD (DCSGD) without (Algorithm 1) and with (Algorithm 2) Error Feedback.
+
+Unbiased vs General Compression Operators. We start with the definition of unbiased and general compression operators (Cordonnier, 2018; Stich et al., 2018; Koloskova et al., 2019).
+
+Definition 1 (Unbiased Compression Operator). A randomized mapping $\mathcal { C } \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } ^ { d } }$ is an unbiased compression operator (unbiased compressor) if there exists $\delta \geq 1$ such that
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ \mathcal { C } ( \boldsymbol { x } ) \right] = \boldsymbol { x } , \qquad \mathrm { E } \left. \mathcal { C } ( \boldsymbol { x } ) \right. ^ { 2 } \leq \delta \left. \boldsymbol { x } \right. ^ { 2 } , \qquad \forall \boldsymbol { x } \in \mathbb { R } ^ { d } . } \end{array}
+$$
+
+If this holds, we will for simplicity write ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ .
+
+Definition 2 (General Compression Operator). A (possibly) randomized mapping $\mathcal { C } \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } ^ { d } }$ is a general compression operator (general compressor) if there exists $\lambda > 0$ and $\delta \geq 1$ such that
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ \left. \lambda \mathcal { C } ( x ) - x \right. ^ { 2 } \right] \leq \left( 1 - \frac { 1 } { \delta } \right) \left. x \right. ^ { 2 } , \qquad \forall x \in \mathbb { R } ^ { d } . } \end{array}
+$$
+
+If this holds, we will for simplicity write $\mathcal { C } \in \mathbb { C } ( \delta )$ .
+
+The following lemma provides a link between these notions (see, e.g. Beznosikov et al. (2020)).
+
+Lemma 1. If ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ , then (3) holds with $\begin{array} { r } { \lambda = \frac { 1 } { \delta } } \end{array}$ , i.e., $\mathcal { C } \in \mathbb { C } ( \delta )$ . That is, $\mathbb { U } ( \delta ) \subset \mathbb { C } ( \delta )$ .
+
+Note that the opposite inclusion to that established in the above lemma does not hold. For instance, the Top- $K$ operator belongs to $\mathbb { C } ( \delta )$ , but does not belong to $\mathbb { U } ( \delta )$ . In the next section we develop a procedure for transforming any mapping $\mathcal { C } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ (and in particular, any general compressor) into a closely related induced unbiased compressor.
+
+Distributed SGD with vs without Error Feedback. In the rest of this section, we compare the convergence rates for DCSGD (Algorithm 1) and DCSGD with EF (Algorithm 2). We do this comparison under standard assumptions (Karimi et al., 2016; Bottou et al., 2018; Necoara et al., 2019; Gower et al., 2019; Stich, 2019b; Stich & Karimireddy, 2020), listed next.
+
+First, we assume throughout that $f$ has a unique minimizer $x ^ { \star }$ , and let $f ^ { \star } = f ( x ^ { \star } ) > - \infty$ .
+
+Assumption 1 ( $\mu$ -quasi convexity). $f$ is $\mu$ -quasi convex, i.e.,
+
+$$
+\begin{array} { r } { f ^ { \star } \geq f ( x ) + \langle \nabla f ( x ) , x ^ { \star } - x \rangle + \frac { \mu } { 2 } \left. x ^ { \star } - x \right. ^ { 2 } , \qquad \forall x \in \mathbb { R } ^ { d } . } \end{array}
+$$
+
+Assumption 2 (unbiased gradient oracle). The stochastic gradient used in Algorithms 1 and 2 satisfies
+
+$$
+\operatorname { E } \left[ g _ { i } ^ { k } \mid x ^ { k } \right] = \nabla f _ { i } ( x ^ { k } ) , \qquad \forall i , k .
+$$
+
+Note that this assumption implies $\begin{array} { r } { \mathrm { E } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } g _ { i } ^ { k } \mid x ^ { k } \right] = \nabla f ( x ^ { k } ) . } \end{array}$
+
+Assumption 3 $( ( L , \sigma ^ { 2 } )$ -expected smoothness). Function $f$ is $( L , \sigma ^ { 2 } )$ -smooth if there exist constants $L > 0$ and $\sigma ^ { 2 } \geq 0$ such that $\forall i \in [ n ]$ and $\forall x ^ { k } \in \mathbb { R } ^ { d }$
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ \left\| g _ { i } ^ { k } \right\| ^ { 2 } \right] \leq 2 L ( f _ { i } ( x ^ { k } ) - f _ { i } ^ { \star } ) + \sigma ^ { 2 } , } \end{array}
+$$
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ \left. \frac { 1 } { n } \sum _ { i = 1 } ^ { n } g _ { i } ^ { k } \right. ^ { 2 } \right] \leq 2 L ( f ( x ^ { k } ) - f ^ { \star } ) + \sigma ^ { 2 } / n , } \end{array}
+$$
+
+where $f _ { i } ^ { \star }$ is the minimum functional value of $f _ { i }$ and $[ n ] = \{ 1 , 2 , \dots , n \}$ .
+
+This assumption generalizes standard smoothness and boundedness of variance assumptions. For more details and discussion, see the works of Gower et al. (2019); Stich (2019b). Equipped with these assumptions, we are ready to proceed with the convergence theory.
+
+Theorem 2 (Convergence of DCSGD). Consider the DCSGD algorithm with $n \geq 1$ nodes. Let Assumptions $_ { I - 3 }$ hold and ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ , where $\begin{array} { r } { \delta _ { n } = \frac { \delta - 1 } { n } + 1 } \end{array}$ . Let $\begin{array} { r } { D : = \frac { 2 L } { n } \sum _ { i = 1 } ^ { n } ( f _ { i } ( x ^ { \star } ) - f _ { i } ^ { \star } ) } \end{array}$ . Then there exist stepsizes $\begin{array} { r } { \eta ^ { k } \leq \frac { 1 } { 2 \delta _ { n } L } } \end{array}$ and weights $w ^ { k } \geq 0$ such that for all $T \geq 1$ we have
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ f ( \bar { x } ^ { T } ) - f ^ { \star } \right] + \mu \mathrm { E } \left[ \left. x ^ { T } - x ^ { \star } \right. ^ { 2 } \right] \leq 6 4 \delta _ { n } L r ^ { 0 } \exp \left[ - \frac { \mu T } { 4 \delta _ { n } L } \right] + 3 6 \frac { ( \delta _ { n } - 1 ) D + \delta \sigma ^ { 2 } / n } { \mu T } , } \end{array}
+$$
+
+$$
+\begin{array} { r } { r ^ { 0 } = \left. x ^ { 0 } - x ^ { \star } \right. ^ { 2 } , W ^ { T } = \sum _ { k = 0 } ^ { T } w ^ { k } , a n d \operatorname* { P r o b } ( \bar { x } ^ { T } = x ^ { k } ) = w ^ { k } / W ^ { T } . } \end{array}
+$$
+
+If $\delta = 1$ (no compression), Theorem 2 recovers the optimal rate of Distributed SGD (Stich, 2019b). If $\delta > 1$ , there is an extra term $( \delta _ { n } - 1 ) D$ in the convergence rate, which appears due to heterogenity of data $\begin{array} { r } { ( \sum _ { i = 1 } ^ { n } \nabla f _ { i } ( x ^ { \star } ) = 0 } \end{array}$ , but $\textstyle \sum _ { i = 1 } ^ { n } { \mathcal { C } } ( \nabla f _ { i } ( x ^ { \star } ) ) \neq { \bar { 0 } }$ in general). In addition, the rate is negatively affected by extra variance due to presence of compression which leads to $L  \delta _ { n } L$ and $\sigma ^ { 2 } / { n }  \delta \sigma ^ { 2 } / n$ .
+
+Next we compare our rate to the best-known result for Error Feedback (Stich & Karimireddy, 2020) $\ R = 1$ ), (Beznosikov et al., 2020) $( n \geq 1 )$ ) used with ${ \mathcal { C } } \in \mathbb { U } ( \delta ) \subset \mathbb { C } ( \delta )$
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ f ( \bar { x } ^ { T } ) - f ^ { \star } \right] = \tilde { \mathcal { O } } \left( \delta L r ^ { 0 } \exp \left[ - \frac { \mu T } { \delta L } \right] + \frac { \delta D + \sigma ^ { 2 } } { \mu T } \right) } \end{array}
+$$
+
+One can note several disadvantages of Error Feedback (Alg. 2) with respect to plain DCSGD (Alg. 1). The first major drawback is that the effect of compression $\delta$ is not reduced with an increasing number of nodes. Another disadvantage is that Theorem 2 implies convergence for both the functional values and the last iterate, rather than for functional values only as it is the case for EF. On top of that, our rate of DCSGD as captured by Theorem 2 does not contain any hidden polylogarithmic factor comparing to EF. Another practical supremacy of DCSGD is that there is no need to store an extra vector for the error, which reduces the storage costs by a factor of two, making Algorithm 1 a viable choice for Deep Learning models with millions of parameters. Finally, one does not need to assume standard $L$ -smoothness in order to prove convergence in Theorem 2, while, one the other hand, $L$ -smoothness is an important building block for proving convergence for general compressors due to the presence of bias (Stich & Karimireddy, 2020; Beznosikov et al., 2020). The only term in which EF might outperform plain DCSGD is $\bar { \mathcal { O } } ( \sigma ^ { 2 } / \mu T )$ for which the corresponding term is $\mathcal { O } ( \delta \sigma ^ { 2 } / n \mu T )$ . This is due to the fact that EF compensates for the error, while standard compression introduces extra variance. Note that this is not major issue as it is reasonable to assume $\delta / { \it n } = \mathcal { O } ( 1 )$ or, in addition, $\sigma ^ { 2 } = 0$ if weak growth condition holds (Vaswani et al., 2019), which is quite standard assumption, or one can remove effect of $\sigma ^ { 2 }$ by either computing full gradient locally or by incorporating variance reduction such as SVRG (Johnson & Zhang, 2013). In Section 4, we also discuss the way how to remove the effect of $D$ in Theorem 2. Putting all together, this suggests that standard DCSGD (Algorithm 1) is strongly preferable, in theory, to DCSGD with Error Feedback (Algorithm 2) for ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ .
+
+# 3 INDUCED COMPRESSOR: FIXING BIAS WITH ERROR-COMPRESSION
+
+In the previous section, we showed that compressed DCSGD is theoretically preferable to DCSGD with Error Feedback for ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ . Unfortunately, $\mathbb { C } ( \delta ) \ \not \subset \ \mathbb { U } ( \delta )$ , an example being the Top- $K$ compressor (Alistarh et al., 2018; Stich et al., 2018). This compressors belongs to $\mathbb { C } ( { \frac { d } { K } } )$ , but does not belong to $\mathbb { U } ( \delta )$ for any $\delta$ . On the other hand, multiple unbiased alternatives to Top- $\bar { K }$ have been proposed in the literature, including gradient sparsification (Wangni et al., 2018) and adaptive random sparsification (Beznosikov et al., 2020).
+
+Induced Compressor. We now propose a general mechanism for constructing an unbiased compressor ${ \mathcal { C } } \in \mathbb { U }$ from any biased compressor ${ \mathcal { C } } _ { 1 } \in \mathbb { C }$ . We shall argue that it is preferable to use this induced compressor within DCSGD, in both theory and practice, to using the original biased compressor $\mathcal { C } _ { 1 }$ within DCSGD $^ +$ Error Feedback.
+
+Theorem 3. For $\mathcal { C } _ { 1 } \in \mathbb { C } ( \delta _ { 1 } )$ with $\lambda = 1$ , choose $\mathcal { C } _ { 2 } \in \mathbb { U } ( \delta _ { 2 } )$ and define the induced compressor via
+
+$$
+{ \mathcal { C } } ( x ) : = { \mathcal { C } } _ { 1 } ( x ) + { \mathcal { C } } _ { 2 } ( x - { \mathcal { C } } _ { 1 } ( x ) ) .
+$$
+
+The induced compression operator satisfies ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ with $\delta = \delta _ { 2 } \left( 1 - \AA ^ { 1 } / \delta _ { 1 } \right) + \delta _ { 2 } \delta _ { 1 }$
+
+To get some intuition about this procedure, recall the structure used in Error Feedback. The gradient estimator is first compressed with $\mathcal { C } _ { 1 } ( g )$ and the error $e = g - \mathcal { C } _ { 1 } ( g )$ is stored in memory and used to modify the gradient in the next iteration. In our proposed approach, instead of storing the error $e$ , we compress it with an unbiased compressor $\mathcal { C } _ { 2 }$ (which can be seen as a parameter allowing flexibility in the design of the induced compressor) and communicate both of these compressed vectors. Note that this procedure results in extra variance as we do not work with the exact error, but with its unbiased estimate only. On the other hand, there is no bias and error accumulation that one needs to correct for. In addition, due to our construction, at least the same amount of information is sent to the master as in the case of plain $\mathcal { C } _ { 1 } ( g )$ : indeed, we send both $\mathcal { C } _ { 1 } ( g )$ and $\mathcal { C } _ { 2 } ( e )$ . The drawback of this is the necessity to send more bits. However, Theorem 3 provides the freedom in generating the induced compressor through the choice of the unbiased compressor $\mathcal { C } _ { 2 }$ . In theory, it makes sense to choose $\mathcal { C } _ { 2 }$ with similar compression factor to the compressor $\mathcal { C } _ { 1 }$ we are transforming as this way the total number of communicated bits per iteration is preserved, up to the factor of two.
+
+Remark: The $\mathrm { r t o p } _ { k _ { 1 } , k _ { 2 } } ( x , y )$ operator proposed by Elibol et al. (2020) can be seen as a special case of our induced compressor with $x = y$ , $\mathcal { C } _ { 1 } = \mathrm { T o p } \mathrm { - } k _ { 1 }$ and $\mathcal { C } _ { 2 } = \mathrm { R a n d } \ – k _ { 2 }$ .
+
+Benefits of Induced Compressor. In the light of the results in Section 2, we argue that one should always prefer unbiased compressors to biased ones as long as their variances $\delta$ and communication complexities are the same, e.g., Rand- $K$ over Top- $K$ . In practice, biased/greedy compressors are in some settings observed to perform better due to their lower empirical variance (Beznosikov et al., 2020). These considerations give a practical significance to Theorem 3 as we demonstrate on the following example. Let us consider two compressors: one biased $\mathcal { C } _ { 1 } \in \mathbb { C } ( \delta _ { 1 } )$ and one unbiased $\mathcal { C } _ { 2 } \in \mathbb { U } ( \delta _ { 2 } )$ , such that $\delta _ { 1 } = \delta _ { 2 } = \delta$ , having identical communication complexity, e.g., Top- $K$ and Rand- $K$ . The induced compressor $\mathcal { C } ( \boldsymbol { x } ) \mathbf { \bar { \Psi } } { = } \mathcal { C } _ { 1 } ( \boldsymbol { x } ) + \mathcal { C } _ { 2 } ( \boldsymbol { x } - \mathcal { C } _ { 1 } ( \boldsymbol { x } ) )$ belongs to $\mathbb { U } ( \delta _ { 3 } )$ , where $\begin{array} { r } { \delta _ { 3 } = \delta - \left( 1 - \frac { 1 } { \delta } \right) < \delta } \end{array}$ . While the size of the transmitted message is doubled, one can use Algorithm 1 since $\mathcal { C }$ is unbiased, which provides better convergence guarantees than Algorithm 2. Based on the construction of the induced compressor, one might expect that we need extra memory as “the error” $e = g - \mathcal { C } _ { 1 } ( g )$ needs to be stored, but during computation only. This is not an issue as compressors for DNNs are always applied layer-wise (Dutta et al., 2019), and hence the size of the extra memory is negligible. It does not help EF, as the error needs to be stored at any time for each layer.
+
+# 4 EXTENSIONS
+
+We now develop several extensions of Algorithm 1 relevant to distributed optimization in general, and to Federated Learning in particular. This is all possible due to the simplicity of our approach. Note that in the case of Error Feedback, these extensions have either not been obtained yet, or similarly to Section 2, the results are worse when compared to our derived bounds for unbiased compressors.
+
+Partial Participation with Arbitrary Distribution over Nodes. In this section, we extend our results to a variant of DCSGD utilizing partial participation, which is of key relevance to Federated
+
+![](images/c5ff1577126c65f6c164d420c126b0d25bcef16cd0af4357823a03704f8ff7de.jpg)  
+Figure 1: Comparison of Top-1 $\mathrm { ( + E F ) }$ and NU Rand-1 on Example 1 from Beznosikov et al. (2020).
+
+Learning. In this framework, only a subset of all nodes communicates to the master node in each communication round. Such framework was analyzed before, but only for the case of uniform subsampling (Sattler et al., 2019; Reisizadeh et al., 2020). In our work, we consider a more general partial participation framework: we assume that the subset of participating clients is determined by a fixed but otherwise arbitrary random set-valued mapping $\mathbb { S }$ (a “sampling”) with values in $2 ^ { [ n ] }$ , where $[ n ] = \{ 1 , 2 , \dots , n \}$ . To the best of our knowledge, this is the first partial participation result for $\mathrm { F L }$ where an arbitrary distribution over the nodes is considered. On the other hand, this is not the first work which makes use of the arbitrary sampling paradigm; this was used before in other contexts, e.g., for obtaining importance sampling guarantees for coordinate descent (Qu et al., 2015), primal-dual methods (Chambolle et al., 2018), and variance reduction (Horváth & Richtárik, 2019).
+
+Note that the sampling $\mathbb { S }$ is uniquely defined by assigning probabilities to all $2 ^ { n }$ subsets of $[ n ]$ . With each sampling $\mathbb { S }$ we associate a probability matrix $\mathbf { P } \in \mathbb { R } ^ { n \times n }$ defined by $\mathbf { P } _ { i j } : = \mathrm { P r o b } ( \{ i , j \} \subseteq \mathbb { S } )$ . The probability vector associated with $\mathbb { S }$ is the vector composed of the diagonal entries of $\mathbf { P }$ : $p = ( p _ { 1 } , \ldots , p _ { n } ) \in \mathbb { R } ^ { n }$ , where $p _ { i } : = { \mathrm { P r o b } } ( i \in \mathbb { S } ) .$ . We say that $\mathbb { S }$ is proper if $p _ { i } > 0$ for all $i$ . It is easy to show that $\begin{array} { r } { b : = \operatorname { E } \left[ | \mathbb { S } | \right] = \operatorname { T r a c e } \left( \mathbf { P } \right) = \sum _ { i = 1 } ^ { n } p _ { i } } \end{array}$ , and hence $b$ can be seen as the expected number of clients participating in each communication round.
+
+There are two algorithmic changes due to this extension: line 4 of Algorithm 1 does not iterate over every node, only over nodes $i \in S ^ { k }$ , where $S ^ { k } \sim \mathbb S$ , and the aggregation step in line 9 is adjusted to lead to an unbiased estimator of the gradient, which gives ∆k = Pi∈Sk 1np .
+
+To prove convergence, we exploit the following lemma.
+
+Lemma 4 (Lemma 1, Horváth & Richtárik (2019)). Let be vectors in $\mathbb { R } ^ { d }$ and let $\textstyle { \bar { \zeta } } : = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \zeta _ { i }$ be their average. Let $\mathbb { S }$ 1 2 n be a proper sampling. Then there exists $v \in \mathbb { R } ^ { n }$ such
+
+$$
+\mathbf { P } - p p ^ { \intercal } \preceq \mathbf { D i a g } \left( p _ { 1 } v _ { 1 } , p _ { 2 } v _ { 2 } , \ldots , p _ { n } v _ { n } \right) .
+$$
+
+Moreover, if $S \sim \mathbb { S } ,$ , then
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ \left. \displaystyle \sum _ { i \in S } \frac { \zeta _ { i } } { n p _ { i } } - \bar { \zeta } \right. ^ { 2 } \right] \leq \frac { 1 } { n ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { n } \frac { v _ { i } } { p _ { i } } \| \zeta _ { i } \| ^ { 2 } . } \end{array}
+$$
+
+The following theorem establishes the convergence rate for Algorithm 1 with partial participation.
+
+Theorem 5. Let Assumptions $_ { I - 3 }$ hold and ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ , then there exist stepsizes $\eta ^ { k } \leq \frac { 1 } { 2 \delta _ { \mathbb { S } } L }$ and weights $w ^ { k } \geq 0$ such that
+
+$$
+\begin{array} { r } { \mathrm { E } \left[ f ( \bar { x } ^ { T } ) - f ^ { \star } \right] + \mu \mathrm { E } \left[ \left. x ^ { T } - x ^ { \star } \right. ^ { 2 } \right] \leq 6 4 \delta _ { \mathrm { S } } L r ^ { 0 } \exp \left[ - \frac { \mu T } { 4 \delta _ { \mathrm { S } } L } \right] + 3 6 \frac { ( \delta _ { \mathrm { S } } - 1 ) D + ( 1 + a _ { \mathrm { S } } ) \delta \sigma ^ { 2 } / n } { \mu T } , } \end{array}
+$$
+
+where $\boldsymbol { r } ^ { 0 } , \boldsymbol { W } ^ { T } , \bar { \boldsymbol { x } } ^ { T }$ , and $D$ are defined in Theorem 2, $a _ { \mathbb { S } } = \operatorname* { m a x } _ { i \in [ n ] } \left\{ v _ { i } / p _ { i } \right\}$ , and $\begin{array} { r } { \delta _ { \mathbb { S } } = \frac { \delta a _ { \mathbb { S } } + ( \delta - 1 ) } { n } + 1 . } \end{array}$
+
+For the case $\mathbb { S } = [ n ]$ with probability 1, one can show that Lemma 4 holds with $v = 0$ , and hence we exactly recover the results of Theorem 2. In addition, we can quantify the slowdown factor with respect to full participation regime (Theorem 2), which is δ maxi∈[n] vipi . While in our framework we assume the distribution $\mathbb { S }$ to be fixed, it can be easily extended to several proper distributions $\mathbb { S } _ { j }$ ’s or we can even handle a block-cyclic structure with each block having an arbitrary proper distribution $\mathbb { S } _ { j }$ over the given block $j$ combining our analysis with the results of Eichner et al. (2019).
+
+![](images/e394c9853ec0c2d7ecf90141a28cfacc4fa5b745b7bbbd645685bb1457585b58.jpg)  
+Figure 2: Algorithm 1 vs. Algorithm 2 on CIFAR10 with ResNet18 (bottom), VGG11 (top) and TernGrad as a compression.
+
+Obtaining Linear Convergence. Note that in all the previous theorems, we can only guarantee a sublinear $\bar { \mathcal { O } } ( 1 / T )$ convergence rate. Linear rate is obtained in the special case when $D = 0$ and $\sigma ^ { 2 } = 0$ . The first condition is satisfied, when $f _ { i } ^ { \star } = f _ { i } ( x ^ { \star } )$ for all $\bar { i } \in [ n ]$ , thus when $x ^ { \star }$ is also minimizer of every local function $f _ { i }$ . Furthermore, the effect od $D$ can be removed using compression of gradient differences, as pioneered in the DIANA algorithm (Mishchenko et al., 2019a). Note that $\sigma ^ { 2 } = 0$ if weak growth condition holds (Vaswani et al., 2019). Moreover, one can remove effect of $\sigma ^ { 2 }$ by either computing full gradients locally or by incorporating variance reduction such as SVRG (Johnson & Zhang, 2013). It was shown by Horváth et al. (2019b) that both $\sigma ^ { 2 }$ and $D$ can be removed for the setting of Theorem 2. These results can be easily extended to partial participation using our proof technique for Theorem 5. Note that this reduction is not possible for Error Feedback as the analysis of the DIANA algorithm is heavily dependent on the unbiasedness property. This points to another advantage of the induced compressor framework introduced in Section 3.
+
+Acceleration. We now comment on the combination of compression and acceleration/momentum. This setting is very important to consider as essentially all state-of-the-art methods for training deep learning models, including Adam (Kingma & Ba, 2015; Reddi et al., 2018), rely on the use of momentum in one form or another. One can treat the unbiased compressed gradient as a stochastic gradient (Gorbunov et al., 2020) and the theory for momentum SGD (Yang et al., 2016; Gadat et al., 2018; Loizou & Richtárik, 2017) would be applicable with an extra smoothness assumption. Moreover, it is possible to remove the variance caused by stochasticity and obtain linear convergence with an accelerated rate, which leads to the Accelerated DIANA method (Li et al., 2020). Similarly to our previous discussion, both of these techniques are heavily dependent on the unbiasedness property. It is an intriguing question, but out of the scope of the paper, to investigate the combined effect of momentum and Error Feedback and see whether these techniques are compatible theoretically.
+
+# 5 EXPERIMENTS
+
+In this section, we compare Algorithms 1 and 2 for several compression operators. If the method contains $^ { 6 6 } + \mathrm { E F } ^ { 5 9 }$ , it means that EF is applied, thus Algorithm 2 is applied. Otherwise, Algorithm 1 is displayed. To be fair, we always compare methods with the same communication complexity per iteration. All experimental details can be found in the Appendix.
+
+Failure of DCSGD with biased Top-1. In this experiment, we present example considered in Beznosikov et al. (2020), which was used as a counterexample to show that some form of error correction is needed in order for biased compressors to work/provably converge. In addition, we run experiments on their construction and show that while Error Feedback fixes divergence, it is still significantly dominated by unbiased non-uniform sparsification(NU Rand-1), which works by only keeping one non-zero coordinate sampled with probability equal to $| \boldsymbol { x } | \big / \Sigma _ { i = 1 } ^ { d } | \boldsymbol { x } | _ { i }$ , where $| x |$ denotes element-wise absolute value, as can be seen in Figure 1. The details can be found in the Appendix.
+
+![](images/dcc348fd8751162a43d222712f5cbc55e96b7721e1ee343225c2c7af4e967f07.jpg)  
+Figure 3: Comparison of different sparsification techniques with and without usage of Error Feedback on CIFAR10 with Resnet18 (top) and VGG11 (bottom). $K = 5 \% * d$ , for Induced compressor $\mathcal { C } _ { 1 }$ is Top- $K / 2$ and $\mathcal { C } _ { 2 }$ is Rand- $\kappa / { } _ { 2 }$ (Wangni et al.).
+
+Error Feedback for Unbiased Compression Operators. In our second experiment, we compare the effect of Error Feedback in the case when an unbiased compressor is used. Note that unbiased compressors are theoretically guaranteed to work both with Algorithm 1 and 2. We can see from Figure 2 that adding Error Feedback can hurt the performance; we use TernGrad (Wen et al., 2017) (coincides with QSGD (Alistarh et al., 2016) and natural dithering (Horváth et al., 2019a) with the infinity norm and one level) as compressors. This agrees with our theoretical findings. In addition, for sparsification techniques such as Random Sparsification or Gradient Sparsification (Wangni et al., 2018), we observed that when sparsity is set to be $10 \%$ , Algorithm 1 converges for all the selected values of step-sizes, but Algorithm 2 diverges and a smaller step-size needs to be used. This is an important observation as many practical works (Li et al., 2014; Wei et al., 2015; Aji & Heafield, 2017; Hsieh et al., 2017; Lin et al., 2018b; Lim et al., 2018) use sparsification techniques mentioned in this section, but proposed to use EF, while our work shows that using unbiasedness property leads not only to better convergence but also to memory savings.
+
+Unbiased Alternatives to Biased Compression. In this section, we investigate candidates for unbiased compressors than can compete with Top- $K$ , one of the most frequently used compressors. Theoretically, Top- $K$ is not guaranteed to work by itself and might lead to divergence (Beznosikov et al., 2020) unless Error Feedback is applied. One would usually compare the performance of Top- $K$ with EF to Rand- $K$ , which keeps $K$ randomly selected coordinates and then scales the output by $d / \kappa$ to preserve unbiasedness. Rather than naively comparing to Rand- $K$ , we propose to use more nuanced unbiased approaches. The first one is Gradient Sparsification proposed by Wagni et al. (Wangni et al., 2018), which we refer to here as Rand- $K$ (Wangni et al.), where the probability of keeping each coordinate scales with its magnitude and communication budget. As the second alternative, we propose to use our induced compressor, where $\mathcal { C } _ { 1 }$ is Top- $a$ and unbiased part $\mathcal { C } _ { 2 }$ is Rand- $( K - a )$ (Wangni et al.) with communication budget $K - a$ . It should be noted that $a$ can be considered as a hyperparameter to tune. For our experiment, we chose it to be $K / 2$ for simplicity. Figure 3 suggests that our induced compressor outperforms all of its competitors as can be seen for both VGG11 and Resnet18. Moreover, induced compressor as well as Rand- $K$ do not require extra memory to store the error vector. Finally, Top- $K$ without EF suffers a significant decrease in performance, which stresses the necessity of error correction.
+
+# 6 CONCLUSION
+
+In this paper, we argue that if compressed communication is required for distributed training due to communication overhead, it is better to use unbiased compressors. We show that this leads to strictly better convergence guarantees with fewer assumptions. In addition, we propose a new construction for transforming any compressor into an unbiased one using a compressed EF-like approach. Besides theoretical superiority, usage of unbiased compressors enjoys lower memory requirements. Our theoretical findings are corroborated with empirical evaluation.
+
+As a future work we plan to investigate the question of the appropriate choice of the inducing compressor $\mathcal { C }$ . Our preliminary studies show that there is much to be discovered here, both in theory and in terms of developing further practical guidelines to those already contained in this work. The question of (theoretically) optimizing for $\mathcal { C } _ { 1 }$ and $\mathcal { C } _ { 2 }$ is difficult, as it necessitates a deeper theoretical understanding of biased compressors, which is currently missing. An alternative is to impose some assumptions on the structure of gradients encountered during the iterative process, or to perform an extensive experimental evaluation on desired tasks to provide guidelines for practitioners.
+
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+
+![](images/f8c2a4b922fece8caa6625f7918e93cc3296f93edaebd2b9944325b55af4f9df.jpg)  
+Figure 4: Algorithm 1 vs. Algorithm 2 on CIFAR10 with ResNet18 (bottom), VGG11 (top) and TernGrad as a compression.
+
+# APPENDIX
+
+# A EXPERIMENTAL DETAILS
+
+To be fair, we always compare methods with the same communication complexity per iteration. We report the number of epochs (passes over the dataset) with respect to training loss and testing accuracy. The test accuracy is obtained by evaluating the best model in terms of validation accuracy. A validation accuracy is computed based on $10 \%$ randomly selected training data. We tune the step-size using based on the training loss. For every experiment, we randomly distributed the training dataset among 8 workers; each worker computes its local gradient-based on its own dataset. We used a local batch size of 32. All the provided figures display the mean performance with one standard error over 5 independent runs. For a fair comparison, we use the same random seed for the compared methods. Our experimental results are based on a Python implementation of all the methods running in PyTorch. All reported quantities are independent of the system architecture and network bandwidth.
+
+Dataset and Models. We do an evaluation on CIFAR10 dataset. We consider VGG11 (Simonyan & Zisserman, 2015) and ResNet18 (He et al., 2016) models and step-sizes 0.1, 0.05 and 0.01.
+
+# A.1 EXTRA EXPERIMENTS
+
+Momentum. In this extra experiment, we look at the effect of momentum on Algorithm 1 and 2. We set momentum to 0.9. Similarly to Figure 2, we work with the unbiased compressor, concretely TernGrad (Wen et al., 2017) (coincides with QSGD (Alistarh et al., 2016) and natural dithering (Horváth et al., 2019a) with the infinity norm and one level), to see the effect of adding Error Feedback. We can see from Figure 4 that adding Error Feedback can hurt the performance, which agrees with our theoretical findings.
+
+# B EXAMPLE 1, BEZNOSIKOV ET AL. (2020)
+
+In this section, we present example considered in Beznosikov et al. (2020), which was used as a counterexample to show that some form of error correction is needed in order for biased compressors to work/provably converge. In addition, we run experiments on their construction and show that while Error Feedback fixes divergence, it is still significantly dominated by unbiased non-uniform sparsification as can be seen in Figure 1. The construction follows.
+
+Consider $n = d = 3$ and define the following smooth and strongly convex quadratic functions
+
+$$
+f _ { 1 } ( x ) = \left. a , x \right. ^ { 2 } + { \frac { 1 } { 4 } } \left\| x \right\| ^ { 2 } , \qquad f _ { 2 } ( x ) = \left. b , x \right. ^ { 2 } + { \frac { 1 } { 4 } } \left\| x \right\| ^ { 2 } , \qquad f _ { 3 } ( x ) = \left. c , x \right. ^ { 2 } + { \frac { 1 } { 4 } } \left\| x \right\| ^ { 2 } ,
+$$
+
+where $a = ( - 3 , 2 , 2 ) , b = ( 2 , - 3 , 2 ) , c = ( 2 , 2 , - 3 )$ . Then, with the initial point $x ^ { 0 } = ( t , t , t ) , t > 0$
+
+$$
+\nabla f _ { 1 } ( x ^ { 0 } ) = \frac { t } { 2 } ( - 1 1 , 9 , 9 ) , \qquad \nabla f _ { 2 } ( x ^ { 0 } ) = \frac { t } { 2 } ( 9 , - 1 1 , 9 ) , \qquad \nabla f _ { 3 } ( x ^ { 0 } ) = \frac { t } { 2 } ( 9 , 9 , - 1 1 ) .
+$$
+
+Using the Top-1 compressor, we get
+
+$$
+\begin{array} { r } { \mathcal { C } ( \nabla f _ { 1 } ( x ^ { 0 } ) ) = \frac { t } { 2 } ( - 1 1 , 0 , 0 ) , \quad \mathcal { C } ( \nabla f _ { 2 } ( x ^ { 0 } ) ) = \frac { t } { 2 } ( 0 , - 1 1 , 0 ) , \quad \mathcal { C } ( \nabla f _ { 3 } ( x ^ { 0 } ) ) = \frac { t } { 2 } ( 0 , 0 , - 1 1 ) . } \end{array}
+$$
+
+The next iterate of DCGD is
+
+$$
+x ^ { 1 } = x ^ { 0 } - \frac { \eta } { 3 } \sum _ { i = 1 } ^ { 3 } \mathcal { C } ( \nabla f _ { i } ( x ^ { 0 } ) ) = \left( 1 + \frac { 1 1 \eta } { 6 } \right) x ^ { 0 } .
+$$
+
+Repeated application gives $\begin{array} { r } { x ^ { k } = \left( 1 + \frac { 1 1 \eta } { 6 } \right) ^ { k } x ^ { 0 } } \end{array}$ , which diverges exponentially fast to $+ \infty$ since $\eta > 0$ .
+
+As a initial point, we use $( 1 , 1 , 1 ) ^ { \top }$ in our experiments and we choose step size $\textstyle { \frac { 1 } { L } }$ , where $L$ is smoothness parameter of $\begin{array} { r } { f = \frac { 1 } { 3 } ( f _ { 1 } + f _ { 2 } + f _ { 3 } ) } \end{array}$ . Note that zero vector is the unique minimizer of $f$ .
+
+# C PROOFS
+
+# C.1 PROOF OF LEMMA 1
+
+We follow (2), which holds for ${ \mathcal { C } } \in \mathbb { U } ( \delta )$ .
+
+$$
+\begin{array} { r l l } { \displaystyle \operatorname { E } \left[ \left\| \frac { 1 } { \delta } \mathcal { C } ^ { k } ( \boldsymbol { x } ) - \boldsymbol { x } \right\| ^ { 2 } \right] } & { = } & { \displaystyle \frac { 1 } { \delta ^ { 2 } } \mathrm { E } \left[ \left\| \mathcal { C } ^ { k } ( \boldsymbol { x } ) \right\| ^ { 2 } \right] - 2 \frac { 1 } { \delta } \left. \mathrm { E } \left[ \mathcal { C } ^ { k } ( \boldsymbol { x } ) \right] , \boldsymbol { x } \right. + \left\| \boldsymbol { x } \right\| ^ { 2 } } \\ & { \leq } & { \displaystyle \left( \frac { 1 } { \delta } - \frac { 2 } { \delta } + 1 \right) \left\| \boldsymbol { x } \right\| ^ { 2 } } \\ & { = } & { \displaystyle \left( 1 - \frac { 1 } { \delta } \right) \left\| \boldsymbol { x } \right\| ^ { 2 } , } \end{array}
+$$
+
+which concludes the proof.
+
+# C.2 PROOF OF THEOREM 2
+
+We use the update of Algorithm 1 to bound the following quantity
+
+$$
+\begin{array} { r l r l } { \mathbb { E }  | | \mathbf { E } ^ { \mu _ { 1 } } - \mathbf { x } ^ { \mu _ { 1 } } | ^ { 2 }  } & { = } & & { \| \mathbf { e } - \mathbf { a } ^ { \nu _ { 1 } } \| ^ { 2 } - \frac { \alpha _ { 2 } ^ { 2 } \nu _ { 1 } } { 4 } \sum _ { k = 1 } ^ { \nu _ { 1 } } \big | \mathcal { E } ( \mathbf { a } ^ { 2 } ( k ) ^ { 2 } , k ^ { 2 } - \frac { \nu _ { 1 } ^ { 2 } } { 4 } ) ^ { 2 } \| ^ { 2 } } \\ & { } & & {  \mathbf { b } ^ { 2 } ( k ) ^ { 2 } , k ^ { 2 }  } \\ & { } &  \underset { \mathrm { i f } } { \sum _ { k = 1 } ^ { \nu _ { 1 } } \Big \{ \sum _ { k = 1 } ^ { \nu _ { 2 } } \Big \| \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } + \mathbf { c } ^ { 2 } \Big \} ^ { 2 } \Big | \mathbf { b } ^ { 2 } \Big \} ^ { 2 } } \\ & { } & & { \| \mathbf { e } ^ { \mu _ { 2 } } + \mathbf { c } ^ { 2 } \Big \| \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } + \mathbf { c } ^ { 2 } \Big \} ^ { 2 } \Big | \mathbf { b } ^ { 2 } \Big \} ^ { 2 } } \\ & { } & & { \| \mathbf { e } ^ { \mu _ { 3 } } + \mathbf { b } ^ { 2 } \Big \| \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } + \mathbf { c } ^ { 2 } \Big \} ^ { 2 } \Big | \mathbf { b } ^ { 2 } \Big \} ^ { 2 } } \\ & { } & & { \| \mathbf { e } ^ { \mu _ { 3 } } + \mathbf { b } ^ { 2 } \Big \| \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } + \mathbf { c } ^ { 2 } \Big \} ^ { 2 } \Big | \mathbf { b } ^ { 2 } \Big \{ \mathbf { b } ^ { 2 } \Big \} ^ { 2 } } \\ & { } & &  \| \mathbf { e } ^ { \mu _ { 3 } } + \mathbf { b } ^ { 2 } \| \mathbf { b } ^ { 2 } \Big  \end{array}
+$$
+
+Taking full expectation and $\begin{array} { r } { \eta ^ { k } \leq \frac { 1 } { 2 \delta _ { n } L } } \end{array}$ , we obtain
+
+$$
+\begin{array} { r } { \tilde { \mathsf { L } } \left[ \left\| x ^ { k + 1 } - x ^ { \star } \right\| ^ { 2 } \right] \le ( 1 - \mu \eta ^ { k } ) \mathrm { E } \left[ \left\| x ^ { k } - x ^ { \star } \right\| ^ { 2 } \right] - \eta ^ { k } \mathrm { E } \left[ f ( x ^ { k } ) - f ^ { \star } \right] + ( \eta ^ { k } ) ^ { 2 } \left( ( \delta _ { n } - 1 ) D + \frac { \delta \sigma ^ { 2 } } { n } \right) } \end{array}
+$$
+
+The rest of the analysis is closely related to the one of Stich (2019b). We would like to point out that similar results to Stich (2019b) were also present in (Lacoste-Julien et al., 2012; Stich et al., 2018; Grimmer, 2019).
+
+We first rewrite the previous inequality to the form
+
+$$
+r ^ { k + 1 } \leq ( 1 - a \eta ^ { k } ) r ^ { k } - \eta ^ { k } s ^ { k } + ( \eta ^ { k } ) ^ { 2 } c ,
+$$
+
+$$
+\begin{array} { r } { r ^ { k } = \operatorname { E } \Big [ \big \| x ^ { k } - x ^ { \star } \big \| ^ { 2 } \Big ] , s ^ { k } = \operatorname { E } \big [ f ( x ^ { k } ) - f ^ { \star } \big ] , a = \mu , c = ( \delta _ { n } - 1 ) D + \frac { \delta \sigma ^ { 2 } } { n } . } \end{array}
+$$
+
+We proceed with lemmas that establish a convergence guarantee for every recursion of type (10).
+
+Lemma 6. Let $\{ r ^ { k } \} _ { k \geq 0 } , \{ s ^ { k } \} _ { k \geq 0 }$ be as in (10) for $a > 0$ and for constant stepsizes $\begin{array} { r } { \eta ^ { k } \equiv \eta : = \frac { 1 } { d } } \end{array}$ $\forall k \geq 0$ . Then it holds for all $T \geq 0$ :
+
+$$
+r ^ { T } \leq r ^ { 0 } \exp \left[ - \frac { a T } { d } \right] + \frac { c } { a d } .
+$$
+
+Proof. This follows by relaxing (10) using $\operatorname { E } \left[ f ( x ^ { k } ) - f ^ { \star } \right] \geq 0$ ,and unrolling the recursion
+
+$$
+r ^ { T } \leq ( 1 - a \eta ) r ^ { T - 1 } + c \gamma ^ { 2 } \leq ( 1 - a \eta ) ^ { T } r ^ { 0 } + c \eta ^ { 2 } \sum _ { k = 0 } ^ { T - 1 } ( 1 - a \eta ) ^ { k } \leq ( 1 - a \eta ) ^ { T } r ^ { 0 } + \frac { c \eta } { a } .
+$$
+
+Lemma 7. Let $\{ r ^ { k } \} _ { k \geq 0 }$ , $\{ s ^ { k } \} _ { k \geq 0 }$ as in (10) for $a > 0$ and for decreasing stepsizes $\begin{array} { r } { \eta ^ { k } : = \frac { 2 } { a ( \kappa + k ) } } \end{array}$ , $\forall k \geq 0$ , with parameter $\textstyle \kappa : = { \frac { 2 d } { a } }$ , and weights $w ^ { k } : = ( \kappa + k )$ . Then
+
+$$
+\frac { 1 } { W ^ { T } } \sum _ { k = 0 } ^ { T } s ^ { k } w ^ { k } + a r ^ { T + 1 } \leq \frac { 2 a \kappa ^ { 2 } r _ { 0 } } { T ^ { 2 } } + \frac { 2 c } { a T } ,
+$$
+
+where $\begin{array} { r } { W ^ { T } : = \sum _ { k = 0 } ^ { T } w ^ { k } } \end{array}$
+
+Proof. We start by re-arranging (10) and multiplying both sides with $w ^ { k }$
+
+$$
+\begin{array} { l } { { s ^ { k } w ^ { k } \leq \displaystyle \frac { w ^ { k } ( 1 - a \eta ^ { k } ) r ^ { k } } { \eta ^ { k } } - \displaystyle \frac { w ^ { k } r ^ { k + 1 } } { \eta ^ { k } } + c \eta ^ { k } w ^ { k } } } \\ { { \quad \quad = a ( \kappa + k ) ( \kappa + k - 2 ) r ^ { k } - a ( \kappa + k ) ^ { 2 } r ^ { k + 1 } + \displaystyle \frac { c } { a } } } \\ { { \quad \quad \leq a ( \kappa + k - 1 ) ^ { 2 } r ^ { k } - a ( \kappa + k ) ^ { 2 } r ^ { k + 1 } + \displaystyle \frac { c } { a } , } } \end{array}
+$$
+
+where the equality follows from the definition of $\eta ^ { k }$ and $w ^ { k }$ and the inequality from $( \kappa + k ) ( \kappa + k -$ $2 ) = ( \kappa + \dot { k } - 1 \dot { ) } ^ { 2 } - 1 \le ( \kappa + k - 1 ) ^ { 2 }$ . Again we have a telescoping sum:
+
+$$
+\frac { 1 } { W ^ { T } } \sum _ { k = 0 } ^ { T } s ^ { k } w ^ { k } + \frac { a ( \kappa + T ) ^ { 2 } r ^ { T + 1 } } { W ^ { T } } \leq \frac { a \kappa ^ { 2 } r ^ { 0 } } { W ^ { T } } + \frac { c ( T + 1 ) } { a W ^ { T } } ,
+$$
+
+with
+
+$$
+\begin{array} { r } { W ^ { T } = \sum _ { k = 0 } ^ { T } w ^ { k } = \sum _ { k = 0 } ^ { T } ( \kappa + k ) = \frac { ( 2 \kappa + T ) ( T + 1 ) } { 2 } \geq \frac { T ( T + 1 ) } { 2 } \geq \frac { T ^ { 2 } } { 2 } , } \end{array}
+$$
+
+$$
+\begin{array} { r } { W ^ { T } = \frac { ( 2 \kappa + T ) ( T + 1 ) } { 2 } \leq \frac { 2 ( \kappa + T ) ( 1 + T ) } { 2 } \leq ( \kappa + T ) ^ { 2 } \mathrm { f o r } \kappa = \frac { 2 d } { a } \geq 1 . } \end{array}
+$$
+
+By applying these two estimates we conclude the proof.
+
+The convergence can be obtained as the combination of these two lemmas.
+
+Lemma 8. Let $\{ r ^ { k } \} _ { k \geq 0 }$ , $\{ s ^ { k } \} _ { k \geq 0 }$ as in (10), $a > 0$ . Then there exists stepsizes $\begin{array} { r } { \eta ^ { k } \leq \frac { 1 } { d } } \end{array}$ and weighs $\textstyle w ^ { k } \geq 0 , W ^ { T } : = \sum _ { k = 0 } ^ { T } w ^ { k }$ , such that
+
+$$
+\frac 1 { W ^ { T } } \sum _ { k = 0 } ^ { T } s ^ { k } w ^ { k } + a r ^ { T + 1 } \leq 3 2 d r _ { 0 } \exp \left[ - \frac { a T } { 2 d } \right] + \frac { 3 6 c } { a T } .
+$$
+
+Proof of Lemma 8. For integer $T \geq 0$ , we choose stepsizes and weights as follows
+
+$$
+\begin{array} { l l l } { { \displaystyle { \mathrm { i f } T \le \frac { d } { a } } , } } & { { \eta ^ { k } = \frac { 1 } { d } , } } & { { w ^ { k } = ( 1 - a \eta ^ { k } ) ^ { - ( k + 1 ) } = \left( 1 - \frac { a } { d } \right) ^ { - ( k + 1 ) } , } } \\ { { \displaystyle { \mathrm { i f } T > \frac { d } { a } } { \mathrm { a n d } k < t _ { 0 } } , } } & { { \eta ^ { k } = \frac { 1 } { d } , } } & { { w ^ { k } = 0 , } } \\ { { \displaystyle { \mathrm { i f } T > \frac { d } { a } } { \mathrm { a n d } k \ge t _ { 0 } } , } } & { { \eta ^ { k } = \frac { 2 } { a ( \kappa + k - t _ { 0 } ) } , } } & { { w ^ { k } = ( \kappa + k - t _ { 0 } ) ^ { 2 } , } } \end{array}
+$$
+
+for $\begin{array} { r } { \kappa = \frac { 2 d } { a } } \end{array}$ and $\begin{array} { r } { t _ { 0 } = \left\lceil \frac { T } { 2 } \right\rceil } \end{array}$ . We will now show that these choices imply the claimed result.
+
+We start with the case $\textstyle T \leq { \frac { d } { a } }$ . For this case, the choice $\begin{array} { r } { \eta = \frac { 1 } { d } } \end{array}$ gives
+
+$$
+\begin{array} { l } { \displaystyle \frac { 1 } { W ^ { T } } \sum _ { k = 0 } ^ { T } s ^ { k } w ^ { k } + a r ^ { T + 1 } \leq ( 1 - a \eta ) ^ { ( T + 1 ) } \displaystyle \frac { r _ { 0 } } { \eta } + c \eta } \\ { \leq \displaystyle \frac { r _ { 0 } } { \eta } \exp \left[ - a \eta ( T + 1 ) \right] + c \eta } \\ { \leq d r _ { 0 } \exp \left[ - \displaystyle \frac { a T } { d } \right] + \displaystyle \frac { c } { a T } . } \end{array}
+$$
+
+If $\textstyle T > { \frac { d } { a } }$ , then we obtain from Lemma 6 that
+
+$$
+r ^ { t _ { 0 } } \leq r ^ { 0 } \exp \left[ - \frac { a T } { 2 d } \right] + \frac { c } { a d } .
+$$
+
+From Lemma 7 we have for the second half of the iterates:
+
+$$
+\frac { 1 } { W ^ { T } } \sum _ { k = 0 } ^ { T } s ^ { k } w ^ { k } + a r ^ { T + 1 } = \frac { 1 } { W ^ { T } } \sum _ { k = t _ { 0 } } ^ { T } s ^ { k } w ^ { k } + a r ^ { T + 1 } \leq \frac { 8 a \kappa ^ { 2 } r ^ { t _ { 0 } } } { T ^ { 2 } } + \frac { 4 c } { a T } .
+$$
+
+Now we observe that the restart condition $r ^ { t _ { 0 } }$ satisfies:
+
+$$
+\frac { a \kappa ^ { 2 } r ^ { t _ { 0 } } } { T ^ { 2 } } = \frac { a \kappa ^ { 2 } r ^ { 0 } \exp \left( - \frac { a T } { 2 d } \right) } { T ^ { 2 } } + \frac { \kappa ^ { 2 } c } { d T ^ { 2 } } \leq 4 a r ^ { 0 } \exp \left[ - \frac { a T } { 2 d } \right] + \frac { 4 c } { a T } ,
+$$
+
+because $\textstyle T > { \frac { d } { a } }$ . These conclude the proof.
+
+Having these general convergence lemmas for the recursion of the form (10), the proof of the theorem follows directly from Lemmas 6 and 8 with $a = \mu$ , $c = \sigma ^ { 2 }$ , $d = 2 \delta _ { n } L$ . It is easy to check that condition $\begin{array} { r } { \eta ^ { k } \leq \frac { 1 } { d } = \frac { 1 } { 2 \delta _ { n } L } } \end{array}$ is satisfied.
+
+# C.3 PROOF OF THEOREM 3
+
+We have to show that our new compression is unbiased and has bounded variance. We start with the first property with $\lambda = 1$ .
+
+$$
+\begin{array} { r l } & { \mathrm { E } \left[ \mathcal { C } _ { 1 } ( x ) + \mathcal { C } _ { 2 } ( x - \mathcal { C } _ { 1 } ( x ) ) \right] = \mathrm { E } _ { \mathcal { C } _ { 1 } } \left[ \mathrm { E } _ { \mathcal { C } _ { 2 } } \left[ \mathcal { C } _ { 1 } ( x ) + \mathcal { C } _ { 2 } ( x - \mathcal { C } _ { 1 } ( x ) ) | \mathcal { C } _ { 1 } ( x ) \right] \right] } \\ & { \qquad = \mathrm { E } _ { \mathcal { C } _ { 1 } } \left[ \mathcal { C } _ { 1 } ( x ) + x - \mathcal { C } _ { 1 } ( x ) \right] = x , } \end{array}
+$$
+
+where the first equality follows from tower property and the second from unbiasedness of $\mathcal { C } _ { 2 }$ . For the second property, we also use tower property
+
+$$
+\begin{array} { r l } & { \displaystyle \operatorname { E } \Big [ \| { \mathcal C } _ { 1 } ( x ) - x + { \mathcal C } _ { 2 } ( x - { \mathcal C } _ { 1 } ( x ) ) \| ^ { 2 } \Big ] = \mathrm { E } _ { { \mathcal C } _ { 1 } } \left[ \mathrm { E } _ { { \mathcal C } _ { 2 } } \left[ \| { \mathcal C } _ { 1 } ( x ) - x + { \mathcal C } _ { 2 } ( x - { \mathcal C } _ { 1 } ( x ) ) \| ^ { 2 } \left| { \mathcal C } _ { 1 } ( x ) \right. \right] \right] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \leq \left( \delta _ { 2 } - 1 \right) \mathrm { E } _ { { \mathcal C } _ { 1 } } \left[ \| { \mathcal C } _ { 1 } ( x ) - x \| ^ { 2 } \right] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \leq \left( \delta _ { 2 } - 1 \right) \left( 1 - \displaystyle \frac { 1 } { \delta _ { 1 } } \right) \left\| x \right\| ^ { 2 } , } \end{array}
+$$
+
+where the first and second inequalities follow directly from (2) and (3).
+
+# C.4 PROOF OF LEMMA 4 (HORVÁTH & RICHTÁRIK, 2019)
+
+For the first part of the claim, it was shown that $\mathbf { P } - p p ^ { \top }$ is positive semidefinite (Richtárik & Takác,ˇ 2016), thus we can bound $\mathbf { P } - p p ^ { \top } \preceq n \mathbf { D i a g } \left( \mathbf { P } \bar { \mathbf { \Lambda } } - p p ^ { \top } \right) \widehat { \mathbf { \Lambda } } = \mathbf { D i a g } \left( p \circ v \right) .$ , where $v _ { i } = n ( 1 - p _ { i } )$ , which implies that (8) holds for this choice of $v$ .
+
+For the second part of the claim, let $1 _ { i \in \mathbb { S } } = 1$ if $i \in \mathbb { S }$ and $1 _ { i \in \mathbb { S } } = 0$ otherwise. Likewise, let $1 _ { i , j \in \mathbb { S } } = 1$ if $i , j \in \mathbb { S }$ and $1 _ { i , j \in \mathbb { S } } = 0$ otherwise. Note that $\operatorname { E } \left[ 1 _ { i \in \mathbb { S } } \right] = p _ { i }$ and $\mathrm { E } \left[ 1 _ { i , j \in \mathbb { S } } \right] = p _ { i j }$ . Next, let us compute the mean of X := Pi∈S ζinpi :
+
+$$
+\operatorname { E } \left[ X \right] = \operatorname { E } \left[ \sum _ { i \in \mathbb { S } } \frac { \zeta _ { i } } { n p _ { i } } \right] = \operatorname { E } \left[ \sum _ { i = 1 } ^ { n } \frac { \zeta _ { i } } { n p _ { i } } 1 _ { i \in \mathbb { S } } \right] = \sum _ { i = 1 } ^ { n } \frac { \zeta _ { i } } { n p _ { i } } \operatorname { E } \left[ 1 _ { i \in \mathbb { S } } \right] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \zeta _ { i } = \bar { \zeta } .
+$$
+
+Let $\mathbf { A } = [ a _ { 1 } , \ldots , a _ { n } ] \in \mathbb { R } ^ { d \times n }$ , where $\begin{array} { r } { a _ { i } = \frac { \zeta _ { i } } { p _ { i } } } \end{array}$ , and let $e$ be the vector of all ones in $\mathbb { R } ^ { n }$ . We now write the variance of $X$ in a form which will be convenient to establish a bound:
+
+$$
+\begin{array} { r l } { \mathbb { E } \left[ \left\| X - \mathbf { E } [ X ] \right\| ^ { 2 } \right] } & { = \phantom { - } \mathbb { E } \Big [ \big \| X \big \| ^ { 2 } \Big ] - \big \| \mathbf { E } [ X ] \big \| ^ { 2 } } \\ & { = \phantom { - } \mathbb { E } \left[ \left\| \sum _ { i \in \overline { \partial } } \underline { { \tilde { S } } } _ { p _ { i } } \right\| ^ { 2 } \right] ^ { 2 } - \big \| \tilde { S } \big \| ^ { 2 } } \\ & { = \phantom { - } \mathbb { E } \left[ \sum _ { i \in \overline { \partial } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } ^ { \top } } { \alpha p _ { i } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } _ { \overline { { S } } } _ { \overline { { S } } } } } { \alpha p _ { i } } \mathrm { i } _ { i , i \in \overline { \partial } } \mathbb { E } \right] - \big \| \tilde { S } \big \| ^ { 2 } } \\ & { = \phantom { - } \sum _ { i \in \overline { \partial } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } ^ { \top } } { \alpha p _ { i } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } } { \alpha p _ { i } } - \sum _ { i \in \overline { \partial } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } ^ { \top } } { \alpha } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } } { \alpha } } \\ & { = \phantom { - } \frac { 1 } { \alpha ^ { 2 } } \sum _ { i , j } \Big ( p _ { i } - p _ { i } p _ { j } \Big ) \alpha ^ { - } \alpha ^ { - } } \\ &  = \phantom { - } \frac { 1 } { \alpha ^ { 2 } } \frac { \tilde { \mathcal { G } } _ { \overline { { S } } } ^ { \top } } { 2 \tilde { \mathcal { G } } ^ { \top } } \big ( \mathbf { ( \mathrm { P } - p _ { i } \mathbf { \tilde { \mathcal { G } } } ) ^ { 2 } \setminus \partial ^ { \top } \Delta ^ { \top } \big ) \epsilon . } \end{array}
+$$
+
+Since by assumption we have $\mathbf { P } - p p ^ { \top } \preceq \mathbf { D i a g } \left( p \circ v \right)$ , we can further bound
+
+$$
+e ^ { \top } \left( \left( \mathbf { P } - p p ^ { \top } \right) \circ \mathbf { A } ^ { \top } \mathbf { A } \right) e \leq e ^ { \top } \left( \mathbf { D i a g } \left( p \circ v \right) \circ \mathbf { A } ^ { \top } \mathbf { A } \right) e = \sum _ { i = 1 } ^ { n } p _ { i } v _ { i } \| a _ { i } \| ^ { 2 } .
+$$
+
+To obtain (9), it remains to combine this with (13).
+
+# C.5 PROOF OF THEOREM 5
+
+Similarly to the proof of Theorem 2, we use the update of Algorithm 1 to bound the following quantity
+
+$$
+\begin{array} { r l } { \Phi ^ { k } - x ^ { k } \Big \Vert ^ { 2 } | x ^ { k } \Big \Vert } & { = - \left. x ^ { k } - x ^ { k } \right. ^ { 2 } - x ^ { k } \displaystyle { \sum _ { i = 1 } ^ { n } \mathbb { E } _ { x } \left( \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { n } \mathbb { E } _ { i } ^ { \lambda } ( x _ { i } ^ { k } ) x ^ { k } - x ^ { k } \right) x ^ { k } } \right. + } \\ & { \qquad \mathbb { E } \left[ \left. \displaystyle \sum _ { i = 1 } ^ { n } \displaystyle \sum _ { j = 1 } ^ { n } \mathbb { E } _ { i } ^ { \lambda } ( x _ { i } ^ { k } ) \right. ^ { 2 } | x ^ { k } \right] } \\ & { \overset { \mathrm { o r s i o j . } } { \geq } \quad \left. \mu ^ { k } - x ^ { k } \right. ^ { 2 } - \mu ^ { k } \langle x ^ { k } ( x ^ { k } ) x ^ { k } - x ^ { k } \rangle - } \\ & { \qquad \Big ( \Phi ^ { k } \rangle ^ { 2 } \left( \mathbb { E } \left[ \left. \displaystyle \sum _ { i = 1 } ^ { n } \displaystyle \sum _ { j = 1 } ^ { n } \mu _ { i } ^ { k } ( x _ { i } ^ { k } ) - \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \alpha _ { i } ^ { k } \Phi ^ { j } \right. \right] ^ { 2 } | x ^ { k } \right) + \mathbb { E } \left[ \left. \displaystyle \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \zeta ^ { j } \right] \right.} \\ & { \overset { \mathrm { o r s i o j . } } { \geq } \quad \left. \mu ^ { k } - x ^ { j } \right. ^ { 2 } - \mu ^ { k } \langle \nabla f ( x ^ { k } ) , x ^ { k } - x ^ { k } \rangle - } \\ & { \qquad \quad \leq \mu ^ { k } \leq x ^ { k } \leq \mu ^ { k } \left. \displaystyle \sum _ { i = 1 } ^ { n } \left. \displaystyle \sum _ { j = 1 } ^ { n } \left( \displaystyle \sum _ { i = 1 } ^ { n } \mu _ { i } ^ { k } ( x ^ { k } ) - \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi \right. ^ { 2 } | x ^ { k } \right. \right. } \\ &  \qquad \quad \left. \mu ^ { k } - x ^ { k } \right.  x ^ { k } \Vert \leq \mu ^ { k } \left. \displaystyle \sum _  i \end{array}
+$$
+
+$$
+\begin{array} { r l } { \overset { ( 4 ) + ( 6 ) + ( 7 ) } { \leq } } & { \left( 1 - \mu \eta ^ { k } \right) \left. x ^ { k } - x ^ { \star } \right. ^ { 2 } - 2 \eta ^ { k } \left( 1 - \eta ^ { k } \delta _ { \mathtt { S } } L \right) \left( f ( x ^ { k } ) - f ^ { \star } \right) + } \\ & { \quad ( \eta ^ { k } ) ^ { 2 } \left( ( \delta _ { \mathtt { S } } - 1 ) D + ( 1 + a _ { \mathtt { S } } ) \frac { \delta \sigma ^ { 2 } } { n } \right) . } \end{array}
+$$
+
+Taking full expectation and $\begin{array} { r } { \eta ^ { k } \leq \frac { 1 } { 2 \delta _ { \mathbb { S } } L } } \end{array}$ , we obtain
+
+E h

parse/train/y4-e1K23GLC/y4-e1K23GLC.md

@@ -0,0 +1,508 @@
+# A LAW OF ROBUSTNESS FOR TWO-LAYERS NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We initiate the study of the inherent tradeoffs between the size of a neural network and its robustness, as measured by its Lipschitz constant. We make a precise conjecture that, for any Lipschitz activation function and for most datasets, any two-layers neural network with $k$ neurons that perfectly fit the data must have its Lipschitz constant larger (up to a constant) than $\sqrt { n / k }$ where $n$ is the number of datapoints. In particular, this conjecture implies that overparametrization is necessary for robustness, since it means that one needs roughly one neuron per datapoint to ensure a $O ( 1 )$ -Lipschitz network, while mere data fitting of $d$ -dimensional data requires only one neuron per $d$ datapoints. We prove a weaker version of this conjecture when the Lipschitz constant is replaced by an upper bound on it based on the spectral norm of the weight matrix. We also prove the conjecture in the highdimensional regime $n \approx d$ (which we also refer to as the undercomplete case, since only $k \leq d$ is relevant here). Finally we prove the conjecture for polynomial activation functions of degree $p$ when $n \approx d ^ { p }$ . We complement these findings with experimental evidence supporting the conjecture.
+
+# 1 INTRODUCTION
+
+We study two-layers neural networks with inputs in $\mathbb { R } ^ { d }$ , $k$ neurons, and Lipschitz non-linearity $\psi : \mathbb { R }  \mathbb { R }$ . These are functions of the form:
+
+$$
+x \mapsto \sum _ { \ell = 1 } ^ { k } a _ { \ell } \psi ( w _ { \ell } \cdot x + b _ { \ell } ) ,
+$$
+
+with $a _ { \ell } , b _ { \ell } \in \mathbb { R }$ and $w _ { \ell } \in \mathbb { R } ^ { d }$ for any $\ell \in \ [ k ]$ . We denote by $\mathcal { F } _ { k } ( \psi )$ the set of functions of the form (1). When $k$ is large enough and $\psi$ is non-polynomial, this set of functions can be used to fit any given data set (Cybenko, 1989; Leshno et al., 1993). That is, given a data set $( x _ { i } , y _ { i } ) _ { i \in [ n ] } \ \in$ $( \mathbb { R } ^ { d } \times \mathbb { R } ) ^ { n }$ , one can find $f \in \mathcal { F } _ { k } ( \psi )$ such that
+
+$$
+f ( x _ { i } ) = y _ { i } , \forall i \in [ n ] .
+$$
+
+In a variety of scenarios one is furthermore interested in fitting the data smoothly. For example, in machine learning, the data fitting model $f$ is used to make predictions at unseen points $x \not \in$ $\{ x _ { 1 } , \ldots , x _ { n } \}$ . It is reasonable to ask for these predictions to be stable, that is a small perturbation of $x$ should result in a small perturbation of $f ( x )$ .
+
+A natural question is: how “costly” is this stability restriction compared to mere data fitting? In practice it seems much harder to find robust models for large scale problems, as first evidenced in the seminal paper (Szegedy et al., 2013). In theory the “cost” of finding robust models has been investigated from a computational complexity perspective in (Bubeck et al., 2019), from a statistical perspective in (Schmidt et al., 2018), and more generally from a model complexity perspective in (Degwekar et al., 2019; Raghunathan et al., 2019; Allen-Zhu and Li, 2020). We propose here a different angle of study within the broad model complexity perspective: does a model have to be larger for it to be robust? Empirical evidence (e.g., (Goodfellow et al., 2015; Madry et al., 2018)) suggests that bigger models (also known as “overparametrization”) do indeed help for robustness.
+
+Our main contribution is a conjecture (Conjecture 1 and Conjecture 2) on the precise tradeoffs between size of the model (i.e., the number of neurons $k$ ) and robustness (i.e., the Lipschitz constant of the data fitting model $f \in { \mathcal { F } } _ { k } ( \psi ) .$ ) for generic data sets. We say that a data set $( x _ { i } , y _ { i } ) _ { i \in [ n ] }$ is generic if it is i.i.d. with $x _ { i }$ uniform (or approximately so, see below) on the sphere $\mathbb { S } ^ { d - 1 } = \{ x \in$ $\mathbf { \mathbb { R } } ^ { d } : \| x \| = 1 \}$ and $y _ { i }$ uniform on $\{ - 1 , + 1 \}$ . We give the precise conjecture in Section 2. We prove several weaker versions of Conjecture 1 and Conjecture 2 respectively in Section 4 and Section 3. We also give empirical evidence for the conjecture in Section 5.
+
+![](images/1a26e7bba59481e5597a361dee2bb6ca3c86075009230029ab1f29e1624b8ef6.jpg)  
+Figure 1: See Section 5 for the details of this experiment.
+
+A corollary of our conjecture. A key fact about generic data, established in Baum (1988); Yun et al. (2019); Bubeck et al. (2020), is that one can memorize arbitrary labels with $k \approx n / d$ , that is merely one neuron per $d$ datapoints. Our conjecture implies that for such optimal-size neural networks it is impossible to be robust, in the sense that the Lipschitz constant must be of order $\sqrt { d }$ . The conjecture also states that to be robust (i.e. attain Lipschitz constant $O ( 1 ) _ { . }$ ) one must necessarily have $k \approx n$ , that is roughly each datapoint must have its own neuron. Therefore, we obtain a trade off between size and robustness, namely to make the network robust it needs to be $d$ times larger than for mere data fitting. We illustrate these two cases in Figure 1. We train a neural network to fit generic data, and plot the maximum gradient over several randomly drawn points (a proxy for the√ Lipschitz constant) for various values of $\sqrt { d }$ , when either $k = n$ (blue dots) or $\textstyle k = { \frac { 1 0 n } { d } }$ (red dots). As predicted, for the large neural network $k = n$ ) the Lipschitz constant remains roughly constant,√ while for the optimally-sized one $\textstyle ( k = { \frac { 1 0 n } { d } }$ ) the Lipschitz constant increases roughly linearly in $\sqrt { d }$ .
+
+Notation. For Ω ⊂ Rd we define LipΩ(f) = supx6=x0∈Ω (if $\Omega = \mathbb { R } ^ { d }$ we omit the subscript and write $\operatorname { L i p } ( f ) )$ , where $\| \cdot \|$ denotes the Euclidean norm. For matrices we use $\parallel \cdot$ $\| \boldsymbol { \mathrm { o p } } , \| \cdot \mathsf { \bar { \| } } _ { \mathrm { o p , * } } , \| \cdot \| _ { \mathrm { F } }$ and $\langle \cdot , \cdot \rangle$ for respectively the operator norm, the nuclear norm (sum of singular values), the Frobenius norm, and the Frobenius inner product. We also use these notations for tensors of higher order, see Appendix A for more details on tensors. We denote $c > 0$ and $C > 0$ for universal numerical constants, respectively small enough and large enough, whose values can change in different occurences. Similarly, by $c _ { p } > 0$ and $C _ { p } > 0$ we denote constants depending only on the parameter $p$ . We also write $\mathrm { R e L U } ( t ) = \operatorname* { m a x } ( t , 0 )$ for the rectified linear unit.
+
+Generic data. We give some flexibility in our definition of “generic data” in order to focus on the essence of the problem, rather than technical details. Namely, in addition to the spherical model mentioned above, where $x _ { i }$ is i.i.d. uniform on the sphere $\mathbb { S } ^ { \bar { d - 1 } } = \{ x \in \mathbb { R } ^ { d } : \lVert x \rVert ^ { \cdot } = 1 \}$ , we also consider the very closely related model where $x _ { i }$ is i.i.d. from a centered Gaussian with covariance $\textstyle { \frac { 1 } { d } } \operatorname { I } _ { d }$ (in particular $\mathbb { E } [ \| x _ { i } \| ^ { 2 } ] = 1$ , and in fact $\| x _ { i } \|$ is tightly concentrated around 1). In both cases we consider $y _ { i }$ to be i.i.d. random signs. We say that a property holds with high probability for generic data, if it holds with high probability either for the spherical model or for the Gaussian model.
+
+# 2 A CONJECTURED LAW OF ROBUSTNESS
+
+Our main contribution is the following conjecture, which asserts that, on generic data sets, increasing the size of a network is necessary to obtain robustness:
+
+Conjecture 1 For generic data sets, with high probability1, any $f \in \mathcal { F } _ { k } ( \psi )$ fitting the data2 (i.e., satisfying (2)) must also satisfy:
+
+$$
+\mathrm { L i p } _ { \mathbb { S } ^ { d - 1 } } ( f ) \geq c \sqrt { \frac { n } { k } } .
+$$
+
+Note that for generic data, with high probability (for $n = { \mathrm { p o l y } } ( d ) )$ , there exists a smooth interpolation. Namely there exists $g : \bar { \mathbb { R } ^ { d } } \overset { \cdot } {  } \mathbb { R }$ with $g ( x _ { i } ) = y _ { i } , \forall i \in [ n ]$ and $\mathrm { L i p } ( g ) = { \cal O } ( 1 )$ . This follows easily from the fact that with high probability (for large $d$ ) one has $\| x _ { i } - x _ { j } \| \geq 1 , \forall i \neq j$ . Conjecture 1 puts restrictions on how smoothly one can interpolate data with small neural networks. A striking consequence of the conjecture is that for a two-layers neural network $f \in \mathcal { F } _ { k } ( \psi )$ to be as robust as this function $g$ (i.e., $\dot { \mathrm { L i p } } ( \dot { \boldsymbol { f } } ) = O ( 1 ) )$ and fit the data, one must have $k = \Omega ( n )$ , i.e., roughly one neuron per data point. On the other hand with that many neurons it is quite trivial to smoothly interpolate the data, as we explain in Section 3.3. Thus the conjecture makes a strong statement that essentially the trivial smooth interpolation is the best thing one can do. In addition to making the prediction that one neuron per datapoint is necessary for optimal smoothness, the conjecture also gives a precise prediction on the possible tradeoff between size of the network and its robustness. We also conjecture that this whole range of tradeoffs is actually achievable:
+
+Conjecture 2 Let $n , d , k$ be such that $C \cdot { \frac { n } { d } } \leq k \leq C \cdot n$ and $n \leq d ^ { C }$ where $C$ is an arbitrarily large constant in the latter occurence. There exists $\psi$ such that, for generic data sets, with high probability, there exists $f \in \mathcal { F } _ { k } ( \psi )$ fitting the data (i.e., satisfying (2)) and such that
+
+$$
+\mathrm { L i p } _ { \mathbb { S } ^ { d - 1 } } ( f ) \leq C \sqrt { \frac { n } { k } } .
+$$
+
+The condition $k \leq C \cdot n$ in Conjecture 2 is necessary, for any interpolation of the data must have Lipschitz constant at least a constant. The other condition on $k$ , namely $k \geq C \cdot \textstyle { \frac { n } { d } }$ , is also necessary, for that many neurons is needed to merely guarantee the existence of a data-fitting neural network with $k$ neurons (see Baum (1988); Yun et al. (2019); Bubeck et al. (2020)). Finally the condition $n \leq d ^ { C }$ is merely used to avoid explicitly stating a logarithmic term in our conjecture (indeed, equivalently one can replace this condition by adding a multiplicative polylogarithmic term in $d$ in the claimed inequality).
+
+Our results around Conjecture 2 (Section 3). We prove Conjecture 2 for both the optimal smoothness regime (which is quite straightforward, see Section 3.3) and for the optimal size regime (here more work is needed, and we use a certain tensor-based construction, see Section 3.4). In the latter case we only prove approximate data fitting (mostly to simplify the proofs), and more importantly we need to assume that $n$ is of order $d ^ { p }$ for some even integer $p$ . It would be interesting to generalize the proof to any $n$ . While the conjecture remains open between these two extreme regimes, we do give a construction in Section 3.3 which has the correct qualitative behavior (namely increasing $k$ improves the Lipschitz constant), albeit the scaling we obtain is $n / k$ instead of $\sqrt { n / k }$ , see Theorem 1.
+
+Our results around Conjecture 1 (Section 4). We prove a weaker version of Conjecture 1 where the Lipschitz constant on the sphere is replaced by a proxy involving the spectral norm of the weight matrix, see Theorem 3. We also prove the conjecture in the optimal size regime, specifically when $n = d ^ { p }$ for an integer $p$ and one uses a polynomial activation function of degree $p$ , see Theorem 6. For $p = 1$ (i.e., $n \approx d$ ) we in fact prove the conjecture for abritrary non-linearities, see Theorem 4.
+
+Further open problems. Our proposed law of robustness is a first mathematical formalization of the broader phenomenon that “overparametrization in neural networks is necessary for robustness”. Ideally one would like a much more refined understanding of the phenomenon than the one given in Conjecture 1. For example, one could imagine that in greater generality, the law would read $\mathrm { L i p } _ { \Omega } ( f ) \geq F ( k , ( x _ { i } , y _ { i } ) _ { i \in [ n ] } , \Omega )$ . That is, we would like to understand how the achievable level of smoothness depends on the particular data set at hand, but also on the set where we expect to be making predictions. Another direction to generalize the law would be to extend it to multilayers neural networks. In particular one could imagine the most general law would replace the parameter $k$ (number of neurons) by the type of architecture being used and in turn predict the best architecture for a given data set and prediction set. Finally note that our proposed law apply to all neural networks, but it would also be interesting to understand how the law interacts with algorithmic considerations (for example in Section 5 we use Adam Kingma and Ba (2015) to find a set of weights that qualitatively match Conjecture 2).
+
+# 3 SMOOTH INTERPOLATION
+
+We start with a warm-up in Section 3.1 where we discuss the simplest case of interpolation with a linear model $( k = 1 , n \leq d )$ and in Section 3.2 for the optimal smoothness regime $( k = n )$ . We generalize the construction of Section 3.2 in Section 3.3 to obtain the whole range of tradeoffs between $k$ and $\operatorname { L i p } ( f )$ , albeit with a suboptimal scaling, see Theorem 1. We also generalize the linear model calculations of Section 3.1 in Section 3.4 to obtain the optimal size regime for larger values of $n$ via a certain tensor construction.
+
+# 3.1 THE SIMPLEST CASE: OPTIMAL SIZE REGIME WHEN $n \leq c \cdot d$
+
+Let us consider $k = 1$ , $n \leq c \cdot d$ and $\psi ( t ) = t$ . Thus we are trying to find $w \in \mathbb { R } ^ { d }$ such that $w \cdot x _ { i } = y _ { i }$ for all $i \in [ n ]$ , or in other words $X w = Y$ with $X$ the $n \times d$ matrix whose $i ^ { t h }$ row is $x _ { i }$ , and $Y = \left( y _ { 1 } , \dots , y _ { n } \right)$ . The smoothest solution to this system (i.e., the one minimizing $\lVert \boldsymbol { w } \rVert .$ ) is
+
+$$
+w = X ^ { \top } ( X X ^ { \top } ) ^ { - 1 } Y ,
+$$
+
+Note that
+
+$$
+\mathrm { L i p } ( x \mapsto w \cdot x ) = \| w \| = \sqrt { w ^ { \top } w } = \sqrt { Y ^ { \top } ( X X ^ { \top } ) ^ { - 1 } Y } .
+$$
+
+Using [Theorem 5.58, Vershynin (2012)] one has with probability at least $1 - \exp ( C - c d )$ (and using that $n \leq c \cdot d )$ that
+
+$$
+X X ^ { \top } \succeq \frac { 1 } { 2 } I _ { n } ,
+$$
+
+and thus $\operatorname { L i p } ( x \mapsto w \cdot x ) \leq { \sqrt { 2 } } \cdot \| Y \| = { \sqrt { 2 n } }$ . This concludes the proof sketch of Conjecture 2 for the simplest case $k = 1$ and $n \leq d$ .
+
+# 3.2 ANOTHER SIMPLE CASE: OPTIMAL SMOOTHNESS REGIME
+
+Next we consider the optimal smoothness regime in Conjecture 2, namely $k = n$ . First note that, for generic data and $n = { \mathrm { p o l y } } ( d )$ , with high probability the caps $C _ { i } : = \lbrace x ^ { \rbrace } \in \mathbb { S } ^ { d - 1 } : x _ { i } \cdot x \geq 0 . 9 \rbrace$ are disjoint sets and moreover they each contain a single data point (namely $x _ { i }$ ). With a single ReLU unit it is then easy to make a smooth function (10-Lipschitz) which is 0 outside of $C _ { i }$ and equal to $+ 1$ at $x _ { i }$ (in other words the neuron activates for a single data point), namely $x \mapsto 1 0$ ·ReLU $( x _ { i } \cdot x - 0 . 9 )$ . Thus one can fit the entire data set with the following ReLU network which is 10-Lipschitz on the sphere:
+
+$$
+f ( x ) = \sum _ { i = 1 } ^ { n } 1 0 y _ { i } \cdot \mathrm { R e L U } \left( x _ { i } \cdot x - 0 . 9 \right) .
+$$
+
+This concludes the proof of Conjecture 2 for the optimal smoothness regime $k = n$
+
+# 3.3 INTERMEDIATE REGIMES VIA RELU NETWORKS
+
+We now combine the two constructions above (the linear model of Section 3.1 and the “isolation” strategy of Section 3.2) to give a construction that can trade off size for robustness (albeit not optimally according to Conjecture 2), see Appendix $\textrm { C }$ for the proof.
+
+Theorem 1 Let $n , d , k$ be such that $\begin{array} { r } { C \cdot \frac { n \log ( n ) } { d } \leq k \leq C \cdot n . } \end{array}$ . For generic data sets, with probability at least $1 - 1 / n ^ { C }$ , there exists $f \in \mathcal { F } _ { k } ( \mathrm { R e L U } )$ fitting the data (i.e., satisfying (2)) and such that
+
+$$
+\mathrm { L i p } _ { \mathbb { S } ^ { d - 1 } } ( f ) \leq C \cdot \frac { n \log ( d ) } { k } .
+$$
+
+3.4 OPTIMAL SIZE NETWORKS VIA TENSOR INTERPOLATION
+
+In this section we essentially prove Conjecture 2 in the optimal size regime (namely $k \cdot d \approx n$ ), with three caveats:
+
+1. We allow a slack of a $\log n$ factor by considering $k \cdot d = C n \log ( n )$ instead of the optimal $k \cdot d = C n$ as in Baum (1988); Bubeck et al. (2020).   
+2. We only prove approximate fit rather than exact fit. It is likely that with more work one can use the core of our argument to obtain exact fit. For that reason we did not make any attempt to optimize the dependency on $\varepsilon$ in Theorem 2. For instance one could probably obtain $\log ( 1 / \varepsilon )$ rather than $1 / \mathrm { p o l y } ( \varepsilon )$ dependency by using an iterative scheme that fits the residuals, as in (Bresler and Nagaraj, 2020; Bubeck et al., 2020).   
+3. We assume that $n$ is of order $d ^ { p }$ for some even integer $p$ . While it might be that one can apply the same proof for odd integers, the whole construction crucially relies on $p$ being an even integer as we essentially do a linear regression over the feature embedding $x \mapsto x ^ { \otimes p }$ . A possible approach to extend the proof to other values of $n$ would be use the scheme of Section 3.3 with the linear regression there replaced by the tensor regression used below.
+
+Theorem 2 Fix $\varepsilon > 0$ , $p$ an even integer, and let $\psi ( t ) = t ^ { p }$ . Let $n , d , k$ be such that $n \log ( n ) =$ $\varepsilon ^ { 2 } \cdot d ^ { p }$ and $k = C _ { p } \cdot d ^ { p - 1 }$ . Then for generic data, with probability at least $1 - 1 / n ^ { C }$ , there exists $f \in \mathcal { F } _ { k } ( \psi )$ such that
+
+and
+
+$$
+\begin{array} { r l } & { \left| f ( x _ { i } ) - y _ { i } \right| \le C _ { p } \cdot \varepsilon , \forall i \in [ n ] , } \\ & { } \\ & { \quad \quad \mathrm { L i p } _ { \mathbb { S } ^ { d - 1 } } ( f ) \le C _ { p } \sqrt { \displaystyle \frac { n } { k } } . } \end{array}
+$$
+
+Proof. We propose to approximately fit with the following neural network:
+
+$$
+f ( { \boldsymbol { x } } ) = \sum _ { i = 1 } ^ { n } y _ { i } ( x _ { i } \cdot x ) ^ { p } .
+$$
+
+Naively one might think that this neural network requires $n$ neurons. However, it turns out that one can always decompose a symmetric tensor of order $p$ into $k = 2 ^ { p } d ^ { p - 1 }$ rank-1 symmetric tensors of order $p$ , so that in fact $f \in \mathcal { F } _ { k } ( \psi )$ . For $p = 2$ this simply follows from eigendecomposition and for general $p$ we give a simple proof in [Appendix A, Lemma 2].
+
+One also has by applying [Appendix B, Lemma 4] with $\tau = C _ { p } \log ( n )$ and doing an union bound, that with probability at least $1 - 1 / n ^ { C }$ , for any $j \in [ n ]$ ,
+
+$$
+\left| \sum _ { i = 1 , i \neq j } ^ { n } y _ { i } ( x _ { i } \cdot x _ { j } ) ^ { p } \right| \leq C _ { p } { \sqrt { \frac { n \log ( n ) } { d ^ { p } } } } \leq C _ { p } \varepsilon .
+$$
+
+In particular this proves (3).
+
+Thus it only remains to estimate the Lipschitz constant, which by [Appendix A, Lemma 1] is reduced to estimating the operator norm of the tensor $\scriptstyle \sum _ { i = 1 } ^ { n } y _ { i } x _ { i } ^ { \otimes p }$ . We do so in [Appendix B, Lemma 5]. 
+
+# 4 PROVABLE WEAKER VERSIONS OF CONJECTURE 1
+
+Conjecture 1 can be made weaker along several directions. For example the quantity of interest $\operatorname { L i p } _ { \mathbb { S } ^ { d - 1 } } ( f )$ can be replaced by various upper bound proxies for the Lipschitz constant. A mild weakening would be to replace it by the Lipschitz constant on the whole space (we shall in fact only consider this notion here). A much more severe weakening is to replace it by a quantity that depends on the spectral norm of the weight matrix (essentially ignoring the pattern of activation functions). For the latter proxy we actually give a complete proof, see Theorem 3, which in particular formally proves that “overparametrization is a law of robustness for generic data sets”. Other interesting directions to weaken the conjecture include specializing it to common activation functions, or simply having a smaller lower bound on the Lipschitz constant. In Section 4.2 we prove the conjecture when $n$ is replaced by $d$ in the lower bound. We say that this inequality is in the “very high-dimensional case”, in the sense that it matches the conjecture for $n \approx d$ (alternatively we also refer to it as the “undercomplete case”, in the sense that only $k \leq d$ is relevant in this very high-dimensional scenario). In the moderately high-dimensional case $\mathit { \Omega } _ { n } \gg d )$ the proof strategy we propose in Section 4.2 cannot work. In Section 4.3 we give another argument for the latter case, specifically in the optimal size regime (i.e., $k \cdot d \approx n$ ) and for a power activation function, see Theorem 5. We generalize this to polynomial activation functions in Section D.1. In the specific case of a quadratic activation function we also show a lower bound that applies for any $k$ and which is in fact larger than the one given in Conjecture 1, see Theorem 7 in Section D.2.
+
+# 4.1 SPECTRAL NORM PROXY FOR THE LIPSCHITZ CONSTANT
+
+We can rewrite (1) as
+
+$$
+f ( x ) = a ^ { \top } \psi ( W x + b ) ,
+$$
+
+where $a = ( a _ { 1 } , \ldots , a _ { k } ) \in \mathbb { R } ^ { k }$ , $b = ( b _ { 1 } , \dots , b _ { k } ) \in \mathbb { R } ^ { k }$ , $W \in \mathbb { R } ^ { k \times d }$ is the matrix whose $\ell ^ { t h }$ row is $w _ { \ell }$ , and $\psi$ is extended from $\mathbb { R } \to \mathbb { R }$ to $\mathbb { R } ^ { k } \to \mathbb { R } ^ { k }$ by applying it coordinate-wise. We prove here the following:
+
+Theorem 3 Assume that $\psi$ is $L$ -Lipschitz. For $f \in \mathcal { F } _ { k } ( \psi )$ one has
+
+$$
+\operatorname { L i p } ( f ) \leq L \cdot \| a \| \cdot \| W \| _ { \mathrm { o p } } .
+$$
+
+For a generic data set, if $f ( x _ { i } ) = y _ { i } , \forall i \in [ n ]$ and $f$ has no bias terms (i.e., $b = 0$ in (4)), then with positive probability one has:
+
+$$
+L \cdot \| a \| \cdot \| W \| _ { \mathrm { o p } } \geq { \sqrt { \frac { n } { k } } } .
+$$
+
+Note that we prove the inequality (6) only with positive probability (i.e., there exists a data set where the inequality is true), but in fact it is easy to derive the statement with high probability using classical concentration inequalities.
+
+Proof. Since $\psi : \mathbb { R }  \mathbb { R }$ is $L$ -Lipschitz, we have:
+
+$$
+\begin{array} { r } { ^ { \mathfrak { r } } ( x ) - f ( x ^ { \prime } ) \le \| a \| \cdot \| \psi ( W x + b ) - \psi ( W x ^ { \prime } + b ) \| \le L \cdot \| a \| \cdot \| W x - W x ^ { \prime } \| \le L \cdot \| a \| \cdot \| W \| _ { \infty } \cdot \| x - x ^ { \prime } \| , } \end{array}
+$$
+
+which directly proves (5).
+
+Next, following the proof of [Proposition 1, Bubeck et al. (2020)] one obtains that for a generic data set, with positive probability, one has (without bias terms):
+
+$$
+\sum _ { \ell = 1 } ^ { k } | a _ { \ell } | \cdot \| w _ { \ell } \| \geq \frac { \sqrt { n } } { L } .
+$$
+
+It only remains to observe that:
+
+$$
+{ \frac { \sqrt { n } } { L } } \leq \sum _ { \ell = 1 } ^ { k } | a _ { \ell } | \cdot \| w _ { \ell } \| \leq { \sqrt { \sum _ { \ell = 1 } ^ { k } | a _ { \ell } | ^ { 2 } \cdot \sum _ { \ell = 1 } ^ { k } \| w _ { \ell } \| ^ { 2 } } } = \| a \| \cdot \| W \| _ { \mathrm { F } } \leq { \sqrt { k } } \cdot \| a \| \cdot \| W \| _ { \mathrm { o p } } ,
+$$
+
+which concludes the proof of (6).
+
+# 4.2 UNDERCOMPLETE CASE
+
+Next we prove the conjecture in the high dimensional case $n \approx d$ . More precisely we replace $n$ by $d$ in the conjectured lower bound. Importantly note that the resulting lower bound then becomes non-trivial only in the regime $k \leq d$ (the “undercomplete case”).
+
+We consider in fact a slightly more general scenario than interpolation with a neural network, namely we simply assume that one interpolates the data with a function $f ( x ) = g ( P x )$ where $P$ is a linear projection on a $k$ -dimensional subspace (this clearly generalizes $f \in \mathcal { F } _ { k } ( \psi )$ , in fact it even allows for the non-linearity $\psi$ to depend on the data3, or to have a different non-linearity for each neuron).
+
+Theorem 4 Let $n \geq d$ . Let $f :  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ be a function such that $f ( x _ { i } ) = y _ { i } , \forall i \in [ n ]$ and moreover $f ( x ) = g ( P x )$ for some differentiable function $g : \mathbb { R } ^ { k }  \mathbb { R }$ and matrix $P \in \dot { \mathbb { R } } ^ { k \times d }$ . Then, for generic data, with probability at least $1 - \exp ( C - c d )$ one must have
+
+$$
+\mathrm { L i p } ( f ) \geq c \sqrt { \frac { d } { k } } .
+$$
+
+Proof. Let us modify $g$ so that $P$ is simply an orthogonal projection operator (i.e., $P P ^ { \top } = \operatorname { I } _ { k } ,$ ). Let us also assume for sake of notational simplicity that we have a balanced data set of size $2 n$ , that is with: $y _ { 1 } , . . . y _ { n } = + 1$ and $y _ { n + 1 } , . . . , y _ { 2 n } = - 1$ . Let us denote $x _ { i } ^ { \prime } = x _ { i } - x _ { n + i }$ for $i \in [ n ]$ . The sequence $x _ { i } ^ { \prime }$ is i.i.d. and satisfies $\begin{array} { r } { \mathbb { E } [ x _ { i } ^ { \prime } x _ { i } ^ { \prime \top } ] = \frac { 2 } { d } { \mathrm { I } } _ { d } } \end{array}$ .
+
+Now observe that on the segment $[ x _ { i } , x _ { n + i } ]$ (whose length is less than 2), the function $f$ changes value from $+ 1$ to $- 1$ , and thus there exists $\dot { z } _ { i } \in [ x _ { i } , x _ { n + i } ]$ such that:
+
+Moreover one has (using that $\nabla f ( x ) = P ^ { \top } \nabla g ( x )$ , and thus $\| \nabla g ( x ) \| = \| P \nabla f ( x ) \| \le \mathrm { L i p } ( f ) )$
+
+$$
+| \nabla f ( z _ { i } ) \cdot x _ { i } ^ { \prime } | = | \nabla g ( P z _ { i } ) \cdot ( P x _ { i } ^ { \prime } ) | \leq \mathrm { L i p } ( f ) \cdot \| P x _ { i } ^ { \prime } \|
+$$
+
+Combining the two above displays one has:
+
+$$
+\frac { n } { \mathrm { L i p } ( f ) } \leq \sum _ { i = 1 } ^ { n } \| P x _ { i } ^ { \prime } \| \leq \sqrt { n \sum _ { i = 1 } ^ { n } \| P x _ { i } ^ { \prime } \| ^ { 2 } } = \sqrt { n \sum _ { i = 1 } ^ { n } { x _ { i } ^ { \prime } } ^ { \top } P ^ { \top } P x _ { i } ^ { \prime } } = \sqrt { n \langle \sum _ { i = 1 } ^ { n } x _ { i } ^ { \prime } x _ { i } ^ { \prime } | ^ { \top } , P ^ { \top } P \rangle _ { \mathrm { H S } } } .
+$$
+
+Using [Theorem 5.39, Vershynin (2012)] (specifically (5.23)) we know that with probability at least $1 - \exp ( C - c d )$ we have $\begin{array} { r } { \left. \sum _ { i = 1 } ^ { n } x _ { i } ^ { \prime } x _ { i } ^ { \prime \top } \right. _ { \mathrm { o p } } ^ { \cdot } \leq C \frac { n } { d } } \end{array}$ (here we use $n \geq d$ too). Moreover we have $\| P ^ { \top } P \| _ { \mathrm { o p } , * } = \operatorname { T r } ( P ^ { \top } P ) = \operatorname { T r } ( P P ^ { \top } ) = k$ . Thus we have $\begin{array} { r } { \langle \sum _ { i = 1 } ^ { n } x _ { i } ^ { \prime } x _ { i } ^ { \prime \top } , P ^ { \top } P \rangle _ { \mathrm { H S } } \leq C \frac { n \cdot k } { d } } \end{array}$ so that with the above display one obtains $\textstyle { \frac { n } { \operatorname { L i p } ( f ) } } \leq n { \sqrt { \frac { C k } { d } } }$ , which concludes the proof. 
+
+# 4.3 POWER ACTIVATION
+
+We prove here the conjecture for the power activation function $\psi ( t ) = t ^ { p }$ with $p$ an integer and with no bias terms (we deal with general polynomials, including with bias, in Appendix D). Without bias such a network can be written as:
+
+$$
+f ( x ) = \sum _ { \ell = 1 } ^ { k } a _ { \ell } ( w _ { \ell } \cdot x ) ^ { p } = \langle T , x ^ { \otimes p } \rangle ,
+$$
+
+where T = Pk\`=1 a\`w⊗\` . As we already saw in the proof of Theorem 2 (see specifically [Appendix A, Lemma 2]), without loss of generality we have $k \leq C _ { p } d ^ { p - 1 }$ . We now prove that tensor networks of the form (7) cannot obtain a Lipschitz constant4 better than $\scriptstyle { \sqrt { n / d ^ { p - 1 } } }$ , in accordance with Conjecture 1 for full rank tensors (where $k \approx d ^ { p - 1 }$ ).
+
+![](images/6ac11ca9bdd3f2eadbb8ee7304075db7fd4adfb764ead07f92e6e5b9135f1ac3.jpg)  
+Figure 2: Scatter plot of maximum random gradient with respect to $\sqrt { \frac { n } { k } }$ with 906 data points (Experiment 1)
+
+![](images/a3537ebd423f7488f75d7e040034a7f61b3c8d6eb72813d1e07a258b38e00e32.jpg)  
+Figure 3: Scatter plot of maximum random gradient with respect to $\sqrt { d }$ in optimal smoothness (blue) and optimal size (red) regimes (Experiment 2)
+
+Theorem 5 Assume that we have a tensor $T$ of order $p$ such that $\langle T , x _ { i } ^ { \otimes p } \rangle = y _ { i } , \forall i \in [ n ]$ . Then, for generic data, with probability at least $1 - C \exp ( - c _ { p } d )$ , one must have
+
+$$
+\| T \| _ { \mathrm { o p } } \geq c _ { p } { \sqrt { \frac { n } { d ^ { p - 1 } } } } .
+$$
+
+Proof. Denoting Ω = Pni=1 yix⊗i , we have (using $y _ { i } ^ { 2 } = 1$ for the first equality and [Appendix A, Lemma 3] for the last inequality):
+
+$$
+n = \langle T , \Omega \rangle \leq \| \Omega \| _ { \mathrm { o p } } \cdot \| T \| _ { \mathrm { o p } , * } \leq d ^ { p - 1 } \cdot \| \Omega \| _ { \mathrm { o p } } \cdot \| T \| _ { \mathrm { o p } } .
+$$
+
+Thus we obtain $\begin{array} { r } { \| T \| _ { \mathrm { o p } } \geq \frac { n } { d ^ { p - 1 } \cdot \| \Omega \| _ { \mathrm { o p } } } } \end{array}$ , and it only remains to apply [Appendix B, Lemma 5] which states that with probability at least $1 - C \exp ( - c _ { p } d )$ one has $\begin{array} { r } { \| \Omega \| _ { \mathrm { o p } } \leq C _ { p } \sqrt { \frac { n } { d ^ { p - 1 } } } } \end{array}$ . 
+
+# 5 EXPERIMENTS
+
+We consider a generic dataset from the Gaussian model (i.e., $x _ { 1 } , \ldots , x _ { n }$ i.i.d. from $\mathcal { N } ( 0 , \textstyle \frac { 1 } { d } I _ { d } )$ and labels $y _ { 1 } , \ldots , y _ { n }$ i.i.d from the uniform distribution over $\{ - 1 , 1 \}$ and independent of $x _ { 1 } , \ldots , x _ { n } )$ . For various values of $( n , d , k )$ we train two-layers neural networks with $k$ ReLU units and batch normalization (see Ioffe and Szegedy (2015)) between the linear layer and ReLU layer, using the Adam optimizer (Kingma and Ba, 2015) on the least squares loss. We keep the values of $( n , k , d )$ where the network successfully memorizes the random labels (possibly after a rounding to $\{ - 1 , + 1 \}$ , and such that prior to rounding the least squares loss is at most some small value $\varepsilon$ to be specified later). Given a triple $( n , d , k )$ , suppose the output of the trained network is $f _ { n , d , k } : \mathbb { R } ^ { d }  \dot { \mathbb { R } }$ . We then generate $z _ { 1 } , \dots , z _ { T }$ (where $T = 1 0 0 0$ ) i.i.d from the distribution $\textstyle { \mathcal { N } } ( 0 , { \frac { 1 } { d } } I _ { d } )$ , independently of everything else and define the “maximum random gradient” to be $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ T ] } \lVert \nabla f _ { n , k , d } ( z _ { i } ) \rVert } \end{array}$ (it is our proxy for the true Lipschitz constant $\begin{array} { r } { \operatorname* { s u p } _ { z \in \mathbb { S } ^ { d - 1 } } \| \nabla f _ { n , d , k } ( z ) \| ) } \end{array}$ . Our experimental results are as follows:
+
+Experiment 1: We ran experiments with $n$ between 100 and 2000, $d$ between $\sim 5 0$ and $\sim n$ , and $k$ between $\sim 1 0$ and $\sim n$ (we also choose $\varepsilon = 0 . 0 2$ for the thresholding). In Figure 2 we give a scatter plot of $\begin{array} { r l } {  { \big ( \sqrt { \frac { n } { k } } , \operatorname* { m a x } _ { i \in [ T ] } \| \nabla f _ { n , k , d } ( z _ { i } ) \| \big ) } \quad } & { { } } \end{array}$ , and as predicted we see a linear trend, thus providing empirical evidence for Conjecture 1.
+
+Experiment 2: In this experiment, we investigate the two extreme cases $k \sim n$ and $k \sim n / d$ . We fix $n = 1 0 ^ { 4 }$ and sweep the value of $d$ between 10 to 5000 (we also choose $\varepsilon = 0 . 1$ for the thresholding). In the first case, we let $k = n$ and in the second case we let $k = 1 0 n / d$ . In Figure 3 we plot $\sqrt { d }$ versus the maximum random gradient (as defined above) for both cases. We observe a linear dependence between the maximum gradient value and $\sqrt { d }$ when we have $k = 1 0 n / d$ , and roughly a constant maximum gradient value when $k = n$ , thus providing again evidence for Conjecture 1
+
+# REFERENCES
+
+James Alexander and Andre Hirschowitz. Polynomial interpolation in several variables. ´ Journal of Algebraic Geometry, 4(2):201–222, 1995.
+
+Zeyuan Allen-Zhu and Yuanzhi Li. Feature purification: How adversarial training performs robust deep learning. arXiv preprint arXiv:2005.10190, 2020.
+
+Eric B Baum. On the capabilities of multilayer perceptrons. Journal of complexity, 4(3):193–215, 1988.
+
+Guy Bresler and Dheeraj Nagaraj. A corrective view of neural networks: Representation, memorization and learning. arXiv preprint arXiv:2002.00274, 2020.
+
+Sebastien Bubeck, Yin Tat Lee, Eric Price, and Ilya Razenshteyn. Adversarial examples from com-´ putational constraints. In International Conference on Machine Learning, pages 831–840, 2019.
+
+Sebastien Bubeck, Ronen Eldan, Yin Tat Lee, and Dan Mikulincer. Network size and weights size ´ for memorization with two-layers neural networks. arXiv preprint arXiv:2006.02855, 2020.
+
+Pierre Comon, Gene Golub, Lek-Heng Lim, and Bernard Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications, 30(3):1254–1279, 2008.
+
+George Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2(4):303–314, 1989.
+
+Akshay Degwekar, Preetum Nakkiran, and Vinod Vaikuntanathan. Computational limitations in robust classification and win-win results. volume 99 of Proceedings of Machine Learning Research (COLT), pages 994–1028, 2019.
+
+Shmuel Friedland and Lek-Heng Lim. Nuclear norm of higher-order tensors. Mathematics of Computation, 87(311):1255–1281, 2018.
+
+Ian Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In International Conference on Learning Representations, 2015.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pages 448– 456, 2015.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.
+
+Moshe Leshno, Vladimir Ya Lin, Allan Pinkus, and Shimon Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural networks, 6(6):861–867, 1993.
+
+Zhening Li, Yuji Nakatsukasa, Tasuku Soma, and Andre Uschmajew. On orthogonal tensors and ´ best rank-one approximation ratio. SIAM Journal on Matrix Analysis and Applications, 39(1): 400–425, 2018.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In International Conference on Learning Representations, 2018.
+
+Arkadi Nemirovski. Interior point polynomial time methods in convex programming. Lecture notes, 2004.
+
+Grigoris Paouris, Petros Valettas, and Joel Zinn. Random version of dvoretzky’s theorem in lpn. Stochastic Processes and their Applications, 127(10):3187–3227, 2017.
+
+Aditi Raghunathan, Sang Michael Xie, Fanny Yang, John C Duchi, and Percy Liang. Adversarial training can hurt generalization. In International Conference on Learning Representations, 2019.
+
+Bruce Arie Reznick. Sum of even powers of real linear forms, volume 463. American Mathematical Soc., 1992.
+
+Ludwig Schmidt, Shibani Santurkar, Dimitris Tsipras, Kunal Talwar, and Aleksander Madry. Adversarially robust generalization requires more data. In Advances in Neural Information Processing Systems, 2018.
+
+Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In International Conference on Learning Representations, 2013.
+
+Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Compressed Sensing: Theory and Practice, pages 210–268. Cambridge University Prteess, 2012.
+
+Chulhee Yun, Suvrit Sra, and Ali Jadbabaie. Small relu networks are powerful memorizers: a tight analysis of memorization capacity. In Advances in Neural Information Processing Systems, pages 15532–15543, 2019.
+
+# A RESULTS ON TENSORS
+
+A tensor of order $p$ is an array $T = ( T _ { i _ { 1 } , \dots , i _ { p } } ) _ { i _ { 1 } , \dots , i _ { p } \in [ d ] }$ . The Frobenius inner product for tensors is defined by:
+
+$$
+\langle T , S \rangle = \sum _ { i _ { 1 } , \dots , i _ { p } = 1 } ^ { d } T _ { i _ { 1 } , \dots , i _ { p } } S _ { i _ { 1 } , \dots , i _ { p } } ,
+$$
+
+with the corresponding norm $\| \cdot \| _ { \mathrm { F } }$ . A tensor is said to be of rank 1 if it can be written as:
+
+$$
+T = u _ { 1 } \otimes \ldots \otimes u _ { p } ,
+$$
+
+for some $u _ { 1 } , \ldots , u _ { p } \in \mathbb { R } ^ { d }$ . The operator norm $\| \cdot \| _ { \mathrm { o p } }$ is defined by:
+
+$$
+\| T \| _ { \mathrm { o p } } = \operatorname* { s u p } _ { S \operatorname { r a n k } 1 , \| S \| _ { \mathrm { F } } \leq 1 } \langle T , S \rangle .
+$$
+
+For symmetric tensors (i.e., such that the entries of the array are invariant under permutation of the $p$ indices), Banach’s Theorem (see e.g., [(2.32), Nemirovski (2004)]) states that in fact one has
+
+$$
+\| T \| _ { \mathrm { o p } } = \operatorname* { s u p } _ { x \in \mathbb { S } ^ { d - 1 } } \langle T , x ^ { \otimes p } \rangle .
+$$
+
+We refer to Friedland and Lim (2018) for more details and background on tensors. We now list a couple of useful results, with short proofs.
+
+Lemma 1 For a tensor $T$ of order $p$ , one has
+
+$$
+\mathrm { L i p } _ { \mathbb { S } ^ { d - 1 } } ( x \mapsto \langle T , x ^ { \otimes p } \rangle ) \leq p \cdot \| T \| _ { \mathrm { o p } } .
+$$
+
+Proof. One has for any $x , y \in \mathbb { S } ^ { d - 1 }$ ,
+
+$$
+\begin{array} { r c l } { \left| \left. T , x ^ { \otimes p } \right. - \left. T , y ^ { \otimes p } \right. \right| } & { \leq } & { \displaystyle \sum _ { q = 1 } ^ { p } \left| \left. T , x ^ { \otimes p - q + 1 } \otimes y ^ { \otimes q - 1 } \right. - \left. T , x ^ { \otimes p - q } \otimes y ^ { \otimes q } \right. \right| } \\ & { \leq } & { p \cdot \left\| x - y \right\| \cdot \operatorname* { s u p } _ { x ^ { 1 } , \ldots , x ^ { p } \in \mathbb { S } ^ { d - 1 } } \left| \left. T , \otimes _ { q = 1 } ^ { p } x ^ { q } \right. \right| } \\ & { = } & { p \cdot \left\| x - y \right\| \cdot \left\| T \right\| _ { \mathrm { l o p } } . } \end{array}
+$$
+
+Lemma 2 For any tensor $T$ of order $p$ , there exists $w _ { 1 } , \ldots , w _ { 2 ^ { p } d ^ { p - 1 } } \in \mathbb { R } ^ { d }$ and $\xi _ { 1 } , \dotsc , \xi _ { 2 ^ { p } d ^ { p - 1 } } \in$ $\{ - 1 , + 1 \}$ such that for all $x \in \mathbb { R } ^ { d }$ ,
+
+$$
+\langle T , x ^ { \otimes p } \rangle = \sum _ { \ell = 1 } ^ { 2 ^ { p } d ^ { p - 1 } } \xi _ { \ell } \cdot ( w _ { \ell } \cdot x ) ^ { p } .
+$$
+
+Results like Lemma 2 go back at least to Reznick (1992). In fact much more precise results on minimal decomposition in rank-1 tensors are known thanks to the work of Alexander and Hirschowitz (1995). We refer to (Comon et al., 2008) for more discussion on this topic.
+
+Proof. First note that trivially $T$ can be written as:
+
+$$
+T = \sum _ { i _ { 1 } , \dots , i _ { p - 1 } = 1 } ^ { d } e _ { i _ { 1 } } \otimes \dots \otimes e _ { i _ { p - 1 } } \otimes T [ i _ { 1 } , \dots , i _ { p - 1 } , 1 : d ] .
+$$
+
+Thus one only needs to prove that a function of the form $\begin{array} { r } { x \mapsto \prod _ { q = 1 } ^ { p } ( w _ { q } \cdot x ) } \end{array}$ can be written as the sum of $2 ^ { p }$ functions of the form $( w ^ { \prime } \cdot x ) ^ { p }$ . To do so note that, with $\varepsilon _ { q }$ i.i.d. random signs,
+
+$$
+\mathbb E \left[ \prod _ { q = 1 } ^ { p } \varepsilon _ { q } \cdot \left( \sum _ { q = 1 } ^ { p } \varepsilon _ { q } w _ { q } \cdot x \right) ^ { p } \right] = \mathbb E \left[ \prod _ { q = 1 } ^ { p } \varepsilon _ { q } \cdot \sum _ { q _ { 1 } , \ldots , q _ { p } = 1 } ^ { p } \left( \prod _ { r = 1 } ^ { p } \varepsilon _ { q _ { r } } w _ { q _ { r } } \cdot x \right) \right] = p ! \prod _ { q = 1 } ^ { p } ( w _ { q } \cdot x ) .
+$$
+
+Lemma 3 For any tensor $T$ of order $p$ one has:
+
+$$
+\| T \| _ { \mathrm { o p } , * } \leq d ^ { p - 1 } \cdot \| T \| _ { \mathrm { o p } } .
+$$
+
+The above result and its proof are directly taken from Li et al. (2018). We only repeat the argument here for sake of completeness.
+
+Proof. Note that the decomposition (10) is orthogonal, and thus for any tensor $S$ of order $p$ one has:
+
+$$
+\begin{array} { r c l } { \langle T , S \rangle } & { \le } & { \sqrt { d ^ { p - 1 } \cdot \displaystyle \sum _ { i _ { 1 } , \ldots , i _ { p - 1 } = 1 } ^ { d } \left. e _ { i _ { 1 } } \otimes \ldots \otimes e _ { i _ { p - 1 } } \otimes T [ i _ { 1 } , \ldots , i _ { p - 1 } , 1 : d ] , S \right. ^ { 2 } } } \\ & { \le } & { \sqrt { d ^ { p - 1 } \cdot \| S \| _ { \mathrm { o p } } ^ { 2 } \cdot \displaystyle \sum _ { i _ { 1 } , \ldots , i _ { p - 1 } = 1 } ^ { d } \| T [ i _ { 1 } , \ldots , i _ { p - 1 } , 1 : d ] \| ^ { 2 } } } \\ & { = } & { d ^ { \frac { p - 1 } { 2 } } \cdot \| S \| _ { \mathrm { o p } } \cdot \| T \| _ { F } . } \end{array}
+$$
+
+Thus one has $\| T \| _ { \mathrm { o p } , * } \leq d ^ { \frac { p - 1 } { 2 } } \cdot \| T \| _ { F }$ . By duality one also has $\| T \| _ { \mathrm { o p } } \geq d ^ { - { \frac { p - 1 } { 2 } } } \cdot \| T \| _ { F }$ , which concludes the proof.
+
+# B RESULTS ON RANDOM TENSORS
+
+Lemma 4 For any fixed $x \in \mathbb { S } ^ { d - 1 }$ and generic data, with probability at least $1 - C \exp ( - c _ { p } \tau )$ one has:
+
+$$
+\left| \sum _ { i = 1 } ^ { n } y _ { i } ( x _ { i } \cdot x ) ^ { p } \right| \leq C _ { p } { \sqrt { \frac { n \tau } { d ^ { p } } } } .
+$$
+
+Proof. Using [Theorem 1, Paouris et al. (2017)] one has, for any fixed $x \in \mathbb { S } ^ { d - 1 }$ and $\tau \leq n$
+
+$$
+\mathbb { P } \left( \left| d ^ { p / 2 } \sum _ { i = 1 } ^ { n } | x _ { i } \cdot x | ^ { p } - n \sigma _ { p } \right| > C _ { p } { \sqrt { n \tau } } \right) \leq C \exp ( - c _ { p } \tau ) ,
+$$
+
+where denotes the $p ^ { t h }$ moment of the standard Gaussian. Let us denote $n ^ { + } = | \{ i \in [ n ] : y _ { i } =$ +1} T + = Pi:yi=+1 and xi , and si arly for $n ^ { - } , T ^ { - }$ . Now with probability $1 - C \exp ( - c \tau )$ $y _ { i } ^ { \prime } s$
+
+$$
+| n ^ { + } - n ^ { - } | \leq { \sqrt { n \tau } } .
+$$
+
+Thus combining the two above displays we obtain with probability at least $1 - C \exp ( - c _ { p } \tau )$ ,
+
+$$
+d ^ { p / 2 } \left| \sum _ { i : y _ { i } = + 1 } | x _ { i } \cdot x | ^ { p } - \sum _ { i : y _ { i } = - 1 } | x _ { i } \cdot x | ^ { p } \right| \leq C _ { p } { \sqrt { n \tau } } + \sigma _ { p } | n ^ { + } - n ^ { - } | \leq C _ { p } { \sqrt { n \tau } } ,
+$$
+
+Lemma 5 For generic data, with probability at least $1 - C \exp ( - c _ { p } d )$ one has:
+
+$$
+\left. \sum _ { i = 1 } ^ { n } y _ { i } x _ { i } ^ { \otimes p } \right. _ { \mathrm { o p } } \leq C _ { p } { \sqrt { \frac { n } { d ^ { p - 1 } } } } .
+$$
+
+Proof. Let $\mathcal { N }$ be an $\frac { 1 } { 2 p }$ -net of $S ^ { d - 1 }$ (in particular $| { \mathcal { N } } | \leq C _ { p } ^ { d } )$ . By an union bound and Lemma 4 one has:
+
+$$
+\mathbb { P } \left( \exists x \in \mathcal { N } _ { \varepsilon } \ : \ \left. \sum _ { i = 1 } ^ { n } y _ { i } \vert x _ { i } \cdot x \vert ^ { p } \right. > C _ { p } \sqrt { \frac { n } { d ^ { p - 1 } } } \right) \leq C \exp ( - c _ { p } d ) ,
+$$
+
+$\textstyle T = \sum _ { i = 1 } ^ { n } y _ { i } x _ { i } ^ { \otimes p }$ $T$
+
+$$
+\| T \| _ { \mathrm { o p } } \leq \operatorname* { m a x } _ { x \in \mathcal { N } } \langle T , x ^ { \otimes p } \rangle + \frac { 1 } { 2 } \| T \| _ { \mathrm { o p } } ,
+$$
+
+and in particular $\begin{array} { r } { \| T \| _ { \mathrm { o p } } \leq 2 \operatorname* { m a x } _ { x \in \mathcal { N } } \langle T , x ^ { \otimes p } \rangle } \end{array}$ , which together with (11) concludes the proof.
+
+# C PROOF OF THEOREM 1
+
+Let $\begin{array} { r } { m = { \frac { n } { k } } } \end{array}$ (by assumption $\begin{array} { r } { m \leq c \cdot \frac { d } { \log ( n ) } ) } \end{array}$ and assume it is an integer. Let us choose $m$ points with the same label, say it is the points $x _ { 1 } , \ldots , x _ { m }$ with label $+ 1$ . As in Section 3.1 let $w \in \mathbb { R } ^ { d }$ be the minimal norm vector that satisfy $w \cdot x _ { i } = 1$ , and thus as we proved there with probability at least $1 - \exp ( C - c d )$ one has $\| w \| \leq \sqrt { 2 m }$ . Crucially for the end of the proof, also note that the distribution of $w$ is rotationally invariant. Next observe that with probability at least $1 - 1 / n ^ { C }$ (with respect to the sampling of $x _ { m + 1 } , \ldots , x _ { n } )$ one has $\begin{array} { r } { \operatorname* { m a x } _ { i \in \{ m + 1 , \ldots , n \} } | w \cdot x _ { i } | \leq C \cdot \| w \| \sqrt { \frac { \log ( n ) } { d } } \leq \frac { 1 } { 2 } } \end{array}$ . In particular the cap $\mathcal { C } : = \{ x \in \mathbb { S } ^ { d - 1 } : w \cdot x \geq \frac { 1 } { 2 } \}$ contains $x _ { 1 } , \ldots , x _ { m }$ but does not contain any $x _ { i }$ , $i > m$ . Thus the neuron
+
+$$
+x \mapsto 2 \cdot { \mathrm { R e L U } } \left( w \cdot x - { \frac { 1 } { 2 } } \right) ~ ,
+$$
+
+computes the value 1 at points $x _ { 1 } , \ldots , x _ { m }$ and the value 0 at the rest of the training set.
+
+One can now repeat this process, and build the neurons $w _ { 1 } , \ldots , w _ { k }$ (all with $\mathrm { n o r m } \leq \sqrt { 2 m } )$ ), so that (with well-chosen signs $\xi _ { \ell } \in \{ - 1 , 1 \}$ ) the data is perfectly fitted by the function:
+
+$$
+f ( x ) = \sum _ { \ell = 1 } ^ { k } 2 \cdot \xi _ { \ell } \cdot \mathrm { { R e L U } } \left( w _ { \ell } \cdot x - \frac 1 2 \right) .
+$$
+
+It only remains to estimate the Lipschitz constant. Note that if a point $x \in \mathbb { S } ^ { d - 1 }$ activates a certain subset $A \subset \{ 1 , \ldots , k \}$ of the neurons, then the gradient at this point is $\textstyle \sum _ { \ell \in A } w _ { \ell } ^ { \prime }$ with $w _ { \ell } ^ { \prime } = 2 \xi _ { \ell } w _ { \ell }$ . Using that the $w _ { \ell ^ { \prime } }$ are rotationally invariant, one also has with probability at least $1 - C n \exp ( - c d )$ that $\begin{array} { r } { \left\| \sum _ { \ell \in A } w _ { \ell } ^ { \prime } \right\| ^ { 2 } \leq C \cdot | A | \cdot m } \end{array}$ for all $A \subset \{ 1 , \ldots , k \}$ . Thus it only remains to control how large $A$ can be. We show below that $| A | \leq C m \log ( d )$ with probability at least $1 - C \exp ( - c d \log ( d ) )$ which will conclude the proof.
+
+$x$ activates neuron $\ell$ then $\begin{array} { r } { w _ { \ell } \cdot x \ge \frac { 1 } { 2 } \ge \frac { \| w _ { \ell } \| } { 4 \sqrt { m } } } \end{array}$ . Now note that for any fixed $x \in \mathbb { S } ^ { d - 1 }$ and fixed $\begin{array} { r } { A \subset [ k ] , \mathbb { P } \left( \forall \ell \in A , w _ { \ell } \cdot x \geq \frac { \| w _ { \ell } \| } { 4 \sqrt { m } } \right) \leq C \exp \left( - c | A | \frac { d } { m } \right) } \end{array}$ , so that
+
+$$
+\mathbb { P } \left( \exists A \subset [ k ] : | A | = a { \mathrm { ~ a n d } } \forall \ell \in A , w _ { \ell } \cdot x \geq { \frac { \| w _ { \ell } \| } { 4 { \sqrt { m } } } } \right) \leq \exp \left( C a \log ( k ) - c a { \frac { d } { m } } \right) .
+$$
+
+In particular we conclude that with $a = C m \log ( d )$ the probability that a fixed point on the sphere activates more than $a$ neuron is exponentially small in $d \log ( d )$ (recall that $m \bar { \log ( k ) } \leq c d$ by assumption). Thus we can conclude via an union bound on an $\varepsilon$ -net that the same holds for the entire sphere simultaneously. This concludes the proof.
+
+# D FURTHER RESULTS AROUND CONJECTURE 1
+
+# D.1 POLYNOMIAL ACTIVATION
+
+We now observe that one can generalize Theorem 5 to handle biases (the parameters $b _ { l }$ in 1), and in fact even general polynomial activation function. Indeed, observe that any polynomial of $\langle w , x \rangle + b$ must also be a polynomial in $\langle w , x \rangle$ , albeit with different coefficients.
+
+Theorem 6 Let $\begin{array} { r } { \psi ( t ) = \sum _ { q = 0 } ^ { p } \alpha _ { q } t ^ { q } } \end{array}$ and assume that we have $f \in \mathcal { F } _ { k } ( \psi )$ such that $f ( x _ { i } ) = y _ { i } , \forall i \in$ $[ n ]$ . Then, for generic data, with probability at least $1 - C \exp ( - c _ { p } d )$ one must have
+
+$$
+\begin{array} { r } { \mathrm { L i p } _ { \{ x : \| x \| \leq 1 \} } ( f ) \geq c _ { p } \sqrt { \frac { n } { d ^ { p - 1 } } } . } \end{array}
+$$
+
+Proof. Note that for $f \in \mathcal { F } _ { k } ( \psi )$ there exists tensors $T _ { 0 } , \ldots , T _ { p }$ , such that $T _ { q }$ is a tensor of order $q$ , and $f$ can be written as:
+
+$$
+f ( x ) = \sum _ { q = 0 } ^ { p } \langle T _ { q } , x ^ { \otimes q } \rangle .
+$$
+
+Now let us define $\begin{array} { r } { \Omega _ { q } = \sum _ { i = 1 } ^ { n } y _ { i } x _ { i } ^ { \otimes q } } \end{array}$ , and observe that
+
+$$
+n = \sum _ { i = 1 } ^ { n } y _ { i } f ( x _ { i } ) = \sum _ { q = 0 } ^ { p } \langle T _ { q } , \Omega _ { q } \rangle ,
+$$
+
+and thus there exists $q \in \{ 1 , \ldots , p \}$ such that $\langle T _ { q } , \Omega _ { q } \rangle \geq c _ { p } n$ (we ignore the term $q \ : = \ : 0$ by considering the largest balanced subset of the data, i.e. we assume $\textstyle \sum _ { i = 1 } ^ { n ^ { - } } y _ { i } = 0 )$ . Now one can repeat the proof of Theorem 5 to obtain that with probability at least $\mathrm { ~ i ~ } \overset { \cdot } { - } C \exp ( - c _ { p } d )$ , one has $\| T _ { q } \| _ { \mathrm { o p } } \geq c _ { p } \sqrt { \frac { n } { d ^ { p - 1 } } }$ . It only remains to observe that the Lipschitz constant of $f$ on the unit ball is lower bounded by $\| T _ { q } \| _ { \mathrm { o p } }$ .
+
+As we mentioned in Section 4.3, without loss of generality we can assume $T _ { q }$ is symmetric, and thus by (9) there exists $x \in \mathbb { S } ^ { d - 1 }$ such that $\| T _ { q } \| _ { \mathrm { o p } } = \langle T _ { q } , x ^ { \otimes q } \rangle$ . Now consider the univariate polynomial $P ( t ) = f ( t x )$ . By Markov brothers’ inequality one has $\begin{array} { r } { \operatorname* { m a x } _ { t \in [ - 1 , 1 ] } P ( t ) \geq | P ^ { ( q ) } ( 0 ) | = } \end{array}$ $q ! \cdot | \langle T _ { q } , x ^ { \otimes q } \rangle | = q ! \cdot \| T _ { q } \| _ { \mathrm { o p } }$ , thus concluding the proof. 
+
+# D.2 QUADRATIC ACTIVATION
+
+In Section 4.3 we obtained a lower bound for tensor networks that match Conjecture 1 only when the rank of the corresponding tensor is maximal. Here we show that for quadratic networks (i.e., $p = 2$ ) we can match Conjecture 1, and in fact even obtain a better bound, for any rank $k$ :
+
+Theorem 7 Assume that we have a matrix $T \in \mathbb { R } ^ { d \times d }$ with rank $k$ such that:
+
+$$
+\langle T , x _ { i } ^ { \otimes 2 } \rangle = y _ { i } , \forall i \in [ n ] .
+$$
+
+Then, for generic data, with probability at least $1 - C \exp ( - c d )$ , one must have
+
+$$
+\| T \| _ { \mathrm { o p } } \geq c { \frac { \sqrt { n d } } { k } } \ \left( \geq c { \sqrt { n / k } } \right) .
+$$
+
+Proof. The proof is exactly the same as for Theorem 5, except that in (8), instead of using Lemma 3 we use the fact that for a matrix $T$ of rank $k$ one has:
+
+$$
+\| T \| _ { \mathrm { o p , * } } \leq k \cdot \| T \| _ { \mathrm { o p } } .
+$$