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+# NEURAL ARCHITECTURE SEARCH ON IMAGENET IN FOUR GPU HOURS: A THEORETICALLY INSPIRED PERSPECTIVE
+
+Wuyang Chen, Xinyu Gong, Zhangyang Wang Department of Electrical and Computer Engineering The University of Texas at Austin, Austin, TX, USA {wuyang.chen,xinyu.gong,atlaswang}@utexas.edu
+
+# ABSTRACT
+
+Neural Architecture Search (NAS) has been explosively studied to automate the discovery of top-performer neural networks. Current works require heavy training of supernet or intensive architecture evaluations, thus suffering from heavy resource consumption and often incurring search bias due to truncated training or approximations. Can we select the best neural architectures without involving any training and eliminate a drastic portion of the search cost? We provide an affirmative answer, by proposing a novel framework called training-free neural architecture search (TE-NAS). TE-NAS ranks architectures by analyzing the spectrum of the neural tangent kernel (NTK) and the number of linear regions in the input space. Both are motivated by recent theory advances in deep networks and can be computed without any training and any label. We show that: (1) these two measurements imply the trainability and expressivity of a neural network; (2) they strongly correlate with the network’s test accuracy. Further on, we design a pruning-based NAS mechanism to achieve a more flexible and superior trade-off between the trainability and expressivity during the search. In NAS-Bench-201 and DARTS search spaces, TE-NAS completes high-quality search but only costs 0.5 and 4 GPU hours with one 1080Ti on CIFAR-10 and ImageNet, respectively. We hope our work inspires more attempts in bridging the theoretical findings of deep networks and practical impacts in real NAS applications. Code is available at: https://github.com/VITA-Group/TENAS.
+
+# 1 INTRODUCTION
+
+The recent development of deep networks significantly contributes to the success of computer vision. Thanks to many efforts by human designers, the performance of deep networks have been significantly boosted (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016; Xie et al., 2017). However, the manual creation of new network architectures not only costs enormous time and resources due to trial-and-error, but also depends on the design experience that does not always scale up. To reduce the human efforts and costs, neural architecture search (NAS) has recently amassed explosive interests, leading to principled and automated discovery for good architectures in a given search space of candidates (Zoph & Le, 2016; Brock et al., 2017; Pham et al., 2018; Liu et al., 2018a; Chen et al., 2018; Bender et al., 2018; Gong et al., 2019; Chen et al., 2020a; Fu et al., 2020).
+
+As an optimization problem, NAS faces two core questions: 1) “how to evaluate”, i.e. the objective function that defines what are good architectures we want; 2) “how to optimize”, i.e. by what means we could effectively optimize the objective function. These two questions are entangled and highly non-trivial, since the search spaces are of extremely high dimension, and the generalization ability of architectures cannot be easily inferred (Dong & Yang, 2020; Dong et al., 2020). Existing NAS methods mainly leverage the validation set and conduct accuracy-driven architecture optimization. They either formulate the search space as a super-network (“supernet”) and make the training loss differentiable through the architecture parameters (Liu et al., 2018b), or treat the architecture selection as a sequential decision making process (Zoph & Le, 2016) or evolution of genetics (Real et al., 2019). However, these NAS algorithms suffer from heavy consumption of both time and GPU resources. Training a supernet till convergence is extremely slow, even with many effective heuristics for sampling or channel approximations (Dong & Yang, 2019; Xu et al., 2019). Approximated proxy inference such as truncated training/early stopping can accelerate the search, but is well known to introduce search bias to the inaccurate results obtained (Pham et al., 2018; Liang et al., 2019; Tan et al., 2020). The heavy search cost not only slows down the discovery of novel architectures, but also blocks us from more meaningfully understanding the NAS behaviors.
+
+On the other hand, the analysis of neural network’s trainability (how effective a network can be optimized via gradient descent) and expressivity (how complex the function a network can represent) has witnessed exciting development recently in the deep learning theory fields. By formulating neural networks as a Gaussian Process (no training involved), the gradient descent training dynamics can be characterized by the Neural Tangent Kernel (NTK) of infinite (Lee et al., 2019) or finite (Yang, 2019) width networks, from which several useful measures can be derived to depict the network trainability at the initialization. Hanin & Rolnick (2019a;b); Xiong et al. (2020) describe another measure of network expressivity, also without any training, by counting the number of unique linear regions that a neural network can divide in its input space. We are therefore inspired to ask:
+
+• How to optimize NAS at network’s initialization without involving any training, thus significantly eliminating a heavy portion of the search cost?   
+• Can we define how to evaluate in NAS by analyzing the trainability and expressivity of architectures, and further benefit our understanding of the search process?
+
+Our answers are yes to both questions. In this work, we propose TE-NAS, a framework for trainingfree neural architecture search. We leverage two indicators, the condition number of NTK and the number of linear regions, that can decouple and effectively characterize the trainability and expressivity of architectures respectively in complex NAS search spaces. Most importantly, these two indicators can be measured in a training-free and label-free manner, thus largely accelerates the NAS search process and benefits the understanding of discovered architectures. To our best knowledge, TE-NAS makes the first attempt to bridge the theoretical findings of deep neural networks and real-world NAS applications. While we intend not to claim that the two indicators we use are the only nor the best options, we hope our work opens a door to theoretically-inspired NAS and inspires the discovery of more deep network indicators. Our contributions are summarized as below:
+
+• We identify and investigate two training-free and label-free indicators to rank the quality of deep architectures: the spectrum of their NTKs, and the number of linear regions in their input space. Our study finds that they reliably indicate the trainability and expressivity of a deep network respectively, and are strongly correlated with the network’s test accuracy.   
+• We leverage the above two theoretically-inspired indicators to establish a training-free NAS framework, TE-NAS, therefore eliminating a drastic portion of the search cost. We further introduce a pruning-based mechanism, to boost search efficiency and to more flexibly trade-off between trainability and expressivity.   
+In NAS-Bench-201/DARTS search spaces, TE-NAS discovers architectures with a strong performance at remarkably lower search costs, compared to previous efforts. With just one $1 0 8 0 \mathrm { T i }$ , it only costs 0.5 GPU hours to search on CIFAR10, and 4 GPU hours on ImageNet, respectively, setting the new record for ultra-efficient yet high-quality NAS.
+
+# 2 RELATED WORKS
+
+Neural architecture search (NAS) is recently proposed to accelerate the principled and automated discovery of high-performance networks. However, most works suffer from heavy search cost, for both weight-sharing based methods (Liu et al., 2018b; Dong & Yang, 2019; Liu et al., 2019; Yu et al., 2020a; Li et al., 2020a; Yang et al., 2020a) and single-path sampling-based methods (Pham et al., 2018; Guo et al., 2019; Real et al., 2019; Tan et al., 2020; Li et al., 2020c; Yang et al., 2020b). A one-shot super network can share its parameters to sampled sub-networks and accelerate the architecture evaluations, but it is very heavy and hard to optimize and suffers from a poor correlation between its accuracy and those of the sub-networks (Yu et al., 2020c). Sampling-based methods achieve more accurate architecture evaluations, but their truncated training still imposes bias to the performance ranking since this is based on the results of early training stages.
+
+Instead of estimating architecture performance by direct training, people also try to predict network’s accuracy (or ranking), called predictor based NAS methods (Liu et al., 2018a; Luo et al., 2018; Dai et al., 2019; Luo et al., 2020). Graph neural network (GNN) is a popular choice as the predictor model (Wen et al., 2019; Chen et al., 2020b). Siems et al. (2020) even propose the first large-scale surrogate benchmark, where most of the architectures’ accuracies are predicted by a pretrained GNN predictor. The learned predictor can achieve highly accurate performance evaluation. However, the data collection step - sampling representative architectures and train them till converge - still requires extremely high cost. People have to sample and train 2,000 to 50,000 architectures to serve as the training data for the predictor. Moreover, none of these works can demonstrate the cross-space transferability of their predictors. This means one has to repeat the data collection and predictor training whenever facing an unseen search space, which is highly nonscalable.
+
+The heavy cost of architecture evaluation hinders the understanding of the NAS search process. Recent pioneer works like Shu et al. (2019) observed that DARTS and ENAS tend to favor architectures with wide and shallow cell structures due to their smooth loss landscape. Siems et al. (2020) studied the distribution of test error for different cell depths and numbers of parameter-free operators. Chen & Hsieh (2020) for the first time regularizes the Hessian norm of the validation loss and visualizes the smoother loss landscape of the supernet. Li et al. (2020b) proposed to approximate the validation loss landscape by learning a mapping from neural architectures to their corresponding validate losses. Still, these analyses cannot be directly leveraged to guide the design of network architectures.
+
+Mellor et al. (2020) recently proposed a NAS framework that does not involve training, which shares the same motivation with us towards training-free architecture search at initialization. They empirically find that the correlation between sample-wise input-output Jacobian can indicate the architecture’s test performance. However, why does the Jacobian work is not well explained and demonstrated. Their search performance on NAS-Bench-201 is still left behind by the state-of-the-art NAS works, and they did not extend to DARTs space.
+
+Meanwhile, we see the evolving development of deep learning theory on neural networks. NTK (neural tangent kernel) is proposed to characterize the gradient descent training dynamics of infinite wide (Jacot et al., 2018) or finite wide deep networks (Hanin & Nica, 2019). Wide networks are also proved to evolve as linear models under gradient descent (Lee et al., 2019). This is further leveraged to decouple the trainability and generalization of networks (Xiao et al., 2019). Besides, a natural measure of ReLU network’s expressivity is the number of linear regions it can separate in its input space (Raghu et al., 2017; Montufar, 2017; Serra et al., 2018; Hanin & Rolnick, 2019a;b; Xiong ´ et al., 2020). In our work, we for the first time discover two important indicators that can effectively rank architectures, thus bridging the theoretic findings and real-world NAS applications. Instead of claiming the two indicators we discover are the best, we believe there are more meaningful properties of deep networks that can benefit the architecture search process. We leave them as open questions and encourage the community to study.
+
+# 3 METHODS
+
+The core motivation of our TE-NAS framework is to achieve architecture evaluation without involving any training, to significantly accelerate the NAS search process and reduce the search cost. In section 3.1 we present our study on two important indicators that reflect the trainability and expressivity of a neural network, and in section 3.2 we design a novel pruning-based method that can achieve a superior trade-off between the two indicators.
+
+# 3.1 ANALYZING TRAINABILITY AND EXPRESSIVITY OF DEEP NETWORKS
+
+Trainability and expressivity are distinct notions regarding a neural network (Xiao et al., 2019). A network can achieve high performance only if the function it can represent is complex enough and at the same time, it can be effectively trained by gradient descent.
+
+# 3.1.1 TRAINABILITY BY CONDITION NUMBER OF NTK
+
+The trainability of a neural network indicates how effective it can be optimized using gradient descent (Burkholz & Dubatovka, 2019; Hayou et al., 2019; Shin & Karniadakis, 2020). Although some heavy networks can theoretically represent complex functions, they not necessarily can be effectively trained by gradient descent. One typical example is that, even with a much more number of parameters, $\mathrm { V g g }$ networks (Simonyan & Zisserman, 2014) usually perform worse and require more special engineering tricks compared with ResNet family (He et al., 2016), whose superior trainability property is studied by Yang & Schoenholz (2017).
+
+Recent work (Jacot et al., 2018; Lee et al., 2019; Chizat et al., 2019) studied the gradient descent training of neural networks using a quantity called the neural tangent kernel (NTK). The finite width NTK is defined by $\hat { \Theta } ( \pmb { x } , \pmb { x } ^ { \prime } ) = \bar { J } ( \pmb { x } ) \bar { J } ( \pmb { x } ^ { \prime } ) ^ { T }$ , where $J _ { i \alpha } ( { \pmb x } ) = \partial _ { \theta _ { \alpha } } z _ { i } ^ { L } ( { \pmb x } )$ is the Jacobian evaluated at a point $_ { \textbf { \em x } }$ for parameter $\theta _ { \alpha }$ , and $z _ { i } ^ { L }$ is the output of the $i$ -th neuron in the last output layer $L$ .
+
+Lee et al. (2019) further proves that wide neural networks evolve as linear models using gradient descent, and their training dynamics is controlled by ODEs that can be solved as
+
+$$
+\mu _ { t } ( X _ { \mathrm { t r a i n } } ) = ( { \bf I } - e ^ { - \eta \hat { \Theta } _ { \mathrm { t r a i n } } t } ) Y _ { \mathrm { t r a i n } }
+$$
+
+for training data. Here $\mu _ { t } ( \pmb { x } ) = \mathbb { E } [ z _ { i } ^ { L } ( \pmb { x } ) ]$ is the expected outputs of an infinitely wide network. $\hat { \Theta } _ { \mathrm { t r a i n } }$ denotes the NTK between the training inputs, and $X _ { \mathrm { t r a i n } }$ and $Y _ { \mathrm { t r a i n } }$ are drawn from the training set $\mathcal { D } _ { \mathrm { t r a i n } }$ . As the training step $t$ tends to infinity we can see that Eq. 1 reduce to $\mu ( X _ { \mathrm { t r a i n } } ) = Y _ { \mathrm { t r a i n } } $ .
+
+The relationship between the conditioning of $\hat { \Theta }$ and the trainability of networks is studied by Xiao et al. (2019), and we brief the conclusion as below. We can write Eq. 1 in terms of the spectrum of $\Theta$ :
+
+$$
+\mu _ { t } ( X _ { \mathrm { t r a i n } } ) _ { i } = ( \mathbf { I } - e ^ { - \eta \lambda _ { i } t } ) Y _ { \mathrm { t r a i n } , i } ,
+$$
+
+where $\lambda _ { i }$ are the eigenvalues of $\hat { \Theta } _ { \mathrm { t r a i n } }$ and we order the eigenvalues $\lambda _ { 0 } \geq \cdots \geq \lambda _ { m }$ . As it has been hypothesized by Lee et al. (2019) that the maximum feasible learning rate scales as $\eta \sim 2 / \lambda _ { 0 }$ , plugging this scaling for $\eta$ into Eq. 2 we see that the $\lambda _ { m }$ will converge exponentially at a rate given by $1 / \kappa$ where $\kappa = \lambda _ { 0 } / \lambda _ { m }$ is the condition number. Then we can conclude that if the $\kappa$ of the NTK associated with a neural network diverges then it will become untrainable, so we use $\kappa$ as a metric for trainability:
+
+![](images/3362e810a2898a515ca6d05037524ddffdc34e876ff29b129e781c046234c2f8.jpg)  
+Figure 1: Condition number of NTK $\kappa _ { \mathcal { N } }$ exhibits negative correlation with the test accuracy of architectures in NAS-Bench201 (Dong & Yang, 2020).
+
+$$
+\kappa _ { \mathcal { N } } = \frac { \lambda _ { 0 } } { \lambda _ { m } } .
+$$
+
+$\kappa _ { \mathcal { N } }$ is calculated without any gradient descent or label. Figure 1 demonstrates that the $\kappa _ { \mathcal { N } }$ is negatively correlated with the architecture’s test accuracy, with the Kendall-tau correlation as $- 0 . 4 2$ . Therefore, minimizing the $\kappa _ { \mathcal { N } }$ during the search will encourage the discovery of architectures with high performance.
+
+# 3.1.2 EXPRESSIVITY BY NUMBER OF LINEAR REGIONS
+
+The expressivity of a neural network indicates how complex the function it can represent (Hornik et al., 1989; Giryes et al., 2016). For ReLU networks, each ReLU function defines a linear boundary and divides its input space into two regions. Since the composition of piecewise linear functions is still piecewise linear, every ReLU network can be seen as a piecewise linear function. The input space of a ReLU network can be partitioned into distinct pieces (i.e. linear regions) (Figure 2), each of which is associated with a set of affine parameters, and the function represented by the network is affine when restricted to each piece. Therefore, it is natural to measure the expressivity of a ReLU network with the number of linear regions it can separate.
+
+![](images/7982522b6a745b07e4a9d64194fc492c3cc1ea03f68b2d71292f0590ce4f610a.jpg)  
+Figure 2: Example of linear regions divided by a ReLU network1
+
+Following Raghu et al. (2017); Montufar (2017); Serra et al. (2018); Hanin & Rolnick´ $( 2 0 1 9 \mathrm { a } ; \mathbf { b } )$ Xiong et al. (2020), we first introduce the following definition of activation patterns and linear regions for ReLU CNNs.
+
+Definition 1. Activation Patterns and Linear Regions (Xiong et al. (2020)) Let $\mathcal { N }$ be a ReLU CNN. An activation pattern of $\mathcal { N }$ is a function $_ { r }$ from the set of neurons to $\{ 1 , - 1 \}$ , i.e., for each neuron $z$ in $\mathcal { N }$ , we have $P ( z ) \in \{ 1 , - 1 \}$ . Let $\theta$ be a fixed set of parameters (weights and biases) in $\mathcal { N }$ , and $_ { r }$ be an activation pattern. The region corresponding to $_ { r }$ and $\theta$ is
+
+$$
+\begin{array} { r } { R ( P ; \theta ) : = \{ { \pmb x } ^ { 0 } \in \mathbb { R } ^ { C \times H \times W } : z ( { \pmb x } ^ { 0 } ; \theta ) \cdot { \pmb P } ( z ) > 0 , \quad \forall z \in \mathcal { N } \} , } \end{array}
+$$
+
+where $z ( \pmb { x } ^ { 0 } ; \theta )$ is the pre-activation of a neuron $z$ . Let $R _ { \mathcal { N } , \theta }$ denote the number of linear regions of $\mathcal { N }$ at $\theta$ , i.e., $R _ { \mathcal { N } , \theta } : = \# \{ R ( P ; \theta ) : R ( P ; \theta ) \neq \emptyset$ for some activation pattern $P \}$ .
+
+Eq. 4 tells us that a linear region in the input space is a set of input data $\mathbf { x } ^ { 0 }$ that satisfies a certain fixed activation pattern $\bar { \boldsymbol { P } } ( z )$ , and therefore the number of linear regions $R _ { \mathcal { N } , \theta }$ measures how many unique activation patterns that can be divided by the network.
+
+In our work, we repeat the measurement of the number of linear regions by sampling network parameters from the Kaiming Norm Initialization (He et al., 2015), and calculate the average as the approximation to its expectation:
+
+$$
+\hat { R } _ { \mathcal { N } } \simeq \mathbb { E } _ { \theta } R _ { \mathcal { N } , \theta }
+$$
+
+![](images/b2d93de6fbb1a383c41afa3257e7e64a8de68570ea962a420a5a43e77c224850.jpg)  
+Figure 3: Number of linear regions $\hat { R } _ { \mathcal { N } }$ of architectures in NAS-Bench201 exhibits positive correlation with test accuracies.
+
+We iterate through all architectures in NAS-Bench-201 (Dong & Yang, 2020), and calculate their numbers of linear regions (without any gradient descent or label). Figure 3 demonstrates that the number of linear regions is positively correlated with the architecture’s test accuracy, with the Kendall-tau correlation as 0.5. Therefore, maximizing the number of linear regions during the search will also encourage the discovery of architectures with high performance.
+
+Finally, in Figure 4 we analyze the operator composition of top $10 \%$ architecture by maximizing $\hat { R } _ { \mathcal { N } }$ and minimizing $\kappa _ { \mathcal { N } }$ , respectively. We can clearly see that $\hat { R } _ { \mathcal { N } }$ and $\kappa _ { \mathcal { N } }$ have different preferences for choosing operators. They both choose a large ratio of conv $3 \times 3$ for high generalization performance. But meanwhile, $\hat { R } _ { \mathcal { N } }$ heavily selects $\mathbf { c o n v 1 } \times 1$ , and $\kappa _ { \mathcal { N } }$ leads to skip-connect, favoring the gradient flow.
+
+![](images/a87c3990390e3008ac1c0860925cb464148f6012c6ea58ca9b73b32873371cb6.jpg)  
+Figure 4: $\kappa { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ prefer different operators in NAS-Bench201.
+
+# 3.2 PRUNING-BY-IMPORTANCE ARCHITECTURE SEARCH
+
+Given the strong correlation between the architecture’s test accuracy and its $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ , how to build an efficient NAS framework on top of them? We motivate this section by addressing two questions:
+
+1. How to combine $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ together, with a good explicit trade-off?
+
+We first need to turn the two measurements $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ into one combined function, based on which we can rank architectures. As seen in Figure 1 and 3, the magnitudes of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ differ much. To avoid one overwhelming the other numerically, one possible remedy is normalization; but we cannot pre-know the ranges nor the value distributions of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ , before computing them over a search space. In order to make our combined function well defined before search and agnostic to the search space, instead of using the numerical values of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ , we could refer to their relative rankings. Specifically, each time by comparing the sampled set of architectures peer-to-peer, we can directly sum up the two relative rankings of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } } ^ { \bar { } }$ as the selection criterion. The equal-weight summation treats trainability and expressivity with the same importance conceptually1 and delivers the best empirical result: we thus choose it as our default combined function. We also tried some other means to combine the two, and the ablation studies can be found in Appendix D.2.
+
+# 2. How to search more efficiently?
+
+Sampling-based methods like reinforcement learning or evolution can use rankings as the reward or filtering metric. However, they are inefficient, especially for complex cell-based search space. Consider a network stacked by repeated cells (directed acyclic graphs) (Zoph et al., 2018; Liu et al., 2018b). Each cell has $E$ edges, and on each edge we only select one operator out of $| \mathcal { O } |$ ( $\mathcal { O }$ is the set of operator candidates). There are $| \mathcal { O } | ^ { E }$ unique cells, and for sampling-based methods, $\gamma \cdot | \mathcal { O } | ^ { E }$ networks have to be sampled during the search. The ratio $\gamma$ can be interpreted as the sampling efficiency: a method with small $\gamma$ can find good architectures faster. However, the search time cost of sampling-based methods still scales up with the size of the search space, i.e., $| \mathcal { O } | ^ { E }$ .
+
+Inspired by recent works on pruning-from-scratch (Lee et al., 2018; Wang et al., 2020), we propose a pruning-by-importance NAS mechanism to quickly shrink the search possibilities and boost the efficiency further, reducing the cost from $| \mathcal { O } | ^ { E }$ to $| \mathcal { O } | \cdot E$ . Specifically, we start the search with a super-network $\mathcal { N } _ { 0 }$ composed of all possible operators and edges. In the outer loop, for every round we prune one operator on each edge. The outer-loop stops when the current supernet $\mathcal { N } _ { t }$ is a single-path network2, i.e., the algorithm will return us the final searched architecture. For the inner-loop, we measure the change of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ before and after pruning each individual operator, and assess its importance using the sum of two ranks. We order all currently available operators in terms of their importance, and prune the lowest-importance operator on each edge.
+
+The whole pruning process is extremely fast. As we will demonstrate later, our approach is principled and can be applied to different spaces without making any modifications. This pruning-by-importance mechanism may also be extended to indicators beyond $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ . We summarize our training-free and pruning-based NAS framework, TE-NAS, in Algorithm 1.
+
+# Algorithm 1: TE-NAS: Training-free Pruning-based NAS via Ranking of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$
+
+1 Input: supernet $\mathcal { N } _ { 0 }$ stacked by cells, each cell has $E$ edges, each edge has $| \mathcal { O } |$ operators, step $t = 0$ .   
+2 while $\mathcal { N } _ { t }$ is not a single-path network do   
+3 for each operator $o _ { j }$ in $\mathcal { N } _ { t }$ do   
+4 $\begin{array} { r l } & { \dot { \bf \Delta } \Delta \kappa _ { t , o _ { j } } = \kappa _ { \mathcal { N } _ { t } } \dot { \bf \Delta } - \kappa _ { \mathcal { N } _ { t } \backslash o _ { j } } } \\ & { \Delta R _ { t , o _ { j } } = R _ { \mathcal { N } _ { t } } - R _ { \mathcal { N } _ { t } \backslash o _ { j } } } \end{array}$ . the higher $\Delta \kappa _ { t , o _ { j } }$ the more likely we will prune $o _ { j }$   
+5 $\triangleright$ the lower $\Delta R _ { t , o _ { j } }$ the more likely we will prune $o _ { j }$   
+6 Get importance by $\kappa _ { \mathcal { N } }$ : $s _ { \kappa } ( o _ { j } ) =$ index of $o _ { j }$ in descendingly sorted list $[ \Delta \kappa _ { t , o _ { 1 } } , . . . , \Delta \kappa _ { t , o _ { | \mathcal { N } _ { t } | } } ]$   
+7 Get importance by $R _ { \mathcal { N } }$ : $s _ { R } ( o _ { j } ) =$ index of $o _ { j }$ in ascendingly sorted list $[ \Delta R _ { t , o _ { 1 } } , . . . , \Delta R _ { t , o _ { | \mathcal { N } _ { t } | } } ]$   
+8 Get importance $s ( o _ { j } ) = s _ { \kappa } ( o _ { j } ) + s _ { R } ( o _ { j } )$   
+9 $\mathcal { N } _ { t + 1 } \dot { = } \mathcal { N } _ { t }$   
+10 for each edge $e _ { i }$ , $i = 1 , . . . , E$ do   
+11 $j ^ { * } = \arg \operatorname* { m i n } _ { j } \{ s ( o _ { j } ) : o _ { j } \in e _ { i } \}$ . find the operator with greatest importance on each edge.   
+12 $\mathcal { N } _ { t + 1 } = \mathcal { N } _ { t + 1 } \backslash o _ { j ^ { * } }$   
+13 $t = t + 1$
+
+# 3.2.1 VISUALIZATION OF SEARCH PROCESS
+
+TE-NAS benefits us towards a better understanding of the search process. We can analyze the trajectory of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ during the search. It is worth noting that our starting point $\mathcal { N } _ { 0 }$ , the un-pruned supernet, is assumed to be of the highest expressivity (as it is composed of all operators in the search space and has the largest number of parameters and ReLU functions). However, it has poor trainability, as people find many engineering techniques are required to effectively training the supernet (Yu et al., 2020a;b). Therefore, during pruning we are expecting to strengthen the trainability of the supernet, while retaining its expressivity as much as possible.
+
+![](images/5eab6423b3234fdb8c4dc99767d6da291ff972a600d6a8e262fa1295d7f97422.jpg)  
+Figure 5: Pruning trajectory on NAS-Bench-201 (top) and DARTs search space (bottom). Number $\cdot _ { 0 } \cdot \mathbf { \vec { \tau } }$ indicates the supernet $\mathcal { N } _ { 0 }$ before any pruning, which is of high expressivity but poor trainability.
+
+As we observe in Figure 5, the supernet $\mathcal { N }$ is first pruned by quickly reducing $\kappa _ { \mathcal { N } }$ , i.e., increasing the network’s trainability. After that, as the improvement of $\kappa _ { \mathcal { N } }$ is almost plateaued, the method carefully fine-tunes the architecture without sacrificing too much expressivity $\hat { R } _ { \mathcal { N } }$ .
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate our TE-NAS on two search spaces: NAS-Bench-201 (Dong & Yang, 2020) and DARTS (Liu et al., 2018b). Search and training protocols are summarized in Appendix A. Our code is available at: https://github.com/VITA-Group/TENAS.
+
+# 4.1 RESULTS ON NAS-BENCH-201
+
+NAS-Bench-201 (Dong & Yang, 2020) provides a standard cell-based search space (containing 15,625 architectures) and a database of architecture’s performance evaluated under a unified protocol. The network’s test accuracy can be directly obtained by querying the database, which facilitates people to focus on studying NAS algorithms without network evaluation. NAS-Bench-201 supports three datasets (CIFAR-10, CIFAR-100, ImageNet-16-120 (Chrabaszcz et al., 2017)). The operation space contains none (zero), skip connection, $c o n \nu 1 \times 1$ , $c o n \nu 3 \times 3$ convolution, and average pooling $3 \times 3$ . We refer to their paper for details of the space. Our search is dataset-specific, i.e. the search and evaluation are conducted on the same dataset.
+
+Table 1: Comparison with state-of-the-art NAS methods on NAS-Bench-201. Test accuracy with mean and deviation are reported. “optimal” indicates the best test accuracy achievable in NAS-Bench-201 search space.   
+
+<table><tr><td>Architecture</td><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet-16-120</td><td>Search Cost (GPU sec.)</td><td>Search Method</td></tr><tr><td>ResNet (He et al.,2016)</td><td>93.97</td><td>70.86</td><td>43.63</td><td>-</td><td>-</td></tr><tr><td>RSPS (Li&amp; Talwalkar,2020)</td><td>87.66(1.69)</td><td>58.33(4.34)</td><td>31.14(3.88)</td><td>8007.13</td><td>random</td></tr><tr><td>ENAS (Pham et al., 2018)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>13314.51</td><td>RL</td></tr><tr><td>DARTS (1st) (Liu et al.,2018b)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>10889.87</td><td>gradient</td></tr><tr><td>DARTS (2nd) (Liu et al.,2018b)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>29901.67</td><td>gradient</td></tr><tr><td>GDAS (Dong&amp; Yang,2019)</td><td>93.61(0.09)</td><td>70.70(0.30)</td><td>41.84(0.90)</td><td>28925.91</td><td>gradient</td></tr><tr><td>NAS w.o. Training (Mellor et al.,2020)</td><td>91.78(1.45)</td><td>67.05(2.89)</td><td>37.07(6.39)</td><td>4.8</td><td>training-free</td></tr><tr><td>TE-NAS (ours)</td><td>93.9(0.47)</td><td>71.24(0.56)</td><td>42.38(0.46)</td><td>1558</td><td>training-free</td></tr><tr><td> Optimal</td><td>94.37</td><td>73.51</td><td>47.31</td><td>-</td><td>-</td></tr></table>
+
+We run TE-NAS for four independent times with different random seeds, and report the mean and standard deviation in Table 1. We can see that TE-NAS achieves the best accuracy on all three datasets, and largely reduces the search cost $( 5 \times \sim 1 9 \times$ reduction). Although Mellor et al. (2020) requires even less search time (by only sampling 25 architectures), they suffer from much inferior accuracy performance, with notably larger deviations across different search rounds.
+
+# 4.2 RESULTS ON CIFAR-10 WITH DARTS SEARCH SPACE
+
+Architecture Space The DARTs operation space $\mathcal { O }$ contains eight choices: none (zero), skip connection, separable convolution $3 \times 3$ and $5 \times 5$ , dilated separable convolution $3 \times 3$ and $5 \times 5$ , max pooling $3 \times 3$ , average pooling $3 \times 3$ . Following previous works (Liu et al., 2018b; Chen et al., 2019; $\mathrm { X u }$ et al., 2019), for evaluation phases, we stack 20 cells to compose the network and set the initial channel number as 36. We place the reduction cells at the 1/3 and 2/3 of the network and each cell consists of six nodes.
+
+Table 2: Comparison with state-of-the-art NAS methods on CIFAR-10.   
+
+<table><tr><td>Architecture</td><td>Test Error (%)</td><td>Params (M)</td><td>Search Cost (GPU days)</td><td>Search Method</td></tr><tr><td>AmoebaNet-A (Real et al., 2019)</td><td>3.34(0.06)</td><td>3.2</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al.,2018a)*</td><td>3.41(0.09)</td><td>3.2</td><td>225</td><td>SMBO</td></tr><tr><td>ENAS (Pham et al., 2018)</td><td>2.89</td><td>4.6</td><td>0.5</td><td>RL</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>2.65</td><td>3.3</td><td>2000</td><td>RL</td></tr><tr><td>DARTS (1st) (Liu et al.,2018b)</td><td>3.00(0.14)</td><td>3.3</td><td>0.4</td><td>gradient</td></tr><tr><td>DARTS (2nd) (Liu et al., 2018b)</td><td>2.76(0.09)</td><td>3.3</td><td>1.0</td><td>gradient</td></tr><tr><td>SNAS (Xie et al.,2018)</td><td>2.85(0.02)</td><td>2.8</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>2.82</td><td>2.5</td><td>0.17</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>2.81(0.04)</td><td>3.4</td><td>0.2</td><td>gradient</td></tr><tr><td>ProxylessNAS (Cai et al.,2018)†</td><td>2.08</td><td>5.7</td><td>4.0</td><td>gradient</td></tr><tr><td>P-DARTS (Chen et al.,2019)</td><td>2.50</td><td>3.4</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (Xu et al.,2019)</td><td>2.57(0.07)</td><td>3.6</td><td>0.1</td><td>gradient</td></tr><tr><td>SDARTS-ADV (Chen &amp; Hsieh,2020)</td><td>2.61(0.02)</td><td>3.3</td><td>1.3</td><td>gradient</td></tr><tr><td>TE-NAS (ours)</td><td>2.63(0.064)</td><td>3.8</td><td>0.05</td><td>training-free</td></tr></table>
+
+? No cutout augmentation. † Different space: PyramidNet (Han et al., 2017) as the backbone. ‡ Recorded on a single GTX 1080Ti GPU.
+
+Results We run TE-NAS for four independent times with different random seeds, and report the mean and standard deviation. Table 2 summarizes the performance of TE-NAS compared with other popular NAS methods. TE-NAS achieves a test error of $2 . 6 3 \%$ , ranking among the top of recent NAS results, but meanwhile largely reduces the search cost to only 0.05 GPU-day. ProxylessNAS achieves the lowest test error, but it searches on a different space with a much longer search time and has a larger model size. Besides, Mellor et al. (2020) did not extend to their Jacobian-based framework to DARTs search space for CIFAR-10 or ImageNet classification.
+
+# 4.3 RESULTS ON IMAGENET WITH DARTS SEARCH SPACE
+
+Architecture Space Following previous works $\mathrm { { X u } }$ et al., 2019; Chen et al., 2019), the architecture for ImageNet is slightly different from that for CIFAR-10. During retraining evaluation, the network is stacked with 14 cells with the initial channel number set to 48, and we follow the mobile setting to control the FLOPs not exceed $6 0 0 { \mathrm { M B } }$ by adjusting the channel number. The spatial resolution is downscaled from $2 2 4 \times 2 2 4$ to $2 8 \times 2 8$ with the first three convolution layers of stride 2.
+
+Results As shown in Table 3, we achieve a top-1/5 test error of $2 4 . 5 \% / 7 . 5 \%$ , achieving competitive performance with recent state-of-the-art works in the ImageNet mobile setting. However, TE-NAS only cost four GPU hours with only one 1080Ti. Searching on ImageNet takes a longer time than on CIFAR-10 due to the larger input size and more network parameters.
+
+Table 3: Comparison with state-of-the-art NAS methods on ImageNet under the mobile setting.   
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Test Error(%)</td><td rowspan="2">Params (M)</td><td rowspan="2">Search Cost (GPU days)</td><td rowspan="2">Search Method</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>26.0</td><td>8.4</td><td>5.3</td><td>2000</td><td>RL</td></tr><tr><td>AmoebaNet-C (Real et al.,2019)</td><td>24.3</td><td>7.6</td><td>6.4</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>25.8</td><td>8.1</td><td>5.1</td><td>225</td><td>SMBO</td></tr><tr><td>MnasNet-92 (Tan et al.,2019)</td><td>25.2</td><td>8.0</td><td>4.4</td><td>1</td><td>RL</td></tr><tr><td>DARTS (2nd) (Liu et al.,2018b)</td><td>26.7</td><td>8.7</td><td>4.7</td><td>4.0</td><td>gradient</td></tr><tr><td>SNAS (mild) (Xie et al.,2018)</td><td>27.3</td><td>9.2</td><td>4.3</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>26.0</td><td>8.5</td><td>5.3</td><td>0.21</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>26.5</td><td>8.9</td><td>3.9</td><td>0.2</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-10) (Chen et al., 2019)</td><td>24.4</td><td>7.4</td><td>4.9</td><td>0.3</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-100)(Chen et al.,2019)</td><td>24.7</td><td>7.5</td><td>5.1</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (CIFAR-1O)(Xu et al.,2019)</td><td>25.1</td><td>7.8</td><td>5.3</td><td>0.1</td><td>gradient</td></tr><tr><td>TE-NAS (ours)</td><td>26.2</td><td>8.3</td><td>6.3</td><td>0.05</td><td>training-free</td></tr><tr><td>PC-DARTS (ImageNet) (Xu et al., 2019)†</td><td>24.2</td><td>7.3</td><td>5.3</td><td>3.8</td><td>gradient</td></tr><tr><td>ProxylessNAS (GPU) (Cai et al.,2018)†</td><td>24.9</td><td>7.5</td><td>7.1</td><td>8.3</td><td>gradient</td></tr><tr><td>TE-NAS (ours)†</td><td>24.5</td><td>7.5</td><td>5.4</td><td>0.17</td><td>training-free</td></tr></table>
+
+† The architecture is searched on ImageNet, otherwise it is searched on CIFAR-10 or CIFAR-100.
+
+# 5 CONCLUSION
+
+The key questions in Neural Architecture Search (NAS) are “what are good architectures” and “how to find them”. Validation loss or accuracy are possible answers but not enough, due to their search bias and heavy evaluation cost. Our work demonstrates that two theoretically inspired indicators, the spectrum of NTK and the number of linear regions, not only strongly correlate with the network’s performance, but also benefit the reduced search cost and decoupled analysis of the network’s trainability and expressivity. Without involving any training, our TE-NAS achieve competitive NAS performance with minimum search time. We for the first time bridge the gap between the theoretic findings of deep neural networks and real-world NAS applications, and we encourage the community to further explore more meaningful network properties so that we will have a better understanding of good architectures and how to search them.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by the NSF Real-Time Machine Learning program (Award Number: 2053279), and the US Army Research Office Young Investigator Award (W911NF2010240).
+
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+
+# A IMPLEMENTATION DETAILS
+
+For $\kappa _ { \mathcal { N } }$ we sample one mini-batch of size 32 from the training set, and calculate $\hat { \Theta } ( \pmb { x } , \pmb { x } ^ { \prime } ) =$ $J ( { \pmb x } ) J ( { \pmb x } ^ { \prime } ) ^ { T }$ . For $\hat { R } _ { \mathcal { N } }$ we sample 5000 images, forward them through the network, and collect the activation patterns from all ReLU layers. The calculation of both $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ are repeated three times in all experiments, where each time the network weights are randomly drawn from Kaiming Norm Initialization (He et al., 2015) without involving any training (network weights are fixed).
+
+Our retraining settings (after search) follow previous works (Xu et al., 2019; Chen et al., 2019; Chen & Hsieh, 2020). On CIFAR-10, we train the searched network with cutout regularization of length 16, drop-path (Zoph et al., 2018) with probability as 0.3, and an auxiliary tower of weight 0.4. On ImageNet, we also use label smoothing during training. On both CIFAR-10 and ImageNet, the network is optimized by an SGD optimizer with cosine annealing, with learning rate initialized as 0.025 and 0.5, respectively.
+
+# B SEARCHED ARCHITECTURE
+
+We visualize the searched normal and reduction cells in figure 6 and 7, which is directly searched on CIFAR-10 and ImageNet respectively.
+
+![](images/3ea788d5aa0e8499282a29795a6e0b6706094d3bcfd951b8e4cd7154e1600ecd.jpg)  
+Figure 6: Normal and Reduction cells discovered by TE-NAS on CIFAR-10.
+
+![](images/0cf482289933baf96e10aac9ca88e6c877afab0dbcbb7c332a85deacc8e0fadf.jpg)  
+Figure 7: Normal and Reduction cells discovered by TE-NAS on imageNet.
+
+C DEPTH AND WIDTH PREFERENCE OF $\kappa _ { \mathcal { N } }$ AND $\hat { R } _ { \mathcal { N } }$ IN DARTS SPACE
+
+To analyze the impact of different architectures on trainability and expressivity in DARTs search space, we visualize $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ with different depths and width. Following Shu et al. (2019), the depth of a cell is defined as the number of connections on the longest path from input nodes to the output node, and the width of a cell is the summation of the edges of the intermediate nodes that are connected to the input nodes. We randomly sample 20,000 architectures in DARTs space, and plot the visualizations in Figure 8. Good architectures should exhibit low $\kappa _ { \mathcal { N } }$ (good trainability, blue dots in Figure 8(a)) and high $\hat { R } _ { \mathcal { N } }$ (powerful expressivity, red dots in Figure 8(b)). Therefore, $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ tell us that in DARTs space shallow but wide cells are preferred to favor both trainability and expressivity. This conclusion matches the findings by Shu et al. (2019): existing NAS algorithms tend to favor architectures with wide and shallow cell structures, which enjoy fast convergence with smooth loss landscape and accurate gradient information.
+
+![](images/68fb7e3b692da7dff551648648324bbc0476d9fbdc1fda9a6ac2d7d65ff25541.jpg)  
+Figure 8: Depth and width preference of (a) $\kappa { \mathcal N }$ and (b) $\hat { R } _ { \mathcal { N } }$ on DARTs Search Space.
+
+# D MORE ABLATION STUDIES
+
+D.1 SEARCH WITH ONLY $\kappa _ { \mathcal { N } }$ OR $\hat { R } _ { \mathcal { N } }$
+
+As we observed in Table 4, searching with only $\kappa _ { \mathcal { N } }$ or $\hat { R } _ { \mathcal { N } }$ leads to inferior performance, which indicates the importance of maintaining both trainability and expressivity during the search.
+
+Table 4: Search with only $\kappa { \mathcal N }$ or $\hat { R } _ { \mathcal { N } }$ on CIFAR-100 in NAS-Bench-201 space.   
+
+<table><tr><td>Methods</td><td>CIFAR-100 Test Accuracy</td></tr><tr><td>Prune with only KN</td><td>69.25 (1.29)</td></tr><tr><td>Prune with only RN</td><td>70.48 (0.29)</td></tr><tr><td>TE-NAS</td><td>71.24 (0.56)</td></tr></table>
+
+# D.2 DIFFERENT COMBINATION OPTIONS FOR $\kappa _ { \mathcal { N } }$ AND $\hat { R } _ { \mathcal { N } }$
+
+Pruning by $s ( o _ { j } ) = s _ { \kappa } ( o _ { j } ) + s _ { R } ( o _ { j } )$ is not the only option (see Algorithm 1). Here in this study we consider more:
+
+1) pruning by $s ( o _ { j } ) = \mathrm { m i n } ( s _ { \kappa } ( o _ { j } ) , s _ { R } ( o _ { j } ) )$ , i.e., pruning by the worst case. 2) Pruninng by $s ( o _ { j } ) = \mathrm { m a x } ( s _ { \kappa } ( o _ { j } ) , s _ { R } ( o _ { j } ) )$ , i.e., pruning by the best case. 3) pruning by summation of changes $\Delta \kappa _ { t , o _ { j } } + \Delta R _ { t , o _ { j } }$ , i.e., directly use the numerical values of the changes.
+
+As we observed in Table 5, our TE-NAS stands out of all options. This means a good trade-off between $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ are important, and also the ranking strategy is better than directly using numerical values.
+
+Table 5: Search with only $\kappa _ { \mathcal { N } }$ or $\hat { R } _ { \mathcal { N } }$ on CIFAR-100 in NAS-Bench-201 space.   
+
+<table><tr><td>Methods</td><td>CIFAR-100 Test Accuracy</td></tr><tr><td>Prune by s(oj)= min(sr(oj),sR(0j))</td><td>70.75 (0.73)</td></tr><tr><td>Prune by s(oj)=max(sκ(oj),sr(0j))</td><td>70.33 (1.09)</td></tr><tr><td>Prune by △Kt,oj+△Rt,oj</td><td>70.47 (0.68)</td></tr><tr><td>Prune by s(oj)= sκ(Oj)+sR(oj) (TE-NAS)</td><td>71.24 (0.56)</td></tr></table>
+
+Figure 9 indicates that by using the summation of the ranking of $\mathsf { b o t h } \kappa _ { \mathcal N }$ and $\hat { R } _ { \mathcal { N } }$ , the combined metric achieves a much higher correlation with the test accuracy. The reason can be explained by Figure 4, as $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ prefers different operators in terms of trainability and Expressivity. Their combination can filter out bad architectures in both aspects and strongly correlate with networks’ final performance.
+
+![](images/d0038d4acd6d5cbad42a776602e3e4b8e5adf3361677add29fc779f187849bbf.jpg)  
+Figure 9: Summation of ranking of $\kappa _ { \mathcal { N } }$ and $\hat { R } _ { \mathcal { N } }$ exhibits stronger (negative) correlation with the test accuracy of architectures in NAS-Bench201 (Dong & Yang, 2020).
+
+# E GENERALIZATION V.S. TEST ACCURACY
+
+Conceptually, the generalization gap is the difference between a model’s performance on training data and its performance on unseen data drawn from the same distribution (e.g., testing set). In comparison, the two indicators $\kappa _ { \mathcal { N } }$ (trainability) and $R _ { \mathcal { N } }$ (expressiveness) of a network determine how well the training set could be fit (i.e., training set accuracy), and do not directly indicate its generalization gap (or test set accuracy). Indeed, probing generalization of an untrained network at its initialization is a daunting, open challenge that seems to go beyond the current theory scope.
+
+![](images/c758a58794154dd00dc45327533a6e9468a026d51804b24738e4702389e3d289.jpg)  
+Figure 10: The correlation between test accuracy and the training accuracy in NAS-Bench201 (Dong & Yang, 2020).
+
+In NAS, we are searching for the architecture with the best test accuracy. As shown in Figure 10, in NAS-Bench201 the training accuracy strongly correlates with test accuracy. This also seems to be a result of the current standard search space design that could have implicitly excluded severe overfitting. This explains why $\kappa _ { \mathcal { N } }$ and $R _ { \mathcal { N } }$ , which only focuses on trainability and expressiveness during training, can still achieve good search results of test accuracy.

parse/train/Cnon5ezMHtu/Cnon5ezMHtu_content_list.json

@@ -0,0 +1,2085 @@
+[
+    {
+        "type": "text",
+        "text": "NEURAL ARCHITECTURE SEARCH ON IMAGENET IN FOUR GPU HOURS: A THEORETICALLY INSPIRED PERSPECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Wuyang Chen, Xinyu Gong, Zhangyang Wang Department of Electrical and Computer Engineering The University of Texas at Austin, Austin, TX, USA {wuyang.chen,xinyu.gong,atlaswang}@utexas.edu ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            275,
+            544,
+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural Architecture Search (NAS) has been explosively studied to automate the discovery of top-performer neural networks. Current works require heavy training of supernet or intensive architecture evaluations, thus suffering from heavy resource consumption and often incurring search bias due to truncated training or approximations. Can we select the best neural architectures without involving any training and eliminate a drastic portion of the search cost? We provide an affirmative answer, by proposing a novel framework called training-free neural architecture search (TE-NAS). TE-NAS ranks architectures by analyzing the spectrum of the neural tangent kernel (NTK) and the number of linear regions in the input space. Both are motivated by recent theory advances in deep networks and can be computed without any training and any label. We show that: (1) these two measurements imply the trainability and expressivity of a neural network; (2) they strongly correlate with the network’s test accuracy. Further on, we design a pruning-based NAS mechanism to achieve a more flexible and superior trade-off between the trainability and expressivity during the search. In NAS-Bench-201 and DARTS search spaces, TE-NAS completes high-quality search but only costs 0.5 and 4 GPU hours with one 1080Ti on CIFAR-10 and ImageNet, respectively. We hope our work inspires more attempts in bridging the theoretical findings of deep networks and practical impacts in real NAS applications. Code is available at: https://github.com/VITA-Group/TENAS. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The recent development of deep networks significantly contributes to the success of computer vision. Thanks to many efforts by human designers, the performance of deep networks have been significantly boosted (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016; Xie et al., 2017). However, the manual creation of new network architectures not only costs enormous time and resources due to trial-and-error, but also depends on the design experience that does not always scale up. To reduce the human efforts and costs, neural architecture search (NAS) has recently amassed explosive interests, leading to principled and automated discovery for good architectures in a given search space of candidates (Zoph & Le, 2016; Brock et al., 2017; Pham et al., 2018; Liu et al., 2018a; Chen et al., 2018; Bender et al., 2018; Gong et al., 2019; Chen et al., 2020a; Fu et al., 2020). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As an optimization problem, NAS faces two core questions: 1) “how to evaluate”, i.e. the objective function that defines what are good architectures we want; 2) “how to optimize”, i.e. by what means we could effectively optimize the objective function. These two questions are entangled and highly non-trivial, since the search spaces are of extremely high dimension, and the generalization ability of architectures cannot be easily inferred (Dong & Yang, 2020; Dong et al., 2020). Existing NAS methods mainly leverage the validation set and conduct accuracy-driven architecture optimization. They either formulate the search space as a super-network (“supernet”) and make the training loss differentiable through the architecture parameters (Liu et al., 2018b), or treat the architecture selection as a sequential decision making process (Zoph & Le, 2016) or evolution of genetics (Real et al., 2019). However, these NAS algorithms suffer from heavy consumption of both time and GPU resources. Training a supernet till convergence is extremely slow, even with many effective heuristics for sampling or channel approximations (Dong & Yang, 2019; Xu et al., 2019). Approximated proxy inference such as truncated training/early stopping can accelerate the search, but is well known to introduce search bias to the inaccurate results obtained (Pham et al., 2018; Liang et al., 2019; Tan et al., 2020). The heavy search cost not only slows down the discovery of novel architectures, but also blocks us from more meaningfully understanding the NAS behaviors. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, the analysis of neural network’s trainability (how effective a network can be optimized via gradient descent) and expressivity (how complex the function a network can represent) has witnessed exciting development recently in the deep learning theory fields. By formulating neural networks as a Gaussian Process (no training involved), the gradient descent training dynamics can be characterized by the Neural Tangent Kernel (NTK) of infinite (Lee et al., 2019) or finite (Yang, 2019) width networks, from which several useful measures can be derived to depict the network trainability at the initialization. Hanin & Rolnick (2019a;b); Xiong et al. (2020) describe another measure of network expressivity, also without any training, by counting the number of unique linear regions that a neural network can divide in its input space. We are therefore inspired to ask: ",
+        "bbox": [
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+            166,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• How to optimize NAS at network’s initialization without involving any training, thus significantly eliminating a heavy portion of the search cost?   \n• Can we define how to evaluate in NAS by analyzing the trainability and expressivity of architectures, and further benefit our understanding of the search process? ",
+        "bbox": [
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+            297,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our answers are yes to both questions. In this work, we propose TE-NAS, a framework for trainingfree neural architecture search. We leverage two indicators, the condition number of NTK and the number of linear regions, that can decouple and effectively characterize the trainability and expressivity of architectures respectively in complex NAS search spaces. Most importantly, these two indicators can be measured in a training-free and label-free manner, thus largely accelerates the NAS search process and benefits the understanding of discovered architectures. To our best knowledge, TE-NAS makes the first attempt to bridge the theoretical findings of deep neural networks and real-world NAS applications. While we intend not to claim that the two indicators we use are the only nor the best options, we hope our work opens a door to theoretically-inspired NAS and inspires the discovery of more deep network indicators. Our contributions are summarized as below: ",
+        "bbox": [
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+            363,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We identify and investigate two training-free and label-free indicators to rank the quality of deep architectures: the spectrum of their NTKs, and the number of linear regions in their input space. Our study finds that they reliably indicate the trainability and expressivity of a deep network respectively, and are strongly correlated with the network’s test accuracy.   \n• We leverage the above two theoretically-inspired indicators to establish a training-free NAS framework, TE-NAS, therefore eliminating a drastic portion of the search cost. We further introduce a pruning-based mechanism, to boost search efficiency and to more flexibly trade-off between trainability and expressivity.   \nIn NAS-Bench-201/DARTS search spaces, TE-NAS discovers architectures with a strong performance at remarkably lower search costs, compared to previous efforts. With just one $1 0 8 0 \\mathrm { T i }$ , it only costs 0.5 GPU hours to search on CIFAR10, and 4 GPU hours on ImageNet, respectively, setting the new record for ultra-efficient yet high-quality NAS. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            696,
+            354,
+            712
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neural architecture search (NAS) is recently proposed to accelerate the principled and automated discovery of high-performance networks. However, most works suffer from heavy search cost, for both weight-sharing based methods (Liu et al., 2018b; Dong & Yang, 2019; Liu et al., 2019; Yu et al., 2020a; Li et al., 2020a; Yang et al., 2020a) and single-path sampling-based methods (Pham et al., 2018; Guo et al., 2019; Real et al., 2019; Tan et al., 2020; Li et al., 2020c; Yang et al., 2020b). A one-shot super network can share its parameters to sampled sub-networks and accelerate the architecture evaluations, but it is very heavy and hard to optimize and suffers from a poor correlation between its accuracy and those of the sub-networks (Yu et al., 2020c). Sampling-based methods achieve more accurate architecture evaluations, but their truncated training still imposes bias to the performance ranking since this is based on the results of early training stages. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Instead of estimating architecture performance by direct training, people also try to predict network’s accuracy (or ranking), called predictor based NAS methods (Liu et al., 2018a; Luo et al., 2018; Dai et al., 2019; Luo et al., 2020). Graph neural network (GNN) is a popular choice as the predictor model (Wen et al., 2019; Chen et al., 2020b). Siems et al. (2020) even propose the first large-scale surrogate benchmark, where most of the architectures’ accuracies are predicted by a pretrained GNN predictor. The learned predictor can achieve highly accurate performance evaluation. However, the data collection step - sampling representative architectures and train them till converge - still requires extremely high cost. People have to sample and train 2,000 to 50,000 architectures to serve as the training data for the predictor. Moreover, none of these works can demonstrate the cross-space transferability of their predictors. This means one has to repeat the data collection and predictor training whenever facing an unseen search space, which is highly nonscalable. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The heavy cost of architecture evaluation hinders the understanding of the NAS search process. Recent pioneer works like Shu et al. (2019) observed that DARTS and ENAS tend to favor architectures with wide and shallow cell structures due to their smooth loss landscape. Siems et al. (2020) studied the distribution of test error for different cell depths and numbers of parameter-free operators. Chen & Hsieh (2020) for the first time regularizes the Hessian norm of the validation loss and visualizes the smoother loss landscape of the supernet. Li et al. (2020b) proposed to approximate the validation loss landscape by learning a mapping from neural architectures to their corresponding validate losses. Still, these analyses cannot be directly leveraged to guide the design of network architectures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Mellor et al. (2020) recently proposed a NAS framework that does not involve training, which shares the same motivation with us towards training-free architecture search at initialization. They empirically find that the correlation between sample-wise input-output Jacobian can indicate the architecture’s test performance. However, why does the Jacobian work is not well explained and demonstrated. Their search performance on NAS-Bench-201 is still left behind by the state-of-the-art NAS works, and they did not extend to DARTs space. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, we see the evolving development of deep learning theory on neural networks. NTK (neural tangent kernel) is proposed to characterize the gradient descent training dynamics of infinite wide (Jacot et al., 2018) or finite wide deep networks (Hanin & Nica, 2019). Wide networks are also proved to evolve as linear models under gradient descent (Lee et al., 2019). This is further leveraged to decouple the trainability and generalization of networks (Xiao et al., 2019). Besides, a natural measure of ReLU network’s expressivity is the number of linear regions it can separate in its input space (Raghu et al., 2017; Montufar, 2017; Serra et al., 2018; Hanin & Rolnick, 2019a;b; Xiong ´ et al., 2020). In our work, we for the first time discover two important indicators that can effectively rank architectures, thus bridging the theoretic findings and real-world NAS applications. Instead of claiming the two indicators we discover are the best, we believe there are more meaningful properties of deep networks that can benefit the architecture search process. We leave them as open questions and encourage the community to study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The core motivation of our TE-NAS framework is to achieve architecture evaluation without involving any training, to significantly accelerate the NAS search process and reduce the search cost. In section 3.1 we present our study on two important indicators that reflect the trainability and expressivity of a neural network, and in section 3.2 we design a novel pruning-based method that can achieve a superior trade-off between the two indicators. ",
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+    {
+        "type": "text",
+        "text": "3.1 ANALYZING TRAINABILITY AND EXPRESSIVITY OF DEEP NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Trainability and expressivity are distinct notions regarding a neural network (Xiao et al., 2019). A network can achieve high performance only if the function it can represent is complex enough and at the same time, it can be effectively trained by gradient descent. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1.1 TRAINABILITY BY CONDITION NUMBER OF NTK ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The trainability of a neural network indicates how effective it can be optimized using gradient descent (Burkholz & Dubatovka, 2019; Hayou et al., 2019; Shin & Karniadakis, 2020). Although some heavy networks can theoretically represent complex functions, they not necessarily can be effectively trained by gradient descent. One typical example is that, even with a much more number of parameters, $\\mathrm { V g g }$ networks (Simonyan & Zisserman, 2014) usually perform worse and require more special engineering tricks compared with ResNet family (He et al., 2016), whose superior trainability property is studied by Yang & Schoenholz (2017). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Recent work (Jacot et al., 2018; Lee et al., 2019; Chizat et al., 2019) studied the gradient descent training of neural networks using a quantity called the neural tangent kernel (NTK). The finite width NTK is defined by $\\hat { \\Theta } ( \\pmb { x } , \\pmb { x } ^ { \\prime } ) = \\bar { J } ( \\pmb { x } ) \\bar { J } ( \\pmb { x } ^ { \\prime } ) ^ { T }$ , where $J _ { i \\alpha } ( { \\pmb x } ) = \\partial _ { \\theta _ { \\alpha } } z _ { i } ^ { L } ( { \\pmb x } )$ is the Jacobian evaluated at a point $_ { \\textbf { \\em x } }$ for parameter $\\theta _ { \\alpha }$ , and $z _ { i } ^ { L }$ is the output of the $i$ -th neuron in the last output layer $L$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lee et al. (2019) further proves that wide neural networks evolve as linear models using gradient descent, and their training dynamics is controlled by ODEs that can be solved as ",
+        "bbox": [
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+            170,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed0db9042e3a1b222d11930ccd3b2a4232d9fe90f88318366a90726f14df51f2.jpg",
+        "text": "$$\n\\mu _ { t } ( X _ { \\mathrm { t r a i n } } ) = ( { \\bf I } - e ^ { - \\eta \\hat { \\Theta } _ { \\mathrm { t r a i n } } t } ) Y _ { \\mathrm { t r a i n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for training data. Here $\\mu _ { t } ( \\pmb { x } ) = \\mathbb { E } [ z _ { i } ^ { L } ( \\pmb { x } ) ]$ is the expected outputs of an infinitely wide network. $\\hat { \\Theta } _ { \\mathrm { t r a i n } }$ denotes the NTK between the training inputs, and $X _ { \\mathrm { t r a i n } }$ and $Y _ { \\mathrm { t r a i n } }$ are drawn from the training set $\\mathcal { D } _ { \\mathrm { t r a i n } }$ . As the training step $t$ tends to infinity we can see that Eq. 1 reduce to $\\mu ( X _ { \\mathrm { t r a i n } } ) = Y _ { \\mathrm { t r a i n } } $ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The relationship between the conditioning of $\\hat { \\Theta }$ and the trainability of networks is studied by Xiao et al. (2019), and we brief the conclusion as below. We can write Eq. 1 in terms of the spectrum of $\\Theta$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6424b013f05b8a7f32fb3081b2b041095692131fcd86a92b8ca8c66a8911cb1c.jpg",
+        "text": "$$\n\\mu _ { t } ( X _ { \\mathrm { t r a i n } } ) _ { i } = ( \\mathbf { I } - e ^ { - \\eta \\lambda _ { i } t } ) Y _ { \\mathrm { t r a i n } , i } ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { i }$ are the eigenvalues of $\\hat { \\Theta } _ { \\mathrm { t r a i n } }$ and we order the eigenvalues $\\lambda _ { 0 } \\geq \\cdots \\geq \\lambda _ { m }$ . As it has been hypothesized by Lee et al. (2019) that the maximum feasible learning rate scales as $\\eta \\sim 2 / \\lambda _ { 0 }$ , plugging this scaling for $\\eta$ into Eq. 2 we see that the $\\lambda _ { m }$ will converge exponentially at a rate given by $1 / \\kappa$ where $\\kappa = \\lambda _ { 0 } / \\lambda _ { m }$ is the condition number. Then we can conclude that if the $\\kappa$ of the NTK associated with a neural network diverges then it will become untrainable, so we use $\\kappa$ as a metric for trainability: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3362e810a2898a515ca6d05037524ddffdc34e876ff29b129e781c046234c2f8.jpg",
+        "image_caption": [
+            "Figure 1: Condition number of NTK $\\kappa _ { \\mathcal { N } }$ exhibits negative correlation with the test accuracy of architectures in NAS-Bench201 (Dong & Yang, 2020). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/566f37c74d36e7a01eccd8299f53023d654b8a5e9e75957a1009a9e099d1dbc9.jpg",
+        "text": "$$\n\\kappa _ { \\mathcal { N } } = \\frac { \\lambda _ { 0 } } { \\lambda _ { m } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\kappa _ { \\mathcal { N } }$ is calculated without any gradient descent or label. Figure 1 demonstrates that the $\\kappa _ { \\mathcal { N } }$ is negatively correlated with the architecture’s test accuracy, with the Kendall-tau correlation as $- 0 . 4 2$ . Therefore, minimizing the $\\kappa _ { \\mathcal { N } }$ during the search will encourage the discovery of architectures with high performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1.2 EXPRESSIVITY BY NUMBER OF LINEAR REGIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The expressivity of a neural network indicates how complex the function it can represent (Hornik et al., 1989; Giryes et al., 2016). For ReLU networks, each ReLU function defines a linear boundary and divides its input space into two regions. Since the composition of piecewise linear functions is still piecewise linear, every ReLU network can be seen as a piecewise linear function. The input space of a ReLU network can be partitioned into distinct pieces (i.e. linear regions) (Figure 2), each of which is associated with a set of affine parameters, and the function represented by the network is affine when restricted to each piece. Therefore, it is natural to measure the expressivity of a ReLU network with the number of linear regions it can separate. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/7982522b6a745b07e4a9d64194fc492c3cc1ea03f68b2d71292f0590ce4f610a.jpg",
+        "image_caption": [
+            "Figure 2: Example of linear regions divided by a ReLU network1 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following Raghu et al. (2017); Montufar (2017); Serra et al. (2018); Hanin & Rolnick´ $( 2 0 1 9 \\mathrm { a } ; \\mathbf { b } )$ Xiong et al. (2020), we first introduce the following definition of activation patterns and linear regions for ReLU CNNs. ",
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+    {
+        "type": "text",
+        "text": "Definition 1. Activation Patterns and Linear Regions (Xiong et al. (2020)) Let $\\mathcal { N }$ be a ReLU CNN. An activation pattern of $\\mathcal { N }$ is a function $_ { r }$ from the set of neurons to $\\{ 1 , - 1 \\}$ , i.e., for each neuron $z$ in $\\mathcal { N }$ , we have $P ( z ) \\in \\{ 1 , - 1 \\}$ . Let $\\theta$ be a fixed set of parameters (weights and biases) in $\\mathcal { N }$ , and $_ { r }$ be an activation pattern. The region corresponding to $_ { r }$ and $\\theta$ is ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/25d1c083efa136bffa44255274ea6a7d64d310b03127e9330f053618164ddd64.jpg",
+        "text": "$$\n\\begin{array} { r } { R ( P ; \\theta ) : = \\{ { \\pmb x } ^ { 0 } \\in \\mathbb { R } ^ { C \\times H \\times W } : z ( { \\pmb x } ^ { 0 } ; \\theta ) \\cdot { \\pmb P } ( z ) > 0 , \\quad \\forall z \\in \\mathcal { N } \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $z ( \\pmb { x } ^ { 0 } ; \\theta )$ is the pre-activation of a neuron $z$ . Let $R _ { \\mathcal { N } , \\theta }$ denote the number of linear regions of $\\mathcal { N }$ at $\\theta$ , i.e., $R _ { \\mathcal { N } , \\theta } : = \\# \\{ R ( P ; \\theta ) : R ( P ; \\theta ) \\neq \\emptyset$ for some activation pattern $P \\}$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Eq. 4 tells us that a linear region in the input space is a set of input data $\\mathbf { x } ^ { 0 }$ that satisfies a certain fixed activation pattern $\\bar { \\boldsymbol { P } } ( z )$ , and therefore the number of linear regions $R _ { \\mathcal { N } , \\theta }$ measures how many unique activation patterns that can be divided by the network. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In our work, we repeat the measurement of the number of linear regions by sampling network parameters from the Kaiming Norm Initialization (He et al., 2015), and calculate the average as the approximation to its expectation: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/0ac51725478ae2c2bbaaa3490e8da3ce5652419ca0641f3a19d0786496f403f8.jpg",
+        "text": "$$\n\\hat { R } _ { \\mathcal { N } } \\simeq \\mathbb { E } _ { \\theta } R _ { \\mathcal { N } , \\theta }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b2d93de6fbb1a383c41afa3257e7e64a8de68570ea962a420a5a43e77c224850.jpg",
+        "image_caption": [
+            "Figure 3: Number of linear regions $\\hat { R } _ { \\mathcal { N } }$ of architectures in NAS-Bench201 exhibits positive correlation with test accuracies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We iterate through all architectures in NAS-Bench-201 (Dong & Yang, 2020), and calculate their numbers of linear regions (without any gradient descent or label). Figure 3 demonstrates that the number of linear regions is positively correlated with the architecture’s test accuracy, with the Kendall-tau correlation as 0.5. Therefore, maximizing the number of linear regions during the search will also encourage the discovery of architectures with high performance. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, in Figure 4 we analyze the operator composition of top $10 \\%$ architecture by maximizing $\\hat { R } _ { \\mathcal { N } }$ and minimizing $\\kappa _ { \\mathcal { N } }$ , respectively. We can clearly see that $\\hat { R } _ { \\mathcal { N } }$ and $\\kappa _ { \\mathcal { N } }$ have different preferences for choosing operators. They both choose a large ratio of conv $3 \\times 3$ for high generalization performance. But meanwhile, $\\hat { R } _ { \\mathcal { N } }$ heavily selects $\\mathbf { c o n v 1 } \\times 1$ , and $\\kappa _ { \\mathcal { N } }$ leads to skip-connect, favoring the gradient flow. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a87c3990390e3008ac1c0860925cb464148f6012c6ea58ca9b73b32873371cb6.jpg",
+        "image_caption": [
+            "Figure 4: $\\kappa { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ prefer different operators in NAS-Bench201. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.2 PRUNING-BY-IMPORTANCE ARCHITECTURE SEARCH ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Given the strong correlation between the architecture’s test accuracy and its $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ , how to build an efficient NAS framework on top of them? We motivate this section by addressing two questions: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. How to combine $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ together, with a good explicit trade-off? ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We first need to turn the two measurements $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ into one combined function, based on which we can rank architectures. As seen in Figure 1 and 3, the magnitudes of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ differ much. To avoid one overwhelming the other numerically, one possible remedy is normalization; but we cannot pre-know the ranges nor the value distributions of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ , before computing them over a search space. In order to make our combined function well defined before search and agnostic to the search space, instead of using the numerical values of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ , we could refer to their relative rankings. Specifically, each time by comparing the sampled set of architectures peer-to-peer, we can directly sum up the two relative rankings of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } } ^ { \\bar { } }$ as the selection criterion. The equal-weight summation treats trainability and expressivity with the same importance conceptually1 and delivers the best empirical result: we thus choose it as our default combined function. We also tried some other means to combine the two, and the ablation studies can be found in Appendix D.2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2. How to search more efficiently? ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Sampling-based methods like reinforcement learning or evolution can use rankings as the reward or filtering metric. However, they are inefficient, especially for complex cell-based search space. Consider a network stacked by repeated cells (directed acyclic graphs) (Zoph et al., 2018; Liu et al., 2018b). Each cell has $E$ edges, and on each edge we only select one operator out of $| \\mathcal { O } |$ ( $\\mathcal { O }$ is the set of operator candidates). There are $| \\mathcal { O } | ^ { E }$ unique cells, and for sampling-based methods, $\\gamma \\cdot | \\mathcal { O } | ^ { E }$ networks have to be sampled during the search. The ratio $\\gamma$ can be interpreted as the sampling efficiency: a method with small $\\gamma$ can find good architectures faster. However, the search time cost of sampling-based methods still scales up with the size of the search space, i.e., $| \\mathcal { O } | ^ { E }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Inspired by recent works on pruning-from-scratch (Lee et al., 2018; Wang et al., 2020), we propose a pruning-by-importance NAS mechanism to quickly shrink the search possibilities and boost the efficiency further, reducing the cost from $| \\mathcal { O } | ^ { E }$ to $| \\mathcal { O } | \\cdot E$ . Specifically, we start the search with a super-network $\\mathcal { N } _ { 0 }$ composed of all possible operators and edges. In the outer loop, for every round we prune one operator on each edge. The outer-loop stops when the current supernet $\\mathcal { N } _ { t }$ is a single-path network2, i.e., the algorithm will return us the final searched architecture. For the inner-loop, we measure the change of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ before and after pruning each individual operator, and assess its importance using the sum of two ranks. We order all currently available operators in terms of their importance, and prune the lowest-importance operator on each edge. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The whole pruning process is extremely fast. As we will demonstrate later, our approach is principled and can be applied to different spaces without making any modifications. This pruning-by-importance mechanism may also be extended to indicators beyond $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ . We summarize our training-free and pruning-based NAS framework, TE-NAS, in Algorithm 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: TE-NAS: Training-free Pruning-based NAS via Ranking of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1 Input: supernet $\\mathcal { N } _ { 0 }$ stacked by cells, each cell has $E$ edges, each edge has $| \\mathcal { O } |$ operators, step $t = 0$ .   \n2 while $\\mathcal { N } _ { t }$ is not a single-path network do   \n3 for each operator $o _ { j }$ in $\\mathcal { N } _ { t }$ do   \n4 $\\begin{array} { r l } & { \\dot { \\bf \\Delta } \\Delta \\kappa _ { t , o _ { j } } = \\kappa _ { \\mathcal { N } _ { t } } \\dot { \\bf \\Delta } - \\kappa _ { \\mathcal { N } _ { t } \\backslash o _ { j } } } \\\\ & { \\Delta R _ { t , o _ { j } } = R _ { \\mathcal { N } _ { t } } - R _ { \\mathcal { N } _ { t } \\backslash o _ { j } } } \\end{array}$ . the higher $\\Delta \\kappa _ { t , o _ { j } }$ the more likely we will prune $o _ { j }$   \n5 $\\triangleright$ the lower $\\Delta R _ { t , o _ { j } }$ the more likely we will prune $o _ { j }$   \n6 Get importance by $\\kappa _ { \\mathcal { N } }$ : $s _ { \\kappa } ( o _ { j } ) =$ index of $o _ { j }$ in descendingly sorted list $[ \\Delta \\kappa _ { t , o _ { 1 } } , . . . , \\Delta \\kappa _ { t , o _ { | \\mathcal { N } _ { t } | } } ]$   \n7 Get importance by $R _ { \\mathcal { N } }$ : $s _ { R } ( o _ { j } ) =$ index of $o _ { j }$ in ascendingly sorted list $[ \\Delta R _ { t , o _ { 1 } } , . . . , \\Delta R _ { t , o _ { | \\mathcal { N } _ { t } | } } ]$   \n8 Get importance $s ( o _ { j } ) = s _ { \\kappa } ( o _ { j } ) + s _ { R } ( o _ { j } )$   \n9 $\\mathcal { N } _ { t + 1 } \\dot { = } \\mathcal { N } _ { t }$   \n10 for each edge $e _ { i }$ , $i = 1 , . . . , E$ do   \n11 $j ^ { * } = \\arg \\operatorname* { m i n } _ { j } \\{ s ( o _ { j } ) : o _ { j } \\in e _ { i } \\}$ . find the operator with greatest importance on each edge.   \n12 $\\mathcal { N } _ { t + 1 } = \\mathcal { N } _ { t + 1 } \\backslash o _ { j ^ { * } }$   \n13 $t = t + 1$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2.1 VISUALIZATION OF SEARCH PROCESS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "TE-NAS benefits us towards a better understanding of the search process. We can analyze the trajectory of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ during the search. It is worth noting that our starting point $\\mathcal { N } _ { 0 }$ , the un-pruned supernet, is assumed to be of the highest expressivity (as it is composed of all operators in the search space and has the largest number of parameters and ReLU functions). However, it has poor trainability, as people find many engineering techniques are required to effectively training the supernet (Yu et al., 2020a;b). Therefore, during pruning we are expecting to strengthen the trainability of the supernet, while retaining its expressivity as much as possible. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5eab6423b3234fdb8c4dc99767d6da291ff972a600d6a8e262fa1295d7f97422.jpg",
+        "image_caption": [
+            "Figure 5: Pruning trajectory on NAS-Bench-201 (top) and DARTs search space (bottom). Number $\\cdot _ { 0 } \\cdot \\mathbf { \\vec { \\tau } }$ indicates the supernet $\\mathcal { N } _ { 0 }$ before any pruning, which is of high expressivity but poor trainability. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As we observe in Figure 5, the supernet $\\mathcal { N }$ is first pruned by quickly reducing $\\kappa _ { \\mathcal { N } }$ , i.e., increasing the network’s trainability. After that, as the improvement of $\\kappa _ { \\mathcal { N } }$ is almost plateaued, the method carefully fine-tunes the architecture without sacrificing too much expressivity $\\hat { R } _ { \\mathcal { N } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate our TE-NAS on two search spaces: NAS-Bench-201 (Dong & Yang, 2020) and DARTS (Liu et al., 2018b). Search and training protocols are summarized in Appendix A. Our code is available at: https://github.com/VITA-Group/TENAS. ",
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+    {
+        "type": "text",
+        "text": "4.1 RESULTS ON NAS-BENCH-201 ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NAS-Bench-201 (Dong & Yang, 2020) provides a standard cell-based search space (containing 15,625 architectures) and a database of architecture’s performance evaluated under a unified protocol. The network’s test accuracy can be directly obtained by querying the database, which facilitates people to focus on studying NAS algorithms without network evaluation. NAS-Bench-201 supports three datasets (CIFAR-10, CIFAR-100, ImageNet-16-120 (Chrabaszcz et al., 2017)). The operation space contains none (zero), skip connection, $c o n \\nu 1 \\times 1$ , $c o n \\nu 3 \\times 3$ convolution, and average pooling $3 \\times 3$ . We refer to their paper for details of the space. Our search is dataset-specific, i.e. the search and evaluation are conducted on the same dataset. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/022c5d2c1baaa61c3cde18d6b48cd9cae1a67897d05c8d02a69dba4843fc4d8e.jpg",
+        "table_caption": [
+            "Table 1: Comparison with state-of-the-art NAS methods on NAS-Bench-201. Test accuracy with mean and deviation are reported. “optimal” indicates the best test accuracy achievable in NAS-Bench-201 search space. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet-16-120</td><td>Search Cost (GPU sec.)</td><td>Search Method</td></tr><tr><td>ResNet (He et al.,2016)</td><td>93.97</td><td>70.86</td><td>43.63</td><td>-</td><td>-</td></tr><tr><td>RSPS (Li&amp; Talwalkar,2020)</td><td>87.66(1.69)</td><td>58.33(4.34)</td><td>31.14(3.88)</td><td>8007.13</td><td>random</td></tr><tr><td>ENAS (Pham et al., 2018)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>13314.51</td><td>RL</td></tr><tr><td>DARTS (1st) (Liu et al.,2018b)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>10889.87</td><td>gradient</td></tr><tr><td>DARTS (2nd) (Liu et al.,2018b)</td><td>54.30(0.00)</td><td>15.61(0.00)</td><td>16.32(0.00)</td><td>29901.67</td><td>gradient</td></tr><tr><td>GDAS (Dong&amp; Yang,2019)</td><td>93.61(0.09)</td><td>70.70(0.30)</td><td>41.84(0.90)</td><td>28925.91</td><td>gradient</td></tr><tr><td>NAS w.o. Training (Mellor et al.,2020)</td><td>91.78(1.45)</td><td>67.05(2.89)</td><td>37.07(6.39)</td><td>4.8</td><td>training-free</td></tr><tr><td>TE-NAS (ours)</td><td>93.9(0.47)</td><td>71.24(0.56)</td><td>42.38(0.46)</td><td>1558</td><td>training-free</td></tr><tr><td> Optimal</td><td>94.37</td><td>73.51</td><td>47.31</td><td>-</td><td>-</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "We run TE-NAS for four independent times with different random seeds, and report the mean and standard deviation in Table 1. We can see that TE-NAS achieves the best accuracy on all three datasets, and largely reduces the search cost $( 5 \\times \\sim 1 9 \\times$ reduction). Although Mellor et al. (2020) requires even less search time (by only sampling 25 architectures), they suffer from much inferior accuracy performance, with notably larger deviations across different search rounds. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ON CIFAR-10 WITH DARTS SEARCH SPACE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Architecture Space The DARTs operation space $\\mathcal { O }$ contains eight choices: none (zero), skip connection, separable convolution $3 \\times 3$ and $5 \\times 5$ , dilated separable convolution $3 \\times 3$ and $5 \\times 5$ , max pooling $3 \\times 3$ , average pooling $3 \\times 3$ . Following previous works (Liu et al., 2018b; Chen et al., 2019; $\\mathrm { X u }$ et al., 2019), for evaluation phases, we stack 20 cells to compose the network and set the initial channel number as 36. We place the reduction cells at the 1/3 and 2/3 of the network and each cell consists of six nodes. ",
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+        "table_caption": [
+            "Table 2: Comparison with state-of-the-art NAS methods on CIFAR-10. "
+        ],
+        "table_footnote": [
+            "? No cutout augmentation. † Different space: PyramidNet (Han et al., 2017) as the backbone. ‡ Recorded on a single GTX 1080Ti GPU. "
+        ],
+        "table_body": "<table><tr><td>Architecture</td><td>Test Error (%)</td><td>Params (M)</td><td>Search Cost (GPU days)</td><td>Search Method</td></tr><tr><td>AmoebaNet-A (Real et al., 2019)</td><td>3.34(0.06)</td><td>3.2</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al.,2018a)*</td><td>3.41(0.09)</td><td>3.2</td><td>225</td><td>SMBO</td></tr><tr><td>ENAS (Pham et al., 2018)</td><td>2.89</td><td>4.6</td><td>0.5</td><td>RL</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>2.65</td><td>3.3</td><td>2000</td><td>RL</td></tr><tr><td>DARTS (1st) (Liu et al.,2018b)</td><td>3.00(0.14)</td><td>3.3</td><td>0.4</td><td>gradient</td></tr><tr><td>DARTS (2nd) (Liu et al., 2018b)</td><td>2.76(0.09)</td><td>3.3</td><td>1.0</td><td>gradient</td></tr><tr><td>SNAS (Xie et al.,2018)</td><td>2.85(0.02)</td><td>2.8</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>2.82</td><td>2.5</td><td>0.17</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>2.81(0.04)</td><td>3.4</td><td>0.2</td><td>gradient</td></tr><tr><td>ProxylessNAS (Cai et al.,2018)†</td><td>2.08</td><td>5.7</td><td>4.0</td><td>gradient</td></tr><tr><td>P-DARTS (Chen et al.,2019)</td><td>2.50</td><td>3.4</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (Xu et al.,2019)</td><td>2.57(0.07)</td><td>3.6</td><td>0.1</td><td>gradient</td></tr><tr><td>SDARTS-ADV (Chen &amp; Hsieh,2020)</td><td>2.61(0.02)</td><td>3.3</td><td>1.3</td><td>gradient</td></tr><tr><td>TE-NAS (ours)</td><td>2.63(0.064)</td><td>3.8</td><td>0.05</td><td>training-free</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Results We run TE-NAS for four independent times with different random seeds, and report the mean and standard deviation. Table 2 summarizes the performance of TE-NAS compared with other popular NAS methods. TE-NAS achieves a test error of $2 . 6 3 \\%$ , ranking among the top of recent NAS results, but meanwhile largely reduces the search cost to only 0.05 GPU-day. ProxylessNAS achieves the lowest test error, but it searches on a different space with a much longer search time and has a larger model size. Besides, Mellor et al. (2020) did not extend to their Jacobian-based framework to DARTs search space for CIFAR-10 or ImageNet classification. ",
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+    {
+        "type": "text",
+        "text": "4.3 RESULTS ON IMAGENET WITH DARTS SEARCH SPACE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Architecture Space Following previous works $\\mathrm { { X u } }$ et al., 2019; Chen et al., 2019), the architecture for ImageNet is slightly different from that for CIFAR-10. During retraining evaluation, the network is stacked with 14 cells with the initial channel number set to 48, and we follow the mobile setting to control the FLOPs not exceed $6 0 0 { \\mathrm { M B } }$ by adjusting the channel number. The spatial resolution is downscaled from $2 2 4 \\times 2 2 4$ to $2 8 \\times 2 8$ with the first three convolution layers of stride 2. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results As shown in Table 3, we achieve a top-1/5 test error of $2 4 . 5 \\% / 7 . 5 \\%$ , achieving competitive performance with recent state-of-the-art works in the ImageNet mobile setting. However, TE-NAS only cost four GPU hours with only one 1080Ti. Searching on ImageNet takes a longer time than on CIFAR-10 due to the larger input size and more network parameters. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be55f27dd7173cf1e3ae32a0bcc20ab312a6fee88d475f850b4a5e120874d7cd.jpg",
+        "table_caption": [
+            "Table 3: Comparison with state-of-the-art NAS methods on ImageNet under the mobile setting. "
+        ],
+        "table_footnote": [
+            "† The architecture is searched on ImageNet, otherwise it is searched on CIFAR-10 or CIFAR-100. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td colspan=\"2\">Test Error(%)</td><td rowspan=\"2\">Params (M)</td><td rowspan=\"2\">Search Cost (GPU days)</td><td rowspan=\"2\">Search Method</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>26.0</td><td>8.4</td><td>5.3</td><td>2000</td><td>RL</td></tr><tr><td>AmoebaNet-C (Real et al.,2019)</td><td>24.3</td><td>7.6</td><td>6.4</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>25.8</td><td>8.1</td><td>5.1</td><td>225</td><td>SMBO</td></tr><tr><td>MnasNet-92 (Tan et al.,2019)</td><td>25.2</td><td>8.0</td><td>4.4</td><td>1</td><td>RL</td></tr><tr><td>DARTS (2nd) (Liu et al.,2018b)</td><td>26.7</td><td>8.7</td><td>4.7</td><td>4.0</td><td>gradient</td></tr><tr><td>SNAS (mild) (Xie et al.,2018)</td><td>27.3</td><td>9.2</td><td>4.3</td><td>1.5</td><td>gradient</td></tr><tr><td>GDAS (Dong &amp; Yang,2019)</td><td>26.0</td><td>8.5</td><td>5.3</td><td>0.21</td><td>gradient</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>26.5</td><td>8.9</td><td>3.9</td><td>0.2</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-10) (Chen et al., 2019)</td><td>24.4</td><td>7.4</td><td>4.9</td><td>0.3</td><td>gradient</td></tr><tr><td>P-DARTS (CIFAR-100)(Chen et al.,2019)</td><td>24.7</td><td>7.5</td><td>5.1</td><td>0.3</td><td>gradient</td></tr><tr><td>PC-DARTS (CIFAR-1O)(Xu et al.,2019)</td><td>25.1</td><td>7.8</td><td>5.3</td><td>0.1</td><td>gradient</td></tr><tr><td>TE-NAS (ours)</td><td>26.2</td><td>8.3</td><td>6.3</td><td>0.05</td><td>training-free</td></tr><tr><td>PC-DARTS (ImageNet) (Xu et al., 2019)†</td><td>24.2</td><td>7.3</td><td>5.3</td><td>3.8</td><td>gradient</td></tr><tr><td>ProxylessNAS (GPU) (Cai et al.,2018)†</td><td>24.9</td><td>7.5</td><td>7.1</td><td>8.3</td><td>gradient</td></tr><tr><td>TE-NAS (ours)†</td><td>24.5</td><td>7.5</td><td>5.4</td><td>0.17</td><td>training-free</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The key questions in Neural Architecture Search (NAS) are “what are good architectures” and “how to find them”. Validation loss or accuracy are possible answers but not enough, due to their search bias and heavy evaluation cost. Our work demonstrates that two theoretically inspired indicators, the spectrum of NTK and the number of linear regions, not only strongly correlate with the network’s performance, but also benefit the reduced search cost and decoupled analysis of the network’s trainability and expressivity. Without involving any training, our TE-NAS achieve competitive NAS performance with minimum search time. We for the first time bridge the gap between the theoretic findings of deep neural networks and real-world NAS applications, and we encourage the community to further explore more meaningful network properties so that we will have a better understanding of good architectures and how to search them. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
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+        "type": "text",
+        "text": "This work is supported in part by the NSF Real-Time Machine Learning program (Award Number: 2053279), and the US Army Research Office Young Investigator Award (W911NF2010240). ",
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+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            441,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For $\\kappa _ { \\mathcal { N } }$ we sample one mini-batch of size 32 from the training set, and calculate $\\hat { \\Theta } ( \\pmb { x } , \\pmb { x } ^ { \\prime } ) =$ $J ( { \\pmb x } ) J ( { \\pmb x } ^ { \\prime } ) ^ { T }$ . For $\\hat { R } _ { \\mathcal { N } }$ we sample 5000 images, forward them through the network, and collect the activation patterns from all ReLU layers. The calculation of both $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ are repeated three times in all experiments, where each time the network weights are randomly drawn from Kaiming Norm Initialization (He et al., 2015) without involving any training (network weights are fixed). ",
+        "bbox": [
+            173,
+            132,
+            825,
+            208
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our retraining settings (after search) follow previous works (Xu et al., 2019; Chen et al., 2019; Chen & Hsieh, 2020). On CIFAR-10, we train the searched network with cutout regularization of length 16, drop-path (Zoph et al., 2018) with probability as 0.3, and an auxiliary tower of weight 0.4. On ImageNet, we also use label smoothing during training. On both CIFAR-10 and ImageNet, the network is optimized by an SGD optimizer with cosine annealing, with learning rate initialized as 0.025 and 0.5, respectively. ",
+        "bbox": [
+            173,
+            215,
+            825,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B SEARCHED ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            319,
+            439,
+            335
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We visualize the searched normal and reduction cells in figure 6 and 7, which is directly searched on CIFAR-10 and ImageNet respectively. ",
+        "bbox": [
+            174,
+            349,
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+            378
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ea788d5aa0e8499282a29795a6e0b6706094d3bcfd951b8e4cd7154e1600ecd.jpg",
+        "image_caption": [
+            "Figure 6: Normal and Reduction cells discovered by TE-NAS on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            396,
+            825,
+            506
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/0cf482289933baf96e10aac9ca88e6c877afab0dbcbb7c332a85deacc8e0fadf.jpg",
+        "image_caption": [
+            "Figure 7: Normal and Reduction cells discovered by TE-NAS on imageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            571,
+            810,
+            676
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C DEPTH AND WIDTH PREFERENCE OF $\\kappa _ { \\mathcal { N } }$ AND $\\hat { R } _ { \\mathcal { N } }$ IN DARTS SPACE ",
+        "bbox": [
+            169,
+            731,
+            772,
+            751
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To analyze the impact of different architectures on trainability and expressivity in DARTs search space, we visualize $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ with different depths and width. Following Shu et al. (2019), the depth of a cell is defined as the number of connections on the longest path from input nodes to the output node, and the width of a cell is the summation of the edges of the intermediate nodes that are connected to the input nodes. We randomly sample 20,000 architectures in DARTs space, and plot the visualizations in Figure 8. Good architectures should exhibit low $\\kappa _ { \\mathcal { N } }$ (good trainability, blue dots in Figure 8(a)) and high $\\hat { R } _ { \\mathcal { N } }$ (powerful expressivity, red dots in Figure 8(b)). Therefore, $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ tell us that in DARTs space shallow but wide cells are preferred to favor both trainability and expressivity. This conclusion matches the findings by Shu et al. (2019): existing NAS algorithms tend to favor architectures with wide and shallow cell structures, which enjoy fast convergence with smooth loss landscape and accurate gradient information. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/68fb7e3b692da7dff551648648324bbc0476d9fbdc1fda9a6ac2d7d65ff25541.jpg",
+        "image_caption": [
+            "Figure 8: Depth and width preference of (a) $\\kappa { \\mathcal N }$ and (b) $\\hat { R } _ { \\mathcal { N } }$ on DARTs Search Space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            126,
+            808,
+            310
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D MORE ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            366,
+            436,
+            382
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 SEARCH WITH ONLY $\\kappa _ { \\mathcal { N } }$ OR $\\hat { R } _ { \\mathcal { N } }$ ",
+        "bbox": [
+            174,
+            396,
+            439,
+            412
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As we observed in Table 4, searching with only $\\kappa _ { \\mathcal { N } }$ or $\\hat { R } _ { \\mathcal { N } }$ leads to inferior performance, which indicates the importance of maintaining both trainability and expressivity during the search. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9dba9f95220e5320e758ddfa22a1ad4f27d40ae47929110a33ea18baa4501bb.jpg",
+        "table_caption": [
+            "Table 4: Search with only $\\kappa { \\mathcal N }$ or $\\hat { R } _ { \\mathcal { N } }$ on CIFAR-100 in NAS-Bench-201 space. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>CIFAR-100 Test Accuracy</td></tr><tr><td>Prune with only KN</td><td>69.25 (1.29)</td></tr><tr><td>Prune with only RN</td><td>70.48 (0.29)</td></tr><tr><td>TE-NAS</td><td>71.24 (0.56)</td></tr></table>",
+        "bbox": [
+            334,
+            492,
+            656,
+            561
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.2 DIFFERENT COMBINATION OPTIONS FOR $\\kappa _ { \\mathcal { N } }$ AND $\\hat { R } _ { \\mathcal { N } }$ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            585,
+            589,
+            602
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Pruning by $s ( o _ { j } ) = s _ { \\kappa } ( o _ { j } ) + s _ { R } ( o _ { j } )$ is not the only option (see Algorithm 1). Here in this study we consider more: ",
+        "bbox": [
+            174,
+            612,
+            825,
+            642
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1) pruning by $s ( o _ { j } ) = \\mathrm { m i n } ( s _ { \\kappa } ( o _ { j } ) , s _ { R } ( o _ { j } ) )$ , i.e., pruning by the worst case. 2) Pruninng by $s ( o _ { j } ) = \\mathrm { m a x } ( s _ { \\kappa } ( o _ { j } ) , s _ { R } ( o _ { j } ) )$ , i.e., pruning by the best case. 3) pruning by summation of changes $\\Delta \\kappa _ { t , o _ { j } } + \\Delta R _ { t , o _ { j } }$ , i.e., directly use the numerical values of the changes. ",
+        "bbox": [
+            204,
+            652,
+            826,
+            720
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As we observed in Table 5, our TE-NAS stands out of all options. This means a good trade-off between $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ are important, and also the ranking strategy is better than directly using numerical values. ",
+        "bbox": [
+            176,
+            731,
+            825,
+            776
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/ecc4f46ada6e994bf0b0a7dc68a4c4ae252d0e89201c8b79aed1f502996f0548.jpg",
+        "table_caption": [
+            "Table 5: Search with only $\\kappa _ { \\mathcal { N } }$ or $\\hat { R } _ { \\mathcal { N } }$ on CIFAR-100 in NAS-Bench-201 space. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>CIFAR-100 Test Accuracy</td></tr><tr><td>Prune by s(oj)= min(sr(oj),sR(0j))</td><td>70.75 (0.73)</td></tr><tr><td>Prune by s(oj)=max(sκ(oj),sr(0j))</td><td>70.33 (1.09)</td></tr><tr><td>Prune by △Kt,oj+△Rt,oj</td><td>70.47 (0.68)</td></tr><tr><td>Prune by s(oj)= sκ(Oj)+sR(oj) (TE-NAS)</td><td>71.24 (0.56)</td></tr></table>",
+        "bbox": [
+            259,
+            813,
+            735,
+            893
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 9 indicates that by using the summation of the ranking of $\\mathsf { b o t h } \\kappa _ { \\mathcal N }$ and $\\hat { R } _ { \\mathcal { N } }$ , the combined metric achieves a much higher correlation with the test accuracy. The reason can be explained by Figure 4, as $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ prefers different operators in terms of trainability and Expressivity. Their combination can filter out bad architectures in both aspects and strongly correlate with networks’ final performance. ",
+        "bbox": [
+            173,
+            130,
+            437,
+            286
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/d0038d4acd6d5cbad42a776602e3e4b8e5adf3361677add29fc779f187849bbf.jpg",
+        "image_caption": [
+            "Figure 9: Summation of ranking of $\\kappa _ { \\mathcal { N } }$ and $\\hat { R } _ { \\mathcal { N } }$ exhibits stronger (negative) correlation with the test accuracy of architectures in NAS-Bench201 (Dong & Yang, 2020). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            504,
+            133,
+            750,
+            276
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E GENERALIZATION V.S. TEST ACCURACY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            358,
+            545,
+            373
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Conceptually, the generalization gap is the difference between a model’s performance on training data and its performance on unseen data drawn from the same distribution (e.g., testing set). In comparison, the two indicators $\\kappa _ { \\mathcal { N } }$ (trainability) and $R _ { \\mathcal { N } }$ (expressiveness) of a network determine how well the training set could be fit (i.e., training set accuracy), and do not directly indicate its generalization gap (or test set accuracy). Indeed, probing generalization of an untrained network at its initialization is a daunting, open challenge that seems to go beyond the current theory scope. ",
+        "bbox": [
+            174,
+            388,
+            446,
+            583
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c758a58794154dd00dc45327533a6e9468a026d51804b24738e4702389e3d289.jpg",
+        "image_caption": [
+            "Figure 10: The correlation between test accuracy and the training accuracy in NAS-Bench201 (Dong & Yang, 2020). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            509,
+            393,
+            754,
+            535
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In NAS, we are searching for the architecture with the best test accuracy. As shown in Figure 10, in NAS-Bench201 the training accuracy strongly correlates with test accuracy. This also seems to be a result of the current standard search space design that could have implicitly excluded severe overfitting. This explains why $\\kappa _ { \\mathcal { N } }$ and $R _ { \\mathcal { N } }$ , which only focuses on trainability and expressiveness during training, can still achieve good search results of test accuracy. ",
+        "bbox": [
+            174,
+            590,
+            825,
+            660
+        ],
+        "page_idx": 14
+    }
+]

parse/train/Hkekl0NFPr/Hkekl0NFPr.md

@@ -0,0 +1,544 @@
+# CONDITIONAL LEARNING OF FAIR REPRESENTATIONS
+
+Han Zhao∗& Amanda Coston Machine Learning Department Carnegie Mellon University han.zhao@cs.cmu.edu acoston@andrew.cmu.edu
+
+Tameem Adel Department of Engineering University of Cambridge tah47@cam.ac.uk
+
+Geoffrey J. Gordon Microsoft Research, Montreal Machine Learning Department Carnegie Mellon University geoff.gordon@microsoft.com
+
+# ABSTRACT
+
+We propose a novel algorithm for learning fair representations that can simultaneously mitigate two notions of disparity among different demographic subgroups in the classification setting. Two key components underpinning the design of our algorithm are balanced error rate and conditional alignment of representations. We show how these two components contribute to ensuring accuracy parity and equalized false-positive and false-negative rates across groups without impacting demographic parity. Furthermore, we also demonstrate both in theory and on two real-world experiments that the proposed algorithm leads to a better utility-fairness trade-off on balanced datasets compared with existing algorithms on learning fair representations for classification.
+
+# 1 INTRODUCTION
+
+High-stakes settings, such as loan approvals, criminal justice, and hiring processes, use machine learning tools to help make decisions. A key question in these settings is whether the algorithm makes fair decisions. In settings that have historically had discrimination, we are interested in defining fairness with respect to a protected group, the group which has historically been disadvantaged. The rapidly growing field of algorithmic fairness has a vast literature that proposes various fairness metrics, characterizes the relationship between fairness metrics, and describes methods to build classifiers that satisfy these metrics (Chouldechova & Roth, 2018; Corbett-Davies & Goel, 2018). Among many recent attempts to achieve algorithmic fairness (Dwork et al., 2012; Hardt et al., 2016; Zemel et al., 2013; Zafar et al., 2015), learning fair representations has attracted increasing attention due to its flexibility in learning rich representations based on advances in deep learning (Edwards & Storkey, 2015; Louizos et al., 2015; Beutel et al., 2017; Madras et al., 2018). The backbone idea underpinning this line of work is very intuitive: if the representations of data from different groups are similar to each other, then any classifier acting on such representations will also be agnostic to the group membership.
+
+However, it has long been empirically observed (Calders et al., 2009) and recently been proved (Zhao & Gordon, 2019) that fairness is often at odds with utility. For example, consider demographic parity, which requires the classifier to be independent of the group membership attribute. It is clear that demographic parity will cripple the utility if the demographic group membership and the target variable are indeed correlated. To escape such inherent trade-off, other notions of fairness, such as equalized odds (Hardt et al., 2016), which asks for equal false positive and negative rates across groups, and accuracy parity (Zafar et al., 2017), which seeks equalized error rates across groups, have been proposed. It is a well-known result that equalized odds is incompatible with demographic parity (Barocas et al., 2017) except in degenerate cases where group membership is independent of the target variable. Accuracy parity and the so-called predictive rate parity (c.f. Definition 2.4) could be simultaneously achieved, e.g., the COMPAS tool (Dieterich et al., 2016). Furthermore, under some conditions, it is also known that demographic parity can lead to accuracy parity (Zhao & Gordon, 2019). However, whether it is possible to simultaneously guarantee equalized odds and accuracy parity remains an open question.
+
+In this work, we provide an affirmative answer to the above question by proposing an algorithm to align the conditional distributions (on the target variable) of representations across different demographic subgroups. The proposed formulation is a minimax problem that admits a simple reduction to cost-sensitive learning. The key component underpinning the design of our algorithm is the balanced error rate (BER, c.f. Section 2) (Feldman et al., 2015; Menon & Williamson, 2018), over the target variable and protected attributes. We demonstrate both in theory and on two real-world experiments that together with the conditional alignment, BER helps our algorithm to simultaneously ensure accuracy parity and equalized odds across groups. Our key contributions are summarized as follows:
+
+• We prove that BER plays a fundamental role in ensuring accuracy parity and a small joint error across groups. Together with the conditional alignment of representations, this implies that we can simultaneously achieve equalized odds and accuracy parity. Furthermore, we also show that when equalized odds is satisfied, BER serves as an upper bound on the error of each subgroup. These results help to justify the design of our algorithm in using BER instead of the marginal error as our loss functions. We provide theoretical results that our method achieves equalized odds without impacting demographic parity. This result shows that we can preserve the demographic parity gap for free while simultaneously achieving equalized odds.   
+• Empirically, among existing fair representation learning methods, we demonstrate that our algorithm is able to achieve a better utility on balanced datasets. On an imbalanced dataset, our algorithm is the only method that achieves accuracy parity; however it does so at the cost of decreased utility.
+
+We believe our theoretical results contribute to the understanding on the relationship between equalized odds and accuracy parity, and the proposed algorithm provides an alternative in real-world scenarios where accuracy parity and equalized odds are desired.
+
+# 2 PRELIMINARY
+
+We first introduce the notations used throughout the paper and formally describe the problem setup and various definitions of fairness explored in the literature.
+
+Notation We use $\mathcal { X } \subseteq \mathbb { R } ^ { d }$ and $\mathcal { V } = \{ 0 , 1 \}$ to denote the input and output space. Accordingly, we use $X$ and $Y$ to denote the random variables which take values in $\mathcal { X }$ and $\mathcal { V }$ , respectively. Lower case letters $\mathbf { x }$ and $y$ are used to denote the instantiation of $X$ and $Y$ . To simplify the presentation, we use $A \in \{ 0 , 1 \}$ as the sensitive attribute, e.g., race, gender, etc. 1 Let $\mathcal { H }$ be the hypothesis class of classifiers. In other words, for $h \in \mathcal H$ , $h : \mathcal { X }  \mathcal { Y }$ is the predictor that outputs a prediction. Note that even if the predictor does not explicitly take the sensitive attribute $A$ as input, this fairness through blindness mechanism can still be biased due to the potential correlations between $X$ and $A$ . In this work we study the stochastic setting where there is a joint distribution $\mathcal { D }$ over $X , Y$ and $A$ from which the data are sampled. To keep the notation consistent, for $a , y \in \{ 0 , 1 \}$ , we use $\mathcal { D } _ { a }$ to mean the conditional distribution of $\mathcal { D }$ given $A = a$ and $\mathcal { D } ^ { y }$ to mean the conditional distribution of $\mathcal { D }$ given $Y = y$ . For an event $E$ , $\mathcal { D } ( E )$ denotes the probability of $E$ under $\mathcal { D }$ . In particular, in the literature of fair machine learning, we call $\mathcal { D } ( Y = 1 )$ the base rate of distribution $\mathcal { D }$ and we use $\Delta _ { \mathrm { B R } } ( \mathcal { D } , \mathcal { D } ^ { \prime } ) : = | \mathcal { D } ( Y = 1 ) - \mathcal { D } ^ { \prime } ( \bar { Y } = 1 ) |$ to denote the difference of the base rates between two distributions $\mathcal { D }$ and $\mathcal { D } ^ { \prime }$ over the same sample space.
+
+Given a feature transformation function $g : \mathcal { X }  \mathcal { Z }$ that maps instances from the input space $\mathcal { X }$ to feature space $\mathcal { Z }$ , we define $g _ { \sharp } \mathcal { D } : = \mathcal { D } \circ g ^ { - 1 }$ to be the induced (pushforward) distribution of $\mathcal { D }$ under $g$ , i.e., for any event $E ^ { \prime } \subseteq { \mathcal { Z } }$ , $g _ { \sharp } \mathcal { D } ( E ^ { \prime } ) : = \mathcal { D } ( g ^ { - 1 } ( E ^ { \prime } ) ) = \mathcal { D } ( \{ x \in \mathcal { X } \mid g ( x ) \in E ^ { \prime } \} )$ . To measure the discrepancy between distributions, we use $d _ { \mathrm { T V } } ( \mathcal { D } , \mathcal { D } ^ { \prime } )$ to denote the total variation between them: $d _ { \mathrm { T V } } ( \mathcal D , \mathcal D ^ { \prime } ) : = \operatorname* { s u p } _ { E } | \mathcal D ( E ) - \mathcal D ^ { \prime } ( E ) |$ . In particular, for binary random variable $Y$ , it can be readily verified that the total variation between the marginal distributions ${ \mathcal { D } } ( Y )$ and $\mathcal { D } ^ { \prime } ( Y )$ reduces to the difference of their base rates, $\Delta _ { \mathrm { B R } } ( \mathcal { D } , \mathcal { D } ^ { \prime } )$ . To see this, realize that $d _ { \mathrm { T V } } ( { \cal D } ( Y ) , { \cal D } ^ { \prime } ( Y ) ) =$ 1 $\begin{array} { r } { \operatorname* { n a x } \{ | \mathcal { D } ( Y = 1 ) - \mathcal { D } ^ { \prime } ( Y = 1 ) | , | \mathcal { D } ( Y = 0 ) - \mathcal { D } ^ { \prime } ( Y = 0 ) | \} = | \mathcal { D } ( Y = 1 ) - \mathcal { D } ^ { \prime } ( Y = 1 ) | = } \end{array}$ $\Delta _ { \mathrm { B R } } ( \mathcal { D } , \mathcal { D } ^ { \prime } )$ by definition.
+
+Given a joint distribution $\mathcal { D }$ , the error of a predictor $h$ under $\mathcal { D }$ is defined as $\mathrm { E r r } _ { \mathcal { D } } ( h ) : = \mathbb { E } _ { \mathcal { D } } [ | Y -$ $h ( X ) | ]$ . Note that for binary classification problems, when $h ( X ) \in \{ 0 , 1 \}$ , $\mathrm { E r r } _ { \mathcal { D } } ( h )$ reduces to the true error rate of binary classification. To make the notation more compact, we may drop the subscript $\mathcal { D }$ when it is clear from the context. Furthermore, we use $\operatorname { C E } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y )$ to denote the crossentropy loss function between the predicted variable $\widehat { Y }$ and the true label distribution $Y$ over the joint distribution $\mathcal { D }$ . For binary random variables $Y$ , we define $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y )$ to be the balanced error rate of predicting $Y$ using $\widehat { Y }$ , e.g., $\operatorname { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) : = \mathcal { D } ( \widehat { Y } = 0 \mid Y = 1 ) + \mathcal { D } ( \widehat { Y } = 1 \mid Y = 0 )$ . Realize that ${ \mathcal { D } } ( \widehat { Y } = 0 \mid Y = 1 )$ is the false negative rate (FNR) of $\widehat { Y }$ and ${ \mathcal { D } } ( \widehat { Y } = 1 \mid Y = 0 )$ corresponds to the false positive rate (FPR). So the balanced error rate can also be understood as the sum of FPR and FNR using the predictor $\widehat { Y }$ . We can similarly define $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { A } \mid \mid A )$ as well.
+
+Problem Setup In this work we focus on group fairness where the group membership is given by the sensitive attribute $A$ . We assume that the sensitive attribute $A$ is available to the learner during training phase, but not inference phase. As a result, post-processing techniques to ensure fairness are not feasible under our setting. In the literature, there are many possible definitions of fairness (Narayanan, 2018), and in what follows we provide a brief review of the ones that are mostly relevant to this work.
+
+Definition 2.1 (Demographic Parity (DP)). Given a joint distribution $\mathcal { D }$ , a classifier $\widehat { Y }$ satisfies demographic parity if $\bar { \hat { Y } }$ is independent of $A$ .
+
+When $\widehat { Y }$ is a deterministic binary classifier, demographic parity reduces to the requirement that $\mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) = \mathcal { D } _ { 1 } ( \widehat { Y } = 1 )$ , i.e., positive outcome is given to the two groups at the same rate. Demographic parity is also known as statistical parity, and it has been adopted as the default definition of fairness in a series of work (Calders & Verwer, 2010; Calders et al., 2009; Edwards & Storkey, 2015; Johndrow et al., 2019; Kamiran & Calders, 2009; Kamishima et al., 2011; Louizos et al., 2015; Zemel et al., 2013; Madras et al., 2018; Adel et al., 2019). It is not surprising that demographic parity may cripple the utility that we hope to achieve, especially in the common scenario where the base rates differ between two groups (Hardt et al., 2016). Formally, the following theorem characterizes the trade-off in terms of the joint error across different groups:
+
+Theorem 2.1. (Zhao & Gordon, 2019) Let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier. Then $\mathrm { E r r } _ { { D _ { 0 } } } ( h \circ g ) +$ $\mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( h \circ g ) \geq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) - d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } , g _ { \sharp } \mathcal { D } _ { 1 } )$ .
+
+In this case of representations that are independent of the sensitive attribute $A$ , then the second term $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } , g _ { \sharp } \mathcal { D } _ { 1 } )$ becomes 0, and this implies:
+
+$$
+\begin{array} { r } { \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( h \circ g ) + \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( h \circ g ) \geq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) . } \end{array}
+$$
+
+At a colloquial level, the above inequality could be understood as an uncertainty principle which says:
+
+For fair representations, it is not possible to construct a predictor that simultaneously minimizes the errors on both demographic subgroups.
+
+More precisely, by the pigeonhole principle, the following corollary holds:
+
+Corollary 2.1. If $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } , g _ { \sharp } \mathcal { D } _ { 1 } ) = 0$ , then for any hypothesis $\begin{array} { r } { h , \operatorname* { m a x } \{ { \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( h \circ g ) , \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( h \circ } } \end{array}$ $g ) \} \} \ge \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) / 2$ .
+
+In words, this means that for fair representations in the demographic parity sense, at least one of the subgroups has to incur a prediction error of at least $\Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) / 2$ which could be large in settings like criminal justice where $\Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } )$ is large. In light of such inherent trade-off, an alternative definition is accuracy parity, which asks for equalized error rates across different groups:
+
+Definition 2.2 (Accuracy Parity). Given a joint distribution $\mathcal { D }$ , a classifier $\widehat { Y }$ satisfies accuracy parity if ${ \mathcal { D } } _ { 0 } ( { \widehat { Y } } \neq Y ) = { \mathcal { D } } _ { 1 } ( { \widehat { Y } } ^ { * } \neq Y )$ .
+
+A violation of accuracy parity is also known as disparate mistreatment (Zafar et al., 2017). Different from the definition of demographic parity, the definition of accuracy parity does not eliminate the perfect predictor even when the base rates differ across groups. Of course, other more refined definitions of fairness also exist in the literature, such as equalized odds (Hardt et al., 2016).
+
+Definition 2.3 (Equalized Odds, a.k.a. Positive Rate Parity). Given a joint distribution $\mathcal { D }$ , a classifier $\widehat { Y }$ satisfies equalized odds if $\widehat { Y }$ is independent of $A$ conditioned on $Y$ .
+
+The definition of equalized odds essentially requires equal true positive and false positive rates between different groups, hence it is also known as positive rate parity. Analogous to accuracy parity, equalized odds does not eliminate the perfect classifier (Hardt et al., 2016), and we will also justify this observation by formal analysis shortly. Last but not least, we have the following definition for predictive rate parity:
+
+Definition 2.4 (Predictive Rate Parity). Given a joint distribution $\mathcal { D }$ , a classifier $\widehat { Y }$ satisfies predictive rate parity if $\mathcal { D } _ { 0 } ( Y = 1 \vert \widehat { Y } = c ) = \mathcal { D } _ { 1 } ( Y = 1 \vert \widehat { Y } = c ) , \forall c \in [ 0 , 1$ .
+
+Note that in the above definition we allow the classifier $\widehat { Y }$ to be probabilistic, meaning that the output of $\widehat { Y }$ could be any value between 0 and 1. For the case where $\widehat { Y }$ is deterministic, Chouldechova (2017) showed that no deterministic classifier can simultaneously satisfy equalized odds and predictive rate parity when the base rates differ across subgroups and the classifier is not perfect.
+
+# 3 ALGORITHM AND ANALYSIS
+
+In this section we first give the proposed optimization formulation and then discuss through formal analysis the motivation of our algorithmic design. Specifically, we show in Section 3.1 why our formulation helps to escape the utility-fairness trade-off. We then in Section 3.2 formally prove that the BERs in the objective function could be used to guarantee a small joint error across different demographic subgroups. In Section 3.3 we establish the relationship between equalized odds and accuracy parity by providing an upper bound of the error gap in terms of both BER and the equalized odds gap. We conclude this section with a brief discussion on the practical implementation of the proposed optimization formulation in Section 3.4. Due to the space limit, we defer all the proofs to the appendix and focus on explaining the high-level intuition and implications of our results.
+
+As briefly discussed in Section 5, a dominant approach in learning fair representations is via adversar ial training. Specifically, the following objective function is optimized:
+
+$$
+\operatorname* { m i n } _ { h , g } \operatorname* { m a x } _ { h ^ { \prime } } \mathrm { C E } _ { \mathcal { D } } ( h ( g ( X ) ) | | Y  ) - \lambda \mathrm { C E } _ { \mathcal { D } } ( h ^ { \prime } ( g ( X ) ) | | \begin{array} { l } { A } \end{array}  )
+$$
+
+In the above formulation, the first term corresponds to minimization of prediction loss of the target variable and the second term represents the loss incurred by the adversary $h ^ { \prime }$ . Overall this minimax optimization problem expresses a trade-off (controlled by the hyperparameter $\lambda > 0$ ) between utility and fairness through the representation learning function $g$ : on one hand $g$ needs to preserve sufficient information related to $Y$ in order to minimize the first term, but on the other hand $g$ also needs to filter out information related to $A$ in order to maximize the second term.
+
+# 3.1 CONDITIONAL LEARNING OF FAIR REPRESENTATIONS
+
+However, as we introduced in Section 2, the above framework is still subjective to the inherent trade-off between utility and fairness. To escape such a trade-off, we advocate for the following optimization formulation instead:
+
+$$
+\operatorname* { m i n } _ { h , g } \ \operatorname* { m a x } _ { h ^ { \prime } , h ^ { \prime \prime } } \quad \mathrm { B E R } _ { \mathcal { D } } ( h ( g ( X ) ) \mid \mid Y ) - \lambda \left( \mathrm { B E R } _ { \mathcal { D } ^ { 0 } } ( h ^ { \prime } ( g ( X ) ) \mid \mid A ) + \mathrm { B E R } _ { \mathcal { D } ^ { 1 } } ( h ^ { \prime \prime } ( g ( X ) ) \mid \mid A ) \right)
+$$
+
+Note that here we optimize over two distinct adversaries, one for each conditional distribution $\mathcal { D } ^ { y }$ , $y \in \{ 0 , 1 \}$ . Intuitively, the main difference between (2) and (1) is that we use BER as our objective function in both terms. By definition, since BER corresponds to the sum of Type-I and Type-II errors in classification, the proposed objective function essentially minimizes the conditional errors instead of the original marginal error:
+
+$$
+\begin{array} { r l } & { \bar { \mathcal { D } } \hat { ( Y \ne Y ) } = \bar { \mathcal { D } } \hat { ( Y = 0 ) } \mathcal { D } \hat { ( Y \ne Y \mid Y = 0 ) } + \bar { \mathcal { D } } \hat { ( Y = 1 ) } \bar { \mathcal { D } } \hat { ( Y \ne Y \mid Y = 1 ) } } \\ & { \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid | Y ) \propto \displaystyle \frac { 1 } { 2 } \bar { \mathcal { D } } ( \widehat { Y } \ne Y \mid Y = 0 ) + \frac { 1 } { 2 } \bar { \mathcal { D } } ( \widehat { Y } \ne Y \mid Y = 1 ) , } \end{array}
+$$
+
+which means that the loss function gives equal importance to the classification error from both groups. Note that the BERs in the second term of (2) are over $\mathcal { D } ^ { y }$ , $y \in \{ 0 , 1 \}$ . Roughly speaking, the second term encourages alignment of the the conditional distributions $g _ { \sharp } \mathcal { D } _ { 0 } ^ { y }$ and $\bar { g } _ { \sharp } \bar { \mathcal { D } } _ { 1 } ^ { y }$ for $y \in \{ 0 , 1 \}$ . The following proposition shows that a perfect conditional alignment of the representations also implies that any classifier based on the representations naturally satisfies the equalized odds criterion:
+
+Proposition 3.1. For $g : \mathcal { X }  \mathcal { Z }$ , if $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } ^ { y } , g _ { \sharp } \mathcal { D } _ { 1 } ^ { y } ) = 0 , \forall y \in \{ 0 , 1 \}$ , then for any classifier $h : \mathcal { Z }  \{ 0 , 1 \}$ , $h \circ g$ satisfies equalized odds.
+
+To understand why we aim for conditional alignment of distributions instead of aligning the marginal feature distributions, the following proposition characterizes why such alignment will help us to escape the previous trade-off:
+
+Proposition 3.2. For $g : \mathcal { X }  \mathcal { Z }$ , if $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } ^ { y } , g _ { \sharp } \mathcal { D } _ { 1 } ^ { y } ) = 0 , \forall y \in \{ 0 , 1 \}$ , then for any classifier $h : \bar { \mathcal { Z } } \to \{ 0 , 1 \}$ , $d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathscr { D } _ { 0 } , ( h \circ g ) _ { \sharp } \mathscr { D } _ { 1 } ) \leq \Delta _ { \mathrm { B R } } ( \mathscr { D } _ { 0 } , \mathscr { D } _ { 1 } )$ .
+
+As a corollary, this implies that the lower bound given in Theorem 2.1 now vanishes if we instead align the conditional distributions of representations:
+
+$$
+\begin{array} { r } { \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( h \circ g ) + \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( h \circ g ) \geq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) - d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathcal { D } _ { 0 } , ( h \circ g ) _ { \sharp } \mathcal { D } _ { 1 } ) = 0 , } \end{array}
+$$
+
+where the first inequality is due to Lemma 3.1 (Zhao & Gordon, 2019) and the triangle inequality by the $d _ { \mathrm { T V } } ( \cdot , \cdot )$ distance. Of course, the above lower bound can only serve as a necessary condition but not sufficient to ensure a small joint error across groups. Later (c.f. Theorem 3.2) we will show that together with a small BER on the target variable, it becomes a sufficient condition as well.
+
+3.2 THE PRESERVATION OF DEMOGRAPHIC PARITY GAP AND SMALL JOINT ERROR
+
+In this section we show that learning representations by aligning the conditional distributions across groups cannot increase the DP gap as compared to the DP gap of $Y$ . Before we proceed, we first introduce a metric to measure the deviation of a predictor from satisfying demographic parity:
+
+Definition 3.1 (DP Gap). Given a joint distribution $\mathcal { D }$ , the demographic parity gap of a classifier $\widehat { Y }$ is $\Delta _ { \mathrm { D P } } ( \widehat { Y } ) : = | \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) |$ .
+
+Clearly, if $\Delta _ { \mathrm { D P } } ( \widehat { Y } ) = 0$ , then $\widehat { Y }$ satisfies demographic parity. To simplify the exposition, let $\gamma _ { a } : = { \mathscr { D } } _ { a } ( Y = 0 ) , \forall a \in \{ 0 , 1 \}$ . We first prove the following lemma:
+
+Lemma 3.1. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\begin{array} { r } { | \mathcal D _ { 0 } ( \widehat { Y } = y ) - \mathcal D _ { 1 } ( \widehat { Y } = y ) | \le | \gamma _ { 0 } - \gamma _ { 1 } | \cdot \big ( \mathcal D ^ { 0 } ( \widehat { Y } = y ) + \mathcal D ^ { 1 } ( \widehat { Y } = y ) \big ) , \forall y \in \{ 0 , 1 \} . } \end{array}$ }.
+
+Lemma 3.1 gives an upper bound on the difference of the prediction probabilities across different subgroups. Applying Lemma 3.1 twice for $y = 0$ and $y = 1$ , we can prove the following theorem:
+
+Theorem 3.1. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\Delta _ { \mathrm { D P } } ( \widehat { Y } ) \leq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) = \Delta _ { \mathrm { D P } } ( Y )$ .
+
+Remark Theorem 3.1 shows that aligning the conditional distributions of representations between groups will not add more bias in terms of the demographic parity gap. In particular, the DP gap of any classifier that satisfies equalized odds will be at most the DP gap of the perfect classifier. This is particularly interesting as it is well-known in the literature (Barocas et al., 2017) that demographic parity is not compatible with equalized odds except in degenerate cases. Despite this result, Theorem 3.1 says that we can still achieve equalized odds and simultaneously preserve the DP gap.
+
+In Section 3.1 we show that aligning the conditional distributions of representations between groups helps reduce the lower bound of the joint error, but nevertheless that is only a necessary condition. In the next theorem we show that together with a small Type-I and Type-II error in inferring the target variable $Y$ , these two properties are also sufficient to ensure a small joint error across different demographic subgroups.
+
+Theorem 3.2. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) + \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \le 2 \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y )$ .
+
+The above bound means that in order to achieve small joint error across groups, it suffices for us to minimize the BER if a classifier satisfies equalized odds. Note that by definition, the BER in the bound equals to the sum of Type-I and Type-II classification errors using $\widehat { Y }$ as a classifier. Theorem 3.2 gives an upper bound of the joint error across groups and it also serves as a motivation for us to design the optimization formulation (2) that simultaneously minimizes the BER and aligns the conditional distributions.
+
+# 3.3 CONDITIONAL ALIGNMENT AND BALANCED ERROR RATES LEAD TO SMALL ERROR
+
+In this section we will see that a small BER and equalized odds together not only serve as a guarantee of a small joint error, but they also lead to a small error gap between different demographic subgroups. Recall that we define $\gamma _ { a } : = \mathscr { D } _ { a } ( Y = 0 ) , \forall a \in \{ 0 , 1 \}$ . Before we proceed, we first formally define the accuracy gap and equalized odds gap of a classifier $\widehat { Y }$ :
+
+Definition 3.2 (Error Gap). Given a joint distribution $\mathcal { D }$ , the error gap of a classifier $\widehat { Y }$ is $\Delta _ { \mathrm { E r r } } ( \widehat { Y } ) : =$ $\vert \mathcal { D } _ { 0 } ( \widehat { Y } \neq Y ) - \mathcal { D } _ { 1 } ( \widehat { Y } \neq Y ) \vert$ .
+
+Definition 3.3 (Equalized Odds Gap). Given a joint distribution $\mathcal { D }$ , the equalized odds gap of a classifier $\widehat { Y }$ is $\begin{array} { r } { \bar { \Delta _ { \mathrm { E O } } ( \widehat { Y } ) } : = \operatorname* { m a x } _ { y \in \{ 0 , 1 \} } \ | \mathcal { D } _ { 0 } ^ { y } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ^ { y } ( \widehat { Y } = 1 ) | } \end{array}$ .
+
+By definition the error gap could also be understood as the accuracy parity gap between different subgroups. The following theorem characterizes the relationship between error gap, equalized odds gap and the difference of base rates across subgroups:
+
+As a direct corollary of Theorem 3.3, if the classifier $\widehat { Y }$ satisfies equalized odds, then $\Delta _ { \mathrm { E O } } ( \widehat { Y } ) = 0$ . In this case since $\Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } )$ is a constant, minimizing the balanced error rate $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y )$ also leads to minimizing the error gap. Furthermore, if we combine Theorem 3.2 and Theorem 3.3 together, we can guarantee that each of the errors cannot be too large:
+
+Corollary 3.1. For any joint distribution $\mathcal { D }$ and classifier $\widehat { Y }$ , if $\widehat { Y }$ satisfies equalized odds, then $\operatorname* { m a x } \{ \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) , \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \} \leq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y ) / 2 + \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y ) .$
+
+Remark It is a well-known fact that out of the three fairness criteria, i.e., demographic parity, equalized odds, and predictive rate parity, any two of them cannot hold simultaneously (Barocas et al., 2017) except in degenerate cases. By contrast, Theorem 3.3 suggests it $i s$ possible to achieve both equalized odds and accuracy parity. In particular, among all classifiers that satisfy equalize odds, it suffices to minimize the sum of Type-I and Type-II error $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y )$ in order to achieve accuracy parity. It is also worth pointing out that Theorem 3.3 provides only an upper bound, but not necessarily the tightest one. In particular, the error gap could still be 0 while $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y )$ is greater than 0. To see this, we have
+
+$$
+\begin{array} { r } { \left\{ \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) = \mathcal { D } _ { 0 } ( Y = 0 ) \cdot \mathcal { D } _ { 0 } ( \widehat { Y } = 1 \mid Y = 0 ) + \mathcal { D } _ { 0 } ( Y = 1 ) \cdot \mathcal { D } _ { 0 } ( \widehat { Y } = 0 \mid Y = 1 ) \right. } \\ { \left. \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) = \mathcal { D } _ { 1 } ( Y = 0 ) \cdot \mathcal { D } _ { 1 } ( \widehat { Y } = 1 \mid Y = 0 ) + \mathcal { D } _ { 1 } ( Y = 1 ) \cdot \mathcal { D } _ { 1 } ( \widehat { Y } = 0 \mid Y = 1 ) . \right. } \end{array}
+$$
+
+Now if the predictor $\widehat { Y }$ satisfies equalized odds, then
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { 0 } \big ( \widehat { Y } = 1 \mid Y = 0 \big ) = \mathcal { D } _ { 1 } \big ( \widehat { Y } = 1 \mid Y = 0 \big ) = \mathcal { D } \big ( \widehat { Y } = 1 \mid Y = 0 \big ) , } \\ & { \mathcal { D } _ { 0 } \big ( \widehat { Y } = 0 \mid Y = 1 \big ) = \mathcal { D } _ { 1 } \big ( \widehat { Y } = 0 \mid Y = 1 \big ) = \mathcal { D } \big ( \widehat { Y } = 0 \mid Y = 1 \big ) . } \end{array}
+$$
+
+Hence the error gap $\Delta _ { \mathrm { E r r } } ( \widehat { Y } )$ admits the following identity:
+
+$$
+\begin{array} { r l } & { \Delta _ { \mathrm { E r r } } ( \widehat { Y } ) = \Big | \mathcal { D } ( \widehat { Y } = 1 \mid Y = 0 ) \big ( \mathcal { D } _ { 0 } ( Y = 0 ) - \mathcal { D } _ { 1 } ( Y = 0 ) \big ) + \mathcal { D } ( \widehat { Y } = 0 \mid Y = 1 ) \big ( \mathcal { D } _ { 0 } ( Y = 1 ) - \mathcal { D } _ { 1 } ( Y = 0 ) \big ) } \\ & { \qquad = \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \Big | \mathcal { D } ( \widehat { Y } = 1 \mid Y = 0 ) - \mathcal { D } ( \widehat { Y } = 0 \mid Y = 1 ) \Big | } \\ & { \qquad = \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \Big | \mathrm { F P R } ( \widehat { Y } ) - \mathrm { F N R } ( \widehat { Y } ) \Big | . } \end{array}
+$$
+
+In other words, if the predictor $\widehat { Y }$ satisfies equalized odds, then in order to have equalized accuracy, $\widehat { Y }$ only needs to have equalized FPR and FNR globally when the base rates differ across groups. This is a much weaker condition to ask for than the one asking $\mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) = 0$ .
+
+# 3.4 PRACTICAL IMPLEMENTATION
+
+We cannot directly optimize the proposed optimization formulation (2) since the binary $0 / 1$ loss is NPhard to optimize, or even approximately optimize over a wide range of hypothesis classes (Ben-David et al., 2003). However, observe that for any classifier $\widehat { Y }$ and $y \in \{ 0 , 1 \}$ , the log-loss (cross-entropy loss) $\mathrm { C E } _ { { D } ^ { y } } ( \widehat { Y } \mid \mid Y )$ is a convex relaxation of the binary loss:
+
+$$
+\mathcal { D } ( \widehat { Y } \neq y \mid Y = y ) = \frac { \mathcal { D } ( \widehat { Y } \neq y , Y = y ) } { \mathcal { D } ( Y = y ) } \leq \frac { \mathrm { C E } _ { \mathcal { D } ^ { y } } ( \widehat { Y } \mid \mid Y ) } { \mathcal { D } ( Y = y ) } .
+$$
+
+Hence in practice we can relax the optimization problem (2) to a cost-sensitive cross-entropy loss minimization problem, where the weight for each class is given by the inverse marginal probability of the corresponding class. This allows us to equivalently optimize the objective function without explicitly computing the conditional distributions.
+
+# 4 EMPIRICAL STUDIES
+
+In light of our theoretic findings, in this section we verify the effectiveness of the proposed algorithm in simultaneously ensuring equalized odds and accuracy parity using real-world datasets. We also analyze the impact of imposing such parity constraints on the utility of the target classifier, as well as its relationship to the intrinsic structure of the binary classification problem, e.g., the difference of base rates across groups, the global imbalance of the target variable, etc. We analyze how this imbalance affects the utility-fairness trade-off. As we shall see shortly, we will empirically demonstrate that, in many cases, especially the ones where the dataset is imbalanced in terms of the target variable, this will inevitably compromise the target utility. While for balanced datasets, this trend is less obvious: the proposed algorithm achieves a better fairness-utility trade-off when compared with existing fair representation learning methods and we can hope to achieve fairness without sacrificing too much on utility.
+
+Table 1: Statistics about the Adult and COMPAS datasets.   
+
+<table><tr><td></td><td>Train /Test</td><td>D(Y = 1)</td><td>D(Y = 1)</td><td>△BR(D0,D1)</td><td>D(Y = 1)</td><td>D(A=1)</td></tr><tr><td>Adult</td><td>30,162/15,060</td><td>0.310</td><td>0.113</td><td>0.196</td><td>0.246</td><td>0.673</td></tr><tr><td>COMPAS</td><td>4,320/1,852</td><td>0.400</td><td>0.529</td><td>0.129</td><td>0.467</td><td>0.514</td></tr></table>
+
+# 4.1 EXPERIMENTAL SETUP
+
+To this end, we perform experiments on two popular real-world datasets in the literature of algorithmic fairness, including an income-prediction dataset, known as the Adult dataset, from the UCI Machine Learning Repository (Dua & Graff, 2017), and the Propublica COMPAS dataset (Dieterich et al., 2016). The basic statistics of these two datasets are listed in Table 1.
+
+Adult Each instance in the Adult dataset describes an adult, e.g., gender, education level, age, etc, from the 1994 US Census. In this dataset we use gender as the sensitive attribute, and the processed data contains 114 attributes. The target variable (income) is also binary: 1 if $\ge 5 0 \mathrm { K }$ /year otherwise 0. For the sensitive attribute $A$ , $A = 0$ means male otherwise female. From Table 1 we can see that the base rates are quite different (0.310 vs. 0.113) across groups in the Adult dataset. The dataset is also imbalanced in the sense that only around $2 4 . 6 \%$ of the instances have target label 1. Furthermore, the group ratio is also imbalanced: roughly $6 7 . 3 \%$ of the data are male.
+
+COMPAS The goal of the COMPAS dataset is binary classification on whether a criminal defendant will recidivate within two years or not. Each instance contains 12 attributes, including age, race, gender, number of prior crimes, etc. For this dataset, we use the race (white $A = 0$ vs. black $A = 1$ ) as our sensitive attribute and target variable is 1 iff recidivism. The base rates are different across groups, but the COMPAS dataset is balanced in both the target variable and the sensitive attribute.
+
+To validate the effect of ensuring equalized odds and accuracy parity, for each dataset, we perform controlled experiments by fixing the baseline network architecture so that it is shared among all the fair representation learning methods. We term the proposed method CFAIR (for conditional fair representations) that minimizes conditional errors both the target variable loss function and adversary loss function. To demonstrate the importance of using BER in the loss function of target variable, we compare with a variant of CFAIR that only uses BER in the loss of adversaries, denoted as CFAIR-EO. To see the relative effect of using cross-entropy loss vs $L _ { 1 }$ loss, we also show the results of LAFTR (Madras et al., 2018), a state-of-the-art method for learning fair representations. Note that LAFTR is closely related to CFAIR-EO but slightly different: LAFTR uses global crossentropy loss for target variable, but conditional $L _ { 1 }$ loss for the adversary. Also, there is only one adversary in LAFTR, while there are two adversaries, one for $\mathcal { D } ^ { 0 }$ and one for $\mathcal { D } ^ { 1 }$ , in both CFAIR and CFAIR-EO. Lastly, we also present baseline results of FAIR (Edwards & Storkey, 2015), which aims for demographic parity representations, and NODEBIAS, the baseline network without any fairness constraint. For all the fair representation learning methods, we use the gradient reversal layer (Ganin et al., 2016) to implement the gradient descent ascent (GDA) algorithm to optimize the minimax problem. All the experimental details, including network architectures, learning rates, batch sizes, etc. are provided in the appendix.
+
+# 4.2 RESULTS AND ANALYSIS
+
+In Figure 1 and Figure 2 we show the error gap $\Delta _ { \mathrm { E r r } }$ , equalized odds gap $\Delta _ { \mathrm { E O } }$ , demographic parity gap $\Delta _ { \mathrm { D P } }$ and the joint error across groups $\mathrm { E r r } _ { 0 } + \mathrm { E r r } _ { 1 }$ of the aforementioned fair representation learning algorithms on both the Adult and the COMPAS datasets. For each algorithm and dataset, we also gradually increase the value of the trade-off parameter $\lambda$ and compute the corresponding metrics.
+
+![](images/5de21dd3a7f7921e963984f5ca3393b4963bb12ae0c8e03867e652dde0c107ec.jpg)  
+Figure 1: The error gap $\Delta _ { \mathrm { E r r } }$ , equalized odds gap $\Delta _ { \mathrm { E O } }$ , demographic parity gap $\Delta _ { \mathrm { D P } }$ and joint error $\mathrm { E r r } _ { 0 } + \mathrm { E r r } _ { 1 }$ on the Adult dataset with $\lambda \in \{ 0 . 1 , 1 . 0 , 1 0 . 0 , 1 0 0 . 0 , 1 0 0 0 . 0 \}$ .
+
+![](images/2e9efab0a2c9e7ed458885c663b08ce1adf75248aabc798703141bda2b97337c.jpg)  
+Figure 2: The error gap $\Delta _ { \mathrm { E r r } }$ , equalized odds gap $\Delta _ { \mathrm { E O } }$ , demographic parity gap $\Delta _ { \mathrm { D P } }$ and joint error $\mathrm { E r r } _ { 0 } + \mathrm { E r r } _ { 1 }$ on the COMPAS dataset with $\lambda \in \{ 0 . 1 , 1 . 0 , 1 0 . 0 \}$ .
+
+Adult Due to the imbalance of $A$ in the Adult dataset, in the first plot of Figure 1 we can see that all the algorithms except CFAIR have a large error gap of around 0.12. As a comparison, observe that the error gap of CFAIR when $\lambda = 1 e 3$ almost reduces to 0, confirming the effectiveness of our algorithm in ensuring accuracy parity. From the second plot, we can verify that all the three methods, including CFAIR, CFAIR-EO and LAFTR successfully ensure a small equalized odds gap, and they also decrease demographic parity gaps (the third plot). FAIR is the most effective one in mitigating $\Delta _ { \mathrm { D P } }$ since its objective function directly optimizes for that goal. Note that from the second plot we can also confirm that CFAIR-EO is more effective than LAFTR in reducing $\Delta _ { \mathrm { E O } }$ . The reason is two-fold. First, CFAIR-EO uses two distinct adversaries and hence it effectively competes with stronger adversaries than LAFTR. Second, CFAIR-EO uses the cross-entropy loss instead of the $L _ { 1 }$ loss for the adversary, and it is well-known that the maximum-likelihood estimator (equivalent to using the cross-entropy loss) is asymptotically efficient and optimal. On the other hand, since the Adult dataset is imbalanced (in terms of $Y$ ), using BER in the loss function of the target variable can thus to a large extent hurt the utility, and this is also confirmed from the last plot, where we show the joint error.
+
+COMPAS The first three plots of Figure 2 once again verify that CFAIR successfully leads to reduced error gap, equalized odds gap and also demographic parity gap. These experimental results are consistent with our theoretical findings where we show that if the representations satisfy equalized odds, then its $\Delta _ { \mathrm { D P } }$ cannot exceed that of the optimal classifier, as shown by the horizontal dashed line in the third plot. In the fourth plot of Figure 2, we can see that as we increase $\lambda$ , all the fair representation learning algorithms sacrifice utility. However, in contrast to Figure 1, here the proposed algorithm CFAIR has the smallest trade-off: this shows that CFAIR is particularly suited in the cases when the dataset is balanced and we would like to simultaneously ensure accuracy parity and equalized odds. As a comparison, while CFAIR-EO is still effective, it is slightly worse than CFAIR in terms of both ensuring parity and achieving small joint error.
+
+# 5 RELATED WORK
+
+Algorithmic Fairness In the literature of algorithmic fairness, two key notions of fairness have been extensively proposed and explored, i.e., group fairness, including various variants defined in Section 2, and individual fairness, which means that similar individuals should be treated similarly. Due to the complexity in defining a distance metric to measure the similarity between individuals (Dwork et al., 2012), most recent research focuses on designing efficient algorithms to achieve group fairness (Zemel et al., 2013; Zafar et al., 2015; Hardt et al., 2016; Zafar et al., 2017; Madras et al., 2018; Creager et al., 2019; Madras et al., 2019). In particular, Hardt et al. (2016) proposed a post-processing technique to achieve equalized odds by taking as input the model’s prediction and the sensitive attribute. However, the post-processing technique requires access to the sensitive attribute during the inference phase, which is often not available in many real-world scenarios. Another line of work uses causal inference to define notions of causal fairness and to formulate procedures for achieving these notions (Zhang et al., 2018; Wang et al., 2019; Madras et al., 2019; Kilbertus et al., 2017; Kusner et al., 2017; Nabi & Shpitser, 2018). These approaches require making untestable assumptions. Of particular note is the observation in Coston et al. (2019) that fairness-adjustment procedures based on $Y$ in settings with treatment effects may lead to adverse outcomes. To apply our method in such settings, we would need to match conditional counterfactual distributions, which could be a direction of future research.
+
+Theoretical Results on Fairness Theoretical work studying the relationship between different kinds of fairness notions are abundant. Motivated by the controversy of the potential discriminatory bias in recidivism prediction instruments, Chouldechova (2017) showed an intrinsic incompatibility between equalized odds and predictive rate parity. In the seminal work of Kleinberg et al. (2016), the authors demonstrated that when the base rates differ between different groups, then a non-perfect classifier cannot simultaneously be statistically calibrated and satisfy equalized odds. In the context of cost-sensitive learning, Menon & Williamson (2018) show that if the optimal decision function is dissimilar to a fair decision, then the fairness constraint will not significantly harm the target utility. The idea of reducing fair classification to cost-sensitive learning is not new. Agarwal et al. (2018) explored the connection between fair classification and a sequence of cost-sensitive learning problems where each stage corresponds to solving a linear minimax saddle point problem. In a recent work (Zhao & Gordon, 2019), the authors proved a lower bound on the joint error across different groups when a classifier satisfies demographic parity. They also showed that when the decision functions are close between groups, demographic parity also implies accuracy parity. The theoretical results in this work establish a relationship between accuracy parity and equalized odds: these two fairness notions are fundamentally related by the base rate gap and the balanced error rate. Furthermore, we also show that for any predictor that satisfies equalized odds, the balanced error rate also serves as an upper bound on the joint error across demographic subgroups.
+
+Fair Representations Through the lens of representation learning, recent advances in building fair algorithmic decision making systems focus on using adversarial methods to learn fair representations that also preserve sufficient information for the prediction task (Edwards & Storkey, 2015; Beutel et al., 2017; Zhang et al., 2018; Madras et al., 2018; Adel et al., 2019). In a nutshell, the key idea is to frame the problem of learning fair representations as a two-player game, where the data owner is competing against an adversary. The goal of the adversary is to infer the group attribute as much as possible while the goal of the data owner is to remove information related to the group attribute and simultaneously to preserve utility-related information for accurate prediction. Apart from using adversarial classifiers to enforce group fairness, other distance metrics have also been used to learn fair representations, e.g., the maximum mean discrepancy (Louizos et al., 2015), and the Wasserstein-1 distance (Jiang et al., 2019). In contrast to these methods, in this paper we advocate for optimizing BER on both the target loss and adversary loss in order to simultaneously achieve accuracy parity and equalized odds. We also show that this leads to better utility-fairness trade-off for balanced datasets.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+In this paper we propose a novel representation learning algorithm that aims to simultaneously ensure accuracy parity and equalized odds. The main idea underlying the design of our algorithm is to align the conditional distributions of representations (rather than marginal distributions) and use balanced error rate (i.e., the conditional error) on both the target variable and the sensitive attribute. Theoretically, we prove how these two concepts together help to ensure accuracy parity and equalized odds without impacting demographic parity, and we also show how these two can be used to give a guarantee on the joint error across different demographic subgroups. Empirically, we demonstrate on two real-world experiments that the proposed algorithm effectively leads to the desired notions of fairness, and it also leads to better utility-fairness trade-off on balanced datasets.
+
+Calibration and Utility Our work takes a step towards better understanding the relationships between different notions of fairness and their corresponding trade-off with utility. In some scenarios, e.g., the COMPAS tool, it is desired to have a decision making system that is also well calibrated. While it is well-known that statistical calibration is not compatible with demographic parity or equalized odds, from a theoretical standpoint it is still not clear whether calibration will harm utility and if so, what is the fundamental limit of a calibrated tool on utility.
+
+Fairness and Privacy Future work could also investigate how to make use of the close relationship between privacy and group fairness. At a colloquial level, fairness constraints require a predictor to be (to some extent) agnostic about the group membership attribute. The membership query attack in privacy asks the same question – is it possible to guarantee that even an optimal adversary cannot steal personal information through inference attacks. Prior work (Dwork et al., 2012) has described the connection between the notion of individual fairness and differential privacy. Hence it would be interesting to exploit techniques developed in the literature of privacy to develop more efficient fairness-aware learning algorithms. On the other hand, results obtained in the algorithmic fairness literature could also potentially lead to better privacy-preserving machine learning algorithms (Zhao et al., 2019).
+
+# ACKNOWLEDGMENTS
+
+HZ and GG would like to acknowledge support from the DARPA XAI project, contract #FA87501720152 and an Nvidia GPU grant. HZ would also like to thank Richard Zemel, Toniann Pitassi, David Madras and Elliot Creager for helpful discussions during HZ’s visit to the Vector Institute.
+
+# REFERENCES
+
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+
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+
+Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness in machine learning. NIPS Tutorial, 2017.
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+
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+
+# A PROOFS
+
+# A.1 PROOF OF PROPOSITION 3.1
+
+Proposition 3.1. For $g : \mathcal { X }  \mathcal { Z }$ , if $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } ^ { y } , g _ { \sharp } \mathcal { D } _ { 1 } ^ { y } ) = 0 , \forall y \in \{ 0 , 1 \}$ , then for any classifier $h : \bar { \mathcal { Z } } \to \{ 0 , 1 \}$ , $h \circ g$ satisfies equalized odds.
+
+Proof. To prove this proposition, we first show that for any pair of distributions $\mathcal { D }$ , $\mathcal { D } ^ { \prime }$ over $\mathcal { Z }$ and any hypothesis $h : \bar { \mathcal { Z } }  \bar { \{ 0 , 1 \} }$ , $d _ { \mathrm { T V } } ( h _ { \sharp } \mathcal { D } , h _ { \sharp } \mathcal { D } ^ { \prime } ) \leq d _ { \mathrm { T V } } ( \boldsymbol { \bar { D } } , \boldsymbol { \bar { D } } ^ { \prime } )$ . Note that since $h$ is a hypothesis, there are only two events in the induced probability space, i.e., $h ( \cdot ) = 0$ or $h ( \cdot ) = 1$ . Hence by definition of the induced (pushforward) distribution, we have:
+
+$$
+\begin{array} { r l } { d _ { \mathrm { T V } } ( h _ { \sharp } \mathcal { D } , h _ { \sharp } \mathcal { D } ^ { \prime } ) = \underset { { E = h ^ { - 1 } ( 0 ) , \mathrm { ~ o r } E = h ^ { - 1 } ( 1 ) } } { \mathrm { m a x } } | \mathcal { D } ( E ) - \mathcal { D } ^ { \prime } ( E ) | } & { } \\ { \leq \underset { { E \mathrm { ~ i s ~ m e a s u r a b l e ~ s u b s e t ~ o f ~ } \mathcal { Z } } } { \mathrm { s u p } } | \mathcal { D } ( E ) - \mathcal { D } ^ { \prime } ( E ) | } & { } \\ { = d _ { \mathrm { T V } } ( \mathcal { D } , \mathcal { D } ^ { \prime } ) . } \end{array}
+$$
+
+Apply the above inequality twice for $y \in \{ 0 , 1 \}$ :
+
+$$
+0 \leq d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathscr { D } _ { 0 } ^ { y } , ( h \circ g ) _ { \sharp } \mathscr { D } _ { 1 } ^ { y } ) \leq d _ { \mathrm { T V } } ( g _ { \sharp } \mathscr { D } _ { 0 } ^ { y } , g _ { \sharp } \mathscr { D } _ { 1 } ^ { y } ) = 0 ,
+$$
+
+meaning
+
+$$
+d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathscr { D } _ { 0 } ^ { y } , ( h \circ g ) _ { \sharp } \mathscr { D } _ { 1 } ^ { y } ) = 0 ,
+$$
+
+which further implies that $h ( g ( X ) )$ is independent of $A$ given $Y = y$ since $h ( g ( X ) )$ is binary.
+
+# A.2 PROOF OF PROPOSITION 3.2
+
+Proposition 3.2. For $g : \mathcal { X }  \mathcal { Z }$ , if $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } ^ { y } , g _ { \sharp } \mathcal { D } _ { 1 } ^ { y } ) = 0 , \forall y \in \{ 0 , 1 \}$ , then for any classifier $h : \mathcal { Z }  \{ 0 , 1 \}$ , $d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathscr { D } _ { 0 } , ( h \circ g ) _ { \sharp } \mathscr { D } _ { 1 } ) \leq \Delta _ { \mathrm { B R } } ( \mathscr { D } _ { 0 } , \mathscr { D } _ { 1 } )$ .
+
+Proof. Let ${ \widehat { Y } } = ( h \circ g ) ( X )$ and note that $\widehat { Y }$ is binary, we have
+
+$$
+d _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathcal { D } _ { 0 } , ( h \circ g ) _ { \sharp } \mathcal { D } _ { 1 } ) = \frac { 1 } { 2 } \left( | \mathcal { D } _ { 0 } ( \widehat { Y } = 0 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 0 ) | + | \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) | \right) .
+$$
+
+Now, by Proposition 3.1, if $d _ { \mathrm { T V } } ( g _ { \sharp } \mathcal { D } _ { 0 } ^ { y } , g _ { \sharp } \mathcal { D } _ { 1 } ^ { y } ) = 0$ , $\forall y \in \{ 0 , 1 \}$ , it follows that $d _ { \mathrm { T V } _ { \perp } } ( ( h \circ g ) _ { \sharp } { \mathcal D } _ { 0 } ^ { y } , ( h \circ$ $g ) _ { \sharp } \mathcal { D } _ { 1 } ^ { y } = 0 , \forall y \in \{ 0 , 1 \}$ as well. Applying Lemma 3.1, we know that $\forall y \in \{ 0 , 1 \}$ ,
+
+$$
+\begin{array} { r } { | D _ { 0 } ( \widehat { Y } = y ) - \mathcal { D } _ { 1 } ( \widehat { Y } = y ) | \le | \mathcal { D } _ { 0 } ( Y = 0 ) - \mathcal { D } _ { 1 } ( Y = 0 ) | \cdot \big ( \mathcal { D } ^ { 0 } ( \widehat { Y } = y ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = y ) \big ) . } \end{array}
+$$
+
+Hence,
+
+$$
+\begin{array} { r l } & { \ell _ { \mathrm { T V } } ( ( h \circ g ) _ { \sharp } \mathcal D _ { 0 } , ( h \circ g ) _ { \sharp } \mathcal D _ { 1 } ) = \frac { 1 } { 2 } \left( \left| \mathcal D _ { 0 } ( \widehat { Y } = 0 ) - \mathcal D _ { 1 } ( \widehat { Y } = 0 ) \right| + \left| \mathcal D _ { 0 } ( \widehat { Y } = 1 ) - \mathcal D _ { 1 } ( \widehat { Y } = 1 ) \right| \right) } \\ & { \qquad \leq \frac { \left| \mathcal D _ { 0 } ( Y = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } { 2 } \left( \left( \mathcal D ^ { 0 } ( \widehat { Y } = 0 ) + \mathcal D ^ { 1 } ( \widehat { Y } = 0 ) \right) + \left( \mathcal D ^ { 0 } ( \widehat { Y } = 1 ) + \mathcal D ^ { 1 } ( \widehat { Y } = 1 ) \right) \right) } \\ & { \qquad = \frac { \left| \mathcal D _ { 0 } ( Y = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } { 2 } \left( \left( \mathcal D ^ { 0 } ( \widehat { Y } = 0 ) + \mathcal D ^ { 0 } ( \widehat { Y } = 1 ) \right) + \left( \mathcal D ^ { 1 } ( \widehat { Y } = 0 ) + \mathcal D ^ { 1 } ( \widehat { Y } = 1 ) \right) \right) } \\ & { \qquad = \frac { \left| \mathcal D _ { 0 } ( Y = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } { 2 } \cdot 2 } \\ & { \qquad = \left| \mathcal D _ { 0 } ( Y = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } \\ & { \qquad = \left| \mathcal D _ { 0 } ( \widehat { Y } = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } \\ & { \qquad = \Delta _ { \mathbb R } ( \mathcal D _ { 0 } , \mathcal D _ { 1 } ) . } \end{array}
+$$
+
+# A.3 PROOF OF LEMMA 3.1
+
+Recall that we define $\gamma _ { a } : = { \mathscr { D } } _ { a } ( Y = 0 ) , \forall a \in \{ 0 , 1 \}$ .
+
+Lemma 3.1. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\begin{array} { r } { | \mathcal { D } _ { 0 } ( \widehat { Y } = y ) - \mathcal { D } _ { 1 } ( \widehat { Y } = y ) | \le | \gamma _ { 0 } - \widehat { \gamma _ { 1 } } | \cdot \big ( \mathcal { D } ^ { 0 } ( \widehat { Y } = y ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = y ) \big ) , \forall y \in \{ 0 , 1 \} . } \end{array}$
+
+Proof. To bound $| \mathcal { D } _ { 0 } ( \widehat { Y } = y ) - \mathcal { D } _ { 1 } ( \widehat { Y } = y ) |$ , for $y \in \{ 0 , 1 \}$ , by the law of total probability, we have:
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { 0 } ( \widehat { Y } = y ) - \mathcal { D } _ { 1 } ( \widehat { Y } = y ) | = } \\ & { \quad = \big | \big ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = y ) \mathcal { D } _ { 0 } ( Y = 0 ) + \mathcal { D } _ { 0 } ^ { 1 } ( \widehat { Y } = y ) \mathcal { D } _ { 0 } ( Y = 1 ) \big ) - \big ( \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = y ) \mathcal { D } _ { 1 } ( Y = 0 ) + \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } = y ) } \\ & { \quad \le \big | \gamma _ { 0 } \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = y ) - \gamma _ { 1 } \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = y ) \big | + \big | ( 1 - \gamma _ { 0 } ) \mathcal { D } _ { 0 } ^ { 1 } ( \widehat { Y } = y ) - ( 1 - \gamma _ { 1 } ) \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } = y ) \big | , } \end{array}
+$$
+
+where the above inequality is due to the triangular inequality. Now apply Proposition 3.1, we know that $\widehat { Y }$ satisfies equalized odds, so we have $\mathcal { D } _ { 0 } ^ { 0 } \big ( \widehat { Y } = y \big ) = \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = y ) = \mathcal { D } ^ { 0 } ( \widehat { Y } = y )$ and $\mathcal { D } _ { 0 } ^ { 1 } ( \widehat { Y } = y ) = \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } = \overset { \cdot } { y } ) = \mathcal { D } ^ { 1 } ( \widehat { Y } = y )$ , leading to:
+
+$$
+\begin{array} { r l } & { = \left| \gamma _ { 0 } - \gamma _ { 1 } \right| \cdot \mathcal { D } ^ { 0 } ( \widehat { Y } = y ) + \left| ( 1 - \gamma _ { 0 } ) - ( 1 - \gamma _ { 1 } ) \right| \cdot \mathcal { D } ^ { 1 } ( \widehat { Y } = y ) } \\ & { = | \gamma _ { 0 } - \gamma _ { 1 } | \cdot \big ( \mathcal { D } ^ { 0 } ( \widehat { Y } = y ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = y ) \big ) , } \end{array}
+$$
+
+which completes the proof.
+
+# A.4 PROOF OF THEOREM 3.1
+
+Theorem 3.1. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\Delta _ { \mathrm { D P } } ( \widehat { Y } ) \leq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) = \Delta _ { \mathrm { D P } } ( Y )$ .
+
+Proof. To bound $\Delta _ { \mathrm { D P } } ( \widehat { Y } )$ , realize that $\begin{array} { r } { | \mathcal { D } _ { 0 } ( \widehat { Y } = 0 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 0 ) | = | \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) | , } \end{array}$ so we can rewrite the DP gap as follows:
+
+$$
+\Delta _ { \mathrm { D P } } ( \widehat { Y } ) = \frac { 1 } { 2 } \left( \left| \mathcal { D } _ { 0 } ( \widehat { Y } = 0 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 0 ) \right| + \left| \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) \right| \right) .
+$$
+
+Now apply Lemma 3.1 twice for $y = 0$ and $y = 1$ , we have:
+
+$$
+\begin{array} { r l } & { | \mathcal { D } _ { 0 } ( \widehat { Y } = 0 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 0 ) | \le | \gamma _ { 0 } - \gamma _ { 1 } | \cdot \big ( \mathcal { D } ^ { 0 } ( \widehat { Y } = 0 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) \big ) } \\ & { | \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) | \le | \gamma _ { 0 } - \gamma _ { 1 } | \cdot \big ( \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 1 ) \big ) . } \end{array}
+$$
+
+Taking sum of the above two inequalities yields
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { 0 } ( \widehat { Y } = 0 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 0 ) | + | \mathcal { D } _ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ( \widehat { Y } = 1 ) | } \\ & { \qquad \le | \gamma _ { 0 } - \gamma _ { 1 } | ( ( \mathcal { D } ^ { 0 } ( \widehat { Y } = 0 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) ) + ( \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 1 ) ) } \\ & { \qquad = | \gamma _ { 0 } - \gamma _ { 1 } | ( ( \mathcal { D } ^ { 0 } ( \widehat { Y } = 0 ) + \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) ) + ( \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 1 ) )  } \\ & { \qquad = 2 | \gamma _ { 0 } - \gamma _ { 1 } | . } \end{array}
+$$
+
+Combining all the inequalities above, we know that
+
+$$
+\begin{array} { r l } & { \Delta _ { \mathrm { D P } } ( \widehat { Y } ) = \displaystyle \frac { 1 } { 2 } \left( \left| \mathcal D _ { 0 } ( \widehat { Y } = 0 ) - \mathcal D _ { 1 } ( \widehat { Y } = 0 ) \right| + \left| \mathcal D _ { 0 } ( \widehat { Y } = 1 ) - \mathcal D _ { 1 } ( \widehat { Y } = 1 ) \right| \right) } \\ & { \qquad \leq \left| \gamma _ { 0 } - \gamma _ { 1 } \right| } \\ & { \qquad = \left| \mathcal D _ { 0 } ( Y = 0 ) - \mathcal D _ { 1 } ( Y = 0 ) \right| } \\ & { \qquad = \left| \mathcal D _ { 0 } ( Y = 1 ) - \mathcal D _ { 1 } ( Y = 1 ) \right| } \\ & { \qquad = \Delta _ { \mathrm { B R } } ( \mathcal D _ { 0 } , \mathcal D _ { 1 } ) = \Delta _ { \mathrm { D P } } ( Y ) , } \end{array}
+$$
+
+completing the proof.
+
+# A.5 PROOF OF THEOREM 3.2
+
+Theorem 3.2. Assume the conditions in Proposition 3.1 hold and let ${ \widehat { Y } } = h ( g ( X ) )$ be the classifier, then $\mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) + \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \le 2 \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y )$ .
+
+Proof. First, by the law of total probability, we have:
+
+$$
+\begin{array} { r l } & { \mathrm { \ddot { g } r r } _ { \mathcal D _ { 0 } } ( \widehat Y ) + \mathrm { E r r } _ { \mathcal D _ { 1 } } ( \widehat Y ) = \mathcal D _ { 0 } ( Y \ne \widehat Y ) + \mathcal D _ { 1 } ( Y \ne \widehat Y ) } \\ & { \qquad = \mathcal D _ { 0 } ^ { 1 } ( \widehat Y = 0 ) \mathcal D _ { 0 } ( Y = 1 ) + \mathcal D _ { 0 } ^ { 0 } ( \widehat Y = 1 ) \mathcal D _ { 0 } ( Y = 0 ) + \mathcal D _ { 1 } ^ { 1 } ( \widehat Y = 0 ) \mathcal D _ { 1 } ( Y = 1 ) + \mathcal D _ { 1 } ^ { 0 } ( \widehat Y = 1 ) . } \end{array}
+$$
+
+Again, by Proposition 3.1, the classifier ${ \widehat { Y } } \ = \ ( h \circ g ) ( X )$ satisfies equalized odds, so we have $\begin{array} { r }  D _ { 0 } ^ { \mathrm { { i } } } ( \widehat { Y } \mathrm { ~  ~ \xi ~ } = 0 ) \mathrm { ~  ~ \psi ~ } = \mathcal { D } ^ { 1 } ( \widehat { Y } \mathrm { ~  ~ \psi ~ } = \mathrm { ~  ~ 0 ~ } ) , \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } \mathrm { ~  ~ \psi ~ } = \mathrm { ~  ~ 1 ~ } ) \mathrm { ~  ~ \psi ~ } = \mathcal { D } ^ { 0 } ( \widehat { Y } \mathrm { ~  ~ \psi ~ } = \mathrm { ~  ~ 1 ~ } ) , \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } \mathrm { ~  ~ \psi ~ } = \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  \psi ~ } \mathrm { ~  ~ \psi ~ } \mathrm { ~  \psi ~ } \mathrm  ~  { ~ \psi ~ } \mathrm { ~   ~ \psi ~ } \mathrm { ~  \psi ~ } \mathrm { ~  \psi ~ \psi ~ } \mathrm  ~ \end{array}$ $\mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } \ = \ 0 ) \ = \ \mathcal { D } ^ { 1 } ( \widehat { Y } \ = \ 0 )$ and $\mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) = \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 )$ :
+
+$$
+\begin{array} { r l } & { \quad = \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) \mathcal { D } _ { 0 } ( Y = 1 ) + \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) \mathcal { D } _ { 0 } ( Y = 0 ) + \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) \mathcal { D } _ { 1 } ( Y = 1 ) + \mathcal { D } ^ { 0 } ( \widehat { Y } = 0 ) } \\ & { \quad = \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) \cdot \big ( \mathcal { D } _ { 0 } ( Y = 1 ) + \mathcal { D } _ { 1 } ( Y = 1 ) \big ) + \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) \cdot \big ( \mathcal { D } _ { 0 } ( Y = 0 ) + \mathcal { D } _ { 1 } ( Y = 0 ) \big ) } \\ & { \quad \le 2 \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) + 2 \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) } \\ & { \quad = 2 \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) , } \end{array}
+$$
+
+which completes the proof.
+
+# A.6 PROOF OF THEOREM 3.3
+
+Theorem 3.3. For any classifier $\widehat { Y } , \Delta _ { \mathrm { E r r } } ( \widehat { Y } ) \leq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y ) + 2 \Delta _ { \mathrm { E O } } ( \widehat { Y } ) .$
+
+Before we give the proof of Theorem 3.3, we first prove the following two lemmas that will be used in the following proof.
+
+Lemma A.1. Define $\gamma _ { a } : = \mathcal { D } _ { a } ( Y = 0 ) , \forall a \in \{ 0 , 1 \}$ , then $| \gamma _ { 0 } { \mathcal { D } } _ { 0 } ^ { 0 } ( { \widehat { Y } } = 1 ) - \gamma _ { 1 } { \mathcal { D } } _ { 1 } ^ { 0 } ( { \widehat { Y } } = 1 ) | \ \leq$ $\vert \gamma _ { 0 } - \gamma _ { 1 } \vert \cdot \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + \gamma _ { 0 } \mathcal { D } ^ { 0 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \gamma _ { 1 } \mathcal { D } ^ { 0 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) .$
+
+Proof. In order to prove the upper bound in the lemma, it suffices if we could give the desired upper bound for the following term
+
+$$
+\begin{array} { r l } & { \left| \big | \gamma _ { 0 } \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \gamma _ { 1 } \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \big | - | ( \gamma _ { 0 } - \gamma _ { 1 } ) \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) | \right| } \\ & { \leq \left| \left( \gamma _ { 0 } \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \gamma _ { 1 } \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \right) - ( \gamma _ { 0 } - \gamma _ { 1 } ) \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) \right| } \\ & { = \Big | \gamma _ { 0 } ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) ) - \gamma _ { 1 } ( \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) ) \Big | , } \end{array}
+$$
+
+following which we will have:
+
+$$
+\begin{array} { r l } & { \gamma _ { 0 } { \mathcal D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \gamma _ { 1 } { \mathcal D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) | \leq | ( \gamma _ { 0 } - \gamma _ { 1 } ) { \mathcal D } ^ { 0 } ( \widehat { Y } = 1 ) | } \\ & { \qquad + \left| \gamma _ { 0 } ( { \mathcal D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - { \mathcal D } ^ { 0 } ( \widehat { Y } = 1 ) ) - \gamma _ { 1 } ( { \mathcal D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - { \mathcal D } ^ { 0 } ( \widehat { Y } = 1 ) ) \right| } \end{array}
+$$
+
+and an application of the Bayes formula could finish the proof. To do so, let us first simplify $\mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = \dot { 1 } ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 )$ . Applying the Bayes’s formula, we know that:
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) = \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \big ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) \mathcal { D } ^ { 0 } ( A = 0 ) + \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \mathcal { D } ^ { 0 } ( A = 1 ) \big ) } \\ & { \qquad = \big ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) \mathcal { D } ^ { 0 } ( A = 0 ) \big ) - \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \mathcal { D } ^ { 0 } ( A = 1 ) } \\ & { \qquad = \mathcal { D } ^ { 0 } ( A = 1 ) \big ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \big ) . } \end{array}
+$$
+
+Similarly, for the second term $\mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 )$ , we can show that:
+
+Plug these two identities into above, we can continue the analysis with
+
+$$
+\begin{array} { r l } & { \Big | \gamma _ { 0 } ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) ) - \gamma _ { 1 } ( \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) ) \Big | } \\ & { = \Big | \gamma _ { 0 } \mathcal { D } ^ { 0 } ( A = 1 ) ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) ) - \gamma _ { 1 } \mathcal { D } ^ { 0 } ( A = 0 ) ( \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) ) \Big | } \\ & { \le \Big | \gamma _ { 0 } \mathcal { D } ^ { 0 } ( A = 1 ) ( \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) ) \Big | + \Big | \gamma _ { 1 } \mathcal { D } ^ { 0 } ( A = 0 ) ( \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) - \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) ) \Big | } \\ & { \le \gamma _ { 0 } \mathcal { D } ^ { 0 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \gamma _ { 1 } \mathcal { D } ^ { 0 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) . } \end{array}
+$$
+
+The first inequality holds by triangular inequality and the second one holds by the definition of equalized odds gap.
+
+Lemma A.2. Define $\gamma _ { a } : = { \mathscr { D } } _ { a } ( Y = 0 ) , \forall a \in \{ 0 , 1 \}$ , then $| ( 1 - \gamma _ { 0 } ) \mathcal { D } _ { 0 } ^ { 1 } ( \widehat { Y } = 0 ) - ( 1 - \gamma _ { 1 } ) \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } =$ $0 ) | \leq | \gamma _ { 0 } - \gamma _ { 1 } | \cdot \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) + ( 1 - \gamma _ { 0 } ) \mathcal { D } ^ { 1 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + ( 1 - \gamma _ { 1 } ) \mathcal { D } ^ { 1 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) .$ .
+
+Proof. The proof of this lemma is symmetric to the previous one, so we omit it here.
+
+Now we are ready to prove Theorem 3.3:
+
+Proof of Theorem 3.3. First, by the law of total probability, it is easy to verify that following identity holds for $a \in \{ 0 , 1 \}$ :
+
+$$
+\begin{array} { c } { { \mathcal { D } _ { a } ( \widehat { Y } \mathrm { ~ \neq ~ } Y ) = \mathcal { D } _ { a } ( Y = 1 , \widehat { Y } = 0 ) + \mathcal { D } _ { a } ( Y = 0 , \widehat { Y } = 1 ) } } \\ { { \mathrm { ~ \neq ~ } ( 1 - \gamma _ { a } ) \mathcal { D } _ { a } ^ { 1 } ( \widehat { Y } = 0 ) + \gamma _ { a } \mathcal { D } _ { a } ^ { 0 } ( \widehat { Y } = 1 ) . } } \end{array}
+$$
+
+Using this identity, to bound the error gap, we have:
+
+$$
+\begin{array} { r l } & { { \mathcal D } _ { 0 } ( Y \ne \hat { Y } ) - { \mathcal D } _ { 1 } ( Y \ne \hat { Y } ) | = \big | ( ( 1 - \gamma _ { 0 } ) { \mathcal D } _ { 0 } ^ { 1 } ( \widehat { Y } = 0 ) + \gamma _ { 0 } { \mathcal D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) ) - ( ( 1 - \gamma _ { 1 } ) { \mathcal D } _ { 1 } ^ { 1 } ( \widehat { Y } = 0 ) + \gamma _ { 1 } } \\ & { \qquad \le \big | \gamma _ { 0 } { \mathcal D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \gamma _ { 1 } { \mathcal D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) \big | + \big | ( 1 - \gamma _ { 0 } ) { \mathcal D } _ { 0 } ^ { 1 } ( \widehat { Y } = 0 ) - ( 1 - \gamma _ { 1 } ) { \mathcal D } _ { 1 } ^ { 1 } ( \widehat { Y } = 0 ) \big | . } \end{array}
+$$
+
+Invoke Lemma A.1 and Lemma A.2 to bound the above two terms:
+
+$$
+\begin{array} { r l } & { | \mathcal { D } _ { 0 } ( Y \neq \widehat { Y } ) - \mathcal { D } _ { 1 } ( Y \neq \widehat { Y } ) | } \\ & { \qquad \leq | \gamma _ { 0 } \mathcal { D } _ { 0 } ^ { 0 } ( \widehat { Y } = 1 ) - \gamma _ { 1 } \mathcal { D } _ { 1 } ^ { 0 } ( \widehat { Y } = 1 ) | + | ( 1 - \gamma _ { 0 } ) \mathcal { D } _ { 0 } ^ { 1 } ( \widehat { Y } = 0 ) - ( 1 - \gamma _ { 1 } ) \mathcal { D } _ { 1 } ^ { 1 } ( \widehat { Y } = 0 ) | } \\ & { \qquad \leq \gamma _ { 0 } \mathcal { D } ^ { 0 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \gamma _ { 1 } \mathcal { D } ^ { 0 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) } \\ & { \qquad + ( 1 - \gamma _ { 0 } ) \mathcal { D } ^ { 1 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + ( 1 - \gamma _ { 1 } ) \mathcal { D } ^ { 1 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) } \\ & { \qquad + | \gamma _ { 0 } - \gamma _ { 1 } | \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + | \gamma _ { 0 } - \gamma _ { 1 } | \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) , } \end{array}
+$$
+
+Realize that both $\gamma _ { 0 } , \gamma _ { 1 } \in [ 0 , 1 ]$ , we have:
+
+$$
+\begin{array} { r l } & { \leq \mathcal { D } ^ { 0 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \mathcal { D } ^ { 0 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \mathcal { D } ^ { 1 } ( A = 1 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \mathcal { D } ^ { 1 } ( A = 0 ) \Delta _ { \mathrm { E O } } ( \widehat { Y } ) } \\ & { + \left. \gamma _ { 0 } - \gamma _ { 1 } \right. \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + \left. \gamma _ { 0 } - \gamma _ { 1 } \right. \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) } \\ & { = 2 \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \left. \gamma _ { 0 } - \gamma _ { 1 } \right. \mathcal { D } ^ { 0 } ( \widehat { Y } = 1 ) + \left. \gamma _ { 0 } - \gamma _ { 1 } \right. \mathcal { D } ^ { 1 } ( \widehat { Y } = 0 ) } \\ & { = 2 \Delta _ { \mathrm { E O } } ( \widehat { Y } ) + \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y ) , } \end{array}
+$$
+
+which completes the proof.
+
+We also provide the proof of Corollary 3.1:
+
+Corollary 3.1. For any joint distribution $\mathcal { D }$ and classifier $\widehat { Y }$ , if $\widehat { Y }$ satisfies equalized odds, then $\operatorname* { m a x } \{ \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) , \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \} \le \Delta _ { \mathrm { B R } } ( { \mathcal { D } _ { 0 } } , { \mathcal { D } _ { 1 } } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid | Y ) / 2 + \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid | Y ) .$
+
+Proof. We first invoke Theorem 3.3, if $\widehat { Y }$ satisfies equalized odds, then $\Delta _ { \mathrm { E O } } ( \widehat { Y } ) = 0$ , which implies:
+
+$$
+\Delta _ { \mathrm { E r r } } ( \widehat { Y } ) = \left| \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) - \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \right| \leq \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \parallel Y ) .
+$$
+
+On the other hand, by Theorem 3.2, we know that
+
+$$
+\mathrm { E r r } _ { \mathscr { D } _ { 0 } } ( \widehat { Y } ) + \mathrm { E r r } _ { \mathscr { D } _ { 1 } } ( \widehat { Y } ) \leq 2 \mathrm { B E R } _ { \mathscr { D } } ( \widehat { Y } \parallel Y ) .
+$$
+
+Combine the above two inequalities and recall that $\begin{array} { r } { \operatorname* { m a x } \{ a , b \} = ( | a + b | + | a - b | ) / 2 , \forall a , b \in \mathbb { R } , } \end{array}$ yielding:
+
+$$
+\begin{array} { r } { \operatorname* { m a x } \{ \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) , \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \} = \frac { \left| \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) - \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \right| + \left| \mathrm { E r r } _ { \mathcal { D } _ { 0 } } ( \widehat { Y } ) + \mathrm { E r r } _ { \mathcal { D } _ { 1 } } ( \widehat { Y } ) \right| } { 2 } } \\ { \leq \frac { \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) + 2 \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) } { 2 } } \\ { = \Delta _ { \mathrm { B R } } ( \mathcal { D } _ { 0 } , \mathcal { D } _ { 1 } ) \cdot \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) / 2 + \mathrm { B E R } _ { \mathcal { D } } ( \widehat { Y } \mid \mid Y ) , } \end{array}
+$$
+
+completing the proof.
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 THE ADULT EXPERIMENT
+
+For the baseline network NODEBIAS, we implement a three-layer neural network with ReLU as the hidden activation function and logistic regression as the target output function. The input layer contains 114 units, and the hidden layer contains 60 hidden units. The output layer only contain one unit, whose output is interpreted as the probability of $\mathcal { D } ( \widehat { Y } = 1 \mid X = x )$ .
+
+For the adversary in FAIR and LAFTR, again, we use a three-layer feed-forward network. Specifically, the input layer of the adversary is the hidden representations of the baseline network that contains 60 units. The hidden layer of the adversary network contains 50 units, with ReLU activation. Finally, the output of the adversary also contains one unit, representing the adversary’s inference probability ${ \mathcal { D } } ( { \widehat { A } } = 1 \mid Z = z )$ . The network structure of the adversaries in both CFAIR and CFAIR-EO are exactly the same as the one used in FAIR and LAFTR, except that there are two adversaries, one for ${ \mathcal { D } } ^ { 0 } ( { \widehat { A } } = 1 \mid Z = z )$ and one for ${ \mathcal { D } } ^ { 1 } ( \widehat { A } = 1 | Z = z )$ .
+
+The hyperparameters used in the experiment are listed in Table 2.
+
+Table 2: Hyperparameters used in the Adult experiment.   
+
+<table><tr><td>Optimization Algorithm</td><td>AdaDelta</td></tr><tr><td>Learning Rate</td><td>1.0</td></tr><tr><td>Batch Size</td><td>512</td></tr><tr><td>Training Ep0chs 入 ∈ {0.1,1.0,10.0,100.0,1000.0}</td><td>100</td></tr></table>
+
+# B.2 THE COMPAS EXPERIMENT
+
+Again, for the baseline network NODEBIAS, we implement a three-layer neural network with ReLU as the hidden activation function and logistic regression as the target output function. The input layer contains 11 units, and the hidden layer contains 10 hidden units. The output layer only contain one unit, whose output is interpreted as the probability of ${ \mathcal { D } } ( { \widehat { Y } } = 1 \mid X = x )$ .
+
+For the adversary in FAIR and LAFTR, again, we use a three-layer feed-forward network. Specifically, the input layer of the adversary is the hidden representations of the baseline network that contains 60 units. The hidden layer of the adversary network contains 10 units, with ReLU activation. Finally, the output of the adversary also contains one unit, representing the adversary’s inference probability ${ \mathcal { D } } ( { \widehat { A } } = 1 \mid Z = z )$ . The network structure of the adversaries in both CFAIR and CFAIR-EO are exactly the same as the one used in FAIR and LAFTR, except that there are two adversaries, one for ${ \mathcal { D } } ^ { 0 } ( { \widehat { A } } = 1 \mid Z = z )$ and one for ${ \mathcal { D } } ^ { 1 } ( \widehat { A } = 1 | Z = z )$ .
+
+The hyperparameters used in the experiment are listed in Table 3.
+
+Table 3: Hyperparameters used in the COMPAS experiment.   
+
+<table><tr><td>Optimization Algorithm Learning Rate Batch Size Training Epochs 入∈ {0.1,1.0} Training Epochs 入= 10.0 15</td><td>AdaDelta 1.0 512 20</td></tr></table>

parse/train/TV9INIrmtWN/TV9INIrmtWN.md

@@ -0,0 +1,391 @@
+# HARD ATTENTION CONTROL BY MUTUAL INFORMATION MAXIMIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Biological agents have adopted the principle of attention to limit the rate of incoming information from the environment. One question that arises is if an artificial agent has access to only a limited view of its surroundings, how can it control its attention to effectively solve tasks? We propose an approach for learning how to control a hard attention window by maximizing the mutual information between the environment state and the attention location at each step. The agent employs an internal world model to make predictions about its state and focuses attention towards where the predictions may be wrong. Attention is trained jointly with a dynamic memory architecture that stores partial observations and keeps track of the unobserved state. We demonstrate that our approach is effective in predicting the full state from a sequence of partial observations. We also show that the agent’s internal representation of the surroundings, a live mental map, can be used for control in two partially observable reinforcement learning tasks. Videos of the trained agent can be found at https://sites.google.com/view/hard-attention-control.
+
+# 1 INTRODUCTION
+
+Reinforcement learning (RL) algorithms have successfully employed neural networks over the past few years, surpassing human level performance in many tasks (Mnih et al., 2015; Silver et al., 2017; Berner et al., 2019; Schulman et al., 2017). But a key difference in the way tasks are performed by humans versus RL algorithms is that humans have the ability to focus on parts of the state at a time, using attention to limit the amount of information gathered at every step. We actively control our attention to build an internal representation of our surroundings over multiple fixations (Fourtassi et al., 2017; Barrouillet et al., 2004; Yarbus, 2013; Itti, 2005). We also use memory and internal world models to predict motions of dynamic objects in the scene when they are not under direct observation (Bosco et al., 2012). By limiting the amount of input information in these two ways, i.e. directing attention only where needed and internally modeling the rest of the environment, we are able to be more efficient in terms of data that needs to be collected from the environment and processed at each time step.
+
+![](images/d6707bda87abc67308d0b23dca6be6c3e5da4d8361981355bd723cb070b529c9.jpg)  
+Figure 1: Illustration of the PhysEnv domain (Du & Narasimhan, 2019). Here, the agent (dark blue) needs to navigate to the goal (red) while avoiding enemies (light blue). Typically the full environment states (top) are provided to the RL algorithm to learn. We consider the case where only observations from under a hard attention window (bottom) are available. The attention’s location (yellow square) at each time step is controllable by the agent. Using only partial observations, the agent must learn to represent its surroundings and complete its task.
+
+By contrast, modern reinforcement learning methods often operate on the entire state. Observing the entire state simultaneously may be difficult in realistic environments. Consider an embodied agent that must actuate its camera to gather visual information about its surroundings learning how to cross a busy street. At every moment, there are different objects in the environment competing for its attention. The agent needs to learn to look left and right to store locations, heading, and speed of nearby vehicles, and perhaps other pedestrians. It must learn to create a live map of its surroundings, frequently checking back on moving objects to update their dynamics. In other words, the agent must learn to selectively move its attention so as to maximize the amount of information it collects from the environment at a time, while internally modeling motions of the other, more predictable parts of the state. Its internal representation, built using successive glimpses, must be sufficient to learn how to complete tasks in this partially observable environment.
+
+We consider the problem of acting in an environment that is only partially observable through a controllable, fixed-size, hard attention window (see figure 1). Only the part of the state that is under the attention window is available to the agent as its observation. The rest must be inferred from previous observations and experience. We assume that the location of the attention window at every time step is under control of the agent and its size compared to the full environment state is known to the agent. We distinguish this task from that of learning soft attention (Vaswani et al., 2017), where the full state is attended to, weighted by a vector, and then fed into subsequent layers. Our system must learn to 1) decide where to place the attention in order to gather more information about its surroundings, 2) record the observation made into an internal memory and model the motion within parts of the state that were unobserved, and 3) use this internal representation to learn how to solve its task within the environment.
+
+Our approach for controlling attention uses RL to maximize an information theoretic objective closely related to the notion of surprise or novelty (Schmidhuber, 1991). It is unsupervised in terms of environment rewards, i.e. it can be trained on offline data (states and actions) without knowing the task or related rewards. We discuss this in more detail in section 5. Memory also plays a crucial role in allowing agents to solve tasks in partially observable environments. We pair our attention control mechanism with a memory architecture inspired largely by Du & Narasimhan (2019)’s SpatialNet, but modified to work in partially observable domains. This is described in section 4. Empirically, we show in section 6.1 that our system is able to reconstruct the full state image including dynamic objects at all time steps given only partial observations. Further, we show in section 6.2 that the internal representation built by our attention control mechanism and memory architecture is sufficient for the agent to learn to solve tasks in this challenging partially observable environment.
+
+# 2 RELATED WORKS
+
+Using hard attention for image classification or object recognition is well studied in computer vision (Alexe et al., 2012; Butko & Movellan, 2009; Larochelle & Hinton, 2010; Paletta et al., 2005; Zoran et al., 2020; Welleck et al., 2017). Attention allows for processing only the salient or interesting parts of the image (Itti et al., 1998). Similarly, attention control has been applied to tracking objects within a video (Denil et al., 2012; Kamkar et al., 2020; Yu et al., 2020). Surprisingly, not a lot of recent work exists on the topic of hard attention control in reinforcement learning domains, where a sequential decision making task has to be solved by using partial observations from under the attention.
+
+Mnih et al. (2014) proposed a framework for hard attention control in the classification setting and a simple reinforcement learning task. Their approach consists of using environment rewards to train an attention control RL agent. Our approach differs mainly in that we train the attention control using our novel information theoretic objective as reward. Mnih et al. (2014)’s approach leads to a task specific policy for attention control, whereas our approach is unsupervised in terms of the task and can be applied generally to downstream tasks in the environment. Our approach also differs in that we use a memory architecture that is more suited to partially observable tasks with 2D images as input, compared to a RNN used by Mnih et al. (2014).
+
+There has been much prior work on memory and world models for reinforcement learning (Ha & Schmidhuber, 2018; Graves et al., 2016; Hausknecht & Stone, 2015; Khan et al., 2017). The work closest to our own is Du & Narasimhan (2019)’s SpatialNet, which attempts to learn a task-agnostic world model for multi-task settings. Our memory architecture is largely inspired by SpatialNet and adapted to work in the partially observable setting. We also use their PhysEnv environment to evaluate our approach. Closely related work is Neural Map (Parisotto & Salakhutdinov, 2017), which uses a structured 2D memory map to store information about the environment. Their approach also applies to partially observable RL tasks, but the attention is fixed to the agent. In contrast, we consider the problem of learning to control an attention window that can move independently of the agent location. Recently, Freeman et al. (2019) showed that world models can be learnt by simply limiting the agent’s ability to observe the environment. They apply observational dropout, where the output of the agent’s world model, rather than the environment state, is occasionally provided to the policy. We consider the related scenario where only a part of the true environment state is provided to the agent at each time step and the rest must be modeled using previous observations.
+
+Finally, mutual information has been used to train self-supervised RL agents. This line of work originates in curiosity driven and intrinsically motivated RL (Schmidhuber, 1991; Pathak et al., 2017; Bellemare et al., 2016). Typically, some notion of predictive error or novelty about an aspect of the environment is optimized in lieu of environment rewards. Multiple papers have successfully used different formulations of mutual information to learn how to efficiently explore the environment without extrinsic rewards (Mohamed & Rezende, 2015; Houthooft et al., 2016; Achiam et al., 2018; Gregor et al., 2016; Eysenbach et al., 2018; Sharma et al., 2019). We apply the idea of using mutual information to the problem of curiosity driven attention control.
+
+# 3 PRELIMINARIES
+
+In this work, we are concerned with tasks solved using reinforcement learning (RL). RL is the study of agents acting in an environment to maximize some notion of long term utility. It is almost always formalized using the language of Markov decision problems (MDPs). States, actions, rewards and transitions form the components of an MDP, often represented as the tuple $\langle S , A , R , T \rangle$ . Maximizing the expected sum of (discounted) rewards over rollouts in the environment is usually set as the objective of learning. Once a problem is formulated as an MDP, such that the components $\langle S , A , R , T \rangle$ are well defined, one can apply a range of model-free RL algorithms to attempt to solve it (Schulman et al., 2017; Mnih et al., 2016; Wu et al., 2017). A solution to an MDP is typically sought as an optimal policy, $\pi ^ { * } : S  A$ , which is a function that maps every state to an action that maximizes the long term expected rewards. In this section, we will attempt to formalize the components of the partially observable reinforcement learning problem under study.
+
+# 3.1 INTERNAL REPRESENTATION
+
+![](images/7d3892123220a2471b3b98f0b35abb357f6151636242067b3385ba55b94cce43.jpg)  
+Figure 2: The agent starts with a representation $\left( \mu _ { t } \right)$ from which it attempts a reconstruction $\left( \tau _ { t } \right)$ of the likely current full (unobserved) state $s _ { t }$ . Then, it picks an attention location $\left( l _ { t } \right)$ and receives an observation of the state from under the attention $\left( o _ { t } \right)$ . We assume that the size of the attention window is known to the agent. The observation is written into $\mu _ { t }$ to form $\mu _ { t } ^ { w }$ . The entire map is then stepped through an internal world model to update the dynamical objects, forming $\mu _ { t + 1 }$ , the representation for the next step.
+
+First, we give a brief description of how the agent stores observations in an internal representation of its surroundings (figure 2). An map $( \mu )$ tracks the full environment state (s), which is never directly observed. The map is empty at the start of the episode and gets sequentially written into as observations are made. The agent also creates a reconstruction of the full state $( \tau )$ at every time step based on its map. This will become useful later for training. For now, it suffices that $\mu _ { t }$ is the map before an observation is made and $\mu _ { t } ^ { w }$ is the one after the observation is made. $\mu _ { t + 1 }$ is the map after the dynamics of the system for the next time step are taken into account.
+
+# 3.2 THE TWO AGENTS
+
+We formulate the solutions to controlling the attention at every time step and completing the environment task as two separate reinforcement learning agents. The attention location is controlled by the glimpse agent (the eye in figure 2), and actions within the environment are taken by the regular agent. These two agents have their own separate MDPs with their $\langle S , A , R \rangle$ tuples defined below.
+
+The glimpse agent’s state at every timestep is $\mu _ { t }$ . Its set of actions is all possible attention locations within the full state (Width $\times$ Height actions). Its reward is based on the information theoretic objective discussed in section 5. Thus, the glimpse agent is provided the map before an observation is made and it must decide where the observation should be made from in order to optimize its reward.
+
+The environment agent acting in the environment receives as input $\mu _ { t } ^ { w }$ , i.e. the internal representation after the observation has been recorded. Its actions are the normal set of actions in the environment and its reward is the normal environment reward. We emphasize that neither agent has access to the full state at any time. They must both act based on the internal representation alone. They also cannot make multiple observations from the same environment state. Once an observation is made, an action must be selected that will change the environment state. In the next section, we will describe in detail how $\mu _ { t } ^ { w }$ , $\mu _ { t + 1 }$ and $\tau _ { t }$ are formed and how the internal map is trained through a sequence of partial observations.
+
+# 4 DYNAMIC MEMORY MAP
+
+Memory plays a critical role in enabling agents to act in partially observable environments. It allows the agent to remember parts of the state that are not under direct observation. Along with memory, the agent must be able to model the dynamics of objects in its environment, as it may not receive observation of them for long periods. Finally, the effect of its own actions must be reflected in its belief about the environment. For this purpose, we design a special recurrent memory unit for visual environments called the Dynamic Memory Map (DMM). DMM is inspired largely by Du & Narasimhan (2019)’s work on SpatialNet, but modified to handle partial observations and work in tandem with the glimpse agent.
+
+# 4.1 MEMORY MODULES
+
+The DMM consists of three major modules: write, step and reconstruct.
+
+Write. This module encodes an incoming observation, $o _ { t }$ , into the current memory representation, $\mu _ { t }$ . The observation is first passed through a series of convolutional operations, $W$ , possibly downsampling it for a more efficient representation. Then it is blended with $\mu$ using a series of convolutions, $B$ . Finally it is written into the memory but only in the locations where the observation was made, leaving the rest of the memory intact. The write operation can be written as:
+
+$$
+\mu _ { t } ^ { w } = C _ { t } * B ( W ( o _ { t } ) , \mu _ { t } ) + ( 1 - C _ { t } ) * \mu _ { t }
+$$
+
+where $C _ { t }$ is a mask which is 1 under the attention window and 0 otherwise.
+
+Step. This module is responsible for modeling the dynamics of the environment and updating the memory to track the full state from one time step to the next. Objects in the environment can be static or dynamic and may be affected by the agent’s own actions. This module is implemented as a residual network, $S$ , similar to SpatialNet (Du & Narasimhan, 2019). We additionally condition $S$ on the agent’s own action $a _ { t }$ . The entire map is updated at each time step to ensure that objects observed previously are also updated to new likely locations.
+
+$$
+\mu _ { t + 1 } = \mu _ { t } + S ( \mu _ { t } ^ { w } , a _ { t } )
+$$
+
+Reconstruct. This module converts the memory representation to a reconstruction of the full state using a series of deconvolutions $( R )$ .
+
+$$
+\tau _ { t } = R ( \mu _ { t } )
+$$
+
+A reconstruction can also be made immediately after the observation, i.e. $\tau _ { t } ^ { w } = R ( \mu _ { t } ^ { w } )$ . Reconstruction is essential for training the other two modules of DMM as we explain in section 4.2. The reconstruction error, or the discrepancy between the observation and the reconstruction under the attention window can be back-propagated through the write and step layers.
+
+# 4.2 TRAINING LOSS
+
+The write loss, $L _ { w }$ is incurred under the current attention window immediately after the observation is made into $\mu _ { t } ^ { w }$ . This ensures that the immediate observation is correctly encoded into the memory and trains the write and reconstruction modules.
+
+$$
+L _ { w } = C _ { t } * \lvert \lvert \tau _ { t } ^ { w } - o _ { t } \rvert \rvert _ { 2 } .
+$$
+
+The step loss, $L _ { s }$ is incurred after stepping the map and under the attention window in the next step.
+
+$$
+L _ { s } = C _ { t + 1 } * \lvert \lvert \tau _ { t + 1 } - o _ { t + 1 } \rvert \rvert _ { 2 } .
+$$
+
+This trains the step module, $S$ , to accurately model the motion of objects in the entire state, so that a faithful representation of the state can be guessed before the next observation is taken. Once the next observation is made, we can calculate the difference between what the agent’s model expected to be at the glimpse location and what actually was there. In figure 2 this is the difference between the images under the yellow rectangle in $\tau _ { t }$ and $o _ { t }$ . This quantity is the amount of surprise in the dynamics model and we will use this again when training the glimpse agent.
+
+In addition to the reconstruction loss, the absolute value of the output by the write and step modules is also penalized in order to regularize the contents of the DMM. The total loss for a single step within the rollout is (where $\alpha = \beta = 0 . 0 1$ here):
+
+$$
+L _ { t } = L _ { w } + L _ { s } + \alpha * C _ { t } * | \mu _ { t } ^ { w } | + \beta * | S ( \mu _ { t } ^ { w } , a _ { t } ) | .
+$$
+
+We sum this loss over the entire rollout and jointly minimize it over all steps. The reconstruction error at every location can be back-propagated to the step an observation was recorded there.
+
+# 5 MAXIMIZING MUTUAL INFORMATION TO CONTROL ATTENTION
+
+The intuition behind our approach is that attention should be used to gather information from the environment when and where it is required. The DMM is constantly modeling its environment and attention should be directed to parts of the world where the model is uncertain. This directly informs the agent about difficult to model parts of the environment, thus reducing uncertainty in the state as a whole. Secondly, by focusing on areas with hard to predict dynamics, more data can be collected from those areas, updating the model for future predictions.
+
+# 5.1 MUTUAL INFORMATION OBJECTIVE
+
+The idea of reducing uncertainty about the environment can be captured in the language of information theory by using mutual information. Specifically, we propose selecting the location of the attention such that its mutual information with the state at the following step is maximized,
+
+$$
+\operatorname* { m a x } _ { l _ { t } } I ( s _ { t + 1 } ; l _ { t } ) .
+$$
+
+Where $l _ { t }$ is the location of the attention window at time $t$ . Eq. 1 can be expanded as
+
+$$
+\operatorname* { m a x } _ { l _ { t } } \mathcal { H } ( s _ { t + 1 } ) - \mathcal { H } ( s _ { t + 1 } | l _ { t } ) ,
+$$
+
+where $\mathcal { H }$ denotes the entropy. This expansion brings out a very intuitive explanation of the objective. We would like to pick an attention location that maximizes the reduction in entropy (uncertainty) of the environment state after an observation is made.
+
+# 5.2 CURIOSITY DRIVEN ATTENTION
+
+In this section, we show how maximizing the amount of surprise can lead to maximizing the mutual information. Let us begin by further expanding eq. 2.
+
+$$
+\begin{array} { r l } & { I ( s _ { t + 1 } ; l _ { t } ) = \mathcal { H } ( s _ { t + 1 } ) - \mathcal { H } ( s _ { t + 1 } | l _ { t } ) } \\ & { \qquad = - \displaystyle \sum _ { s _ { t + 1 } } p ( s _ { t + 1 } ) \log p ( s _ { t + 1 } ) + \displaystyle \sum _ { l _ { t } } p ( l _ { t } ) \sum _ { s _ { t + 1 } } p ( s _ { t + 1 } | l _ { t } ) \log p ( s _ { t + 1 } | l _ { t } ) } \\ & { \qquad = - \displaystyle \sum _ { s _ { t + 1 } \sim p ( s _ { t + 1 } ) } [ \log p ( s _ { t + 1 } ) ] + \displaystyle \sum _ { l _ { t } } \sum _ { s _ { t + 1 } } p ( s _ { t + 1 } , l _ { t } ) \log p ( s _ { t + 1 } | l _ { t } ) } \\ & { \qquad = - \displaystyle \operatorname* { l i m } _ { s _ { t + 1 } \sim p ( s _ { t + 1 } ) } [ \log p ( s _ { t + 1 } ) ] + \displaystyle \sum _ { s _ { t + 1 } , l _ { t } \sim p ( s _ { t + 1 } , l _ { t } ) } [ \log p ( s _ { t + 1 } | l _ { t } ) ] } \end{array}
+$$
+
+So, maximizing mutual information between the attention location and the environment state is equivalent to minimizing $\log p ( s _ { t + 1 } )$ and maximizing $\log p ( s _ { t + 1 } | l _ { t } )$ under expectation. The first term is the log-likelihood of the next state prior to selecting $l _ { t }$ . Computing it requires marginalization over all possible $l _ { t }$ , which is prohibited by our environment as only a single partial observation from the full state can be provided to the agent. The second term, $\log p ( s _ { t + 1 } | l _ { t } )$ , is computable at a single attention location from a single state and hence we will focus on maximizing this term.
+
+Now, assume the agent’s belief over the true environment state at the next step, $s _ { t + 1 }$ , is represented by a Gaussian with unit variance around the reconstruction, $\tau _ { t + 1 }$ . So, $s _ { t + 1 } \sim \mathcal { N } ( \tau _ { t + 1 } , \mathcal { T } )$ and
+
+$$
+\begin{array} { l } { \log p ( s _ { t + 1 } | \tau _ { t + 1 } ) \propto \log e x p [ - ( s _ { t + 1 } - \tau _ { t + 1 } ) ^ { T } ( s _ { t + 1 } - \tau _ { t + 1 } ) / 2 ] } \\ { \quad \quad \quad = - \displaystyle \sum _ { i } ( s _ { t + 1 } ^ { i } - \tau _ { t + 1 } ^ { i } ) ^ { 2 } / 2 } \end{array}
+$$
+
+So, minimizing the squared difference between $s _ { t + 1 }$ and $\tau _ { t + 1 }$ leads to maximizing $\log p ( s _ { t + 1 } | \tau _ { t + 1 } )$ . Maximizing $\log p ( s _ { t + 1 } \vert \tau _ { t + 1 } )$ in turn means maximizing $\log p ( s _ { t + 1 } | l _ { t } )$ , since $\tau _ { t + 1 }$ is constructed using the deterministic functions $W$ , $S$ , and $R$ once $l _ { t }$ has been picked (see section 4).
+
+In order to minimize the squared difference between $s _ { t + 1 }$ and $\tau _ { t + 1 }$ , the agent needs to pick a location $l _ { t }$ that observes the maximum error between $s _ { t }$ and its reconstruction $\tau _ { t }$ . This is because at step $t$ , if the agent observes the location with the highest reconstruction error, i.e. the least likelihood $p ( s _ { t } | \tau _ { t } )$ , it will ensure that at the next time step $t + 1$ the agent has the most recent information on that region and can make a good reconstruction of it. Hence, maximizing $\log p ( s _ { t + 1 } | l _ { t } )$ is equivalent to picking an $l _ { t }$ that will lead to the highest reconstruction error at step $t$ .
+
+$$
+\operatorname* { m a x } _ { l _ { t } } \log p ( s _ { t + 1 } | l _ { t } ) \implies \operatorname* { m a x } _ { l _ { t } } \sum _ { i \in I _ { t } } ( s _ { t } ^ { i } - \tau _ { t } ^ { i } ) ^ { 2 } / 2
+$$
+
+where $I _ { t }$ forms the set of indices under the window at location $l _ { t }$ . Directly optimizing this quantity with respect to $l _ { t }$ is not possible as it can only be computed once an observation has been made and only at a single location. Hence, the glimpse agent must learn to predict the location that is most likely to result in high reconstruction error before an observation is made. Our approach is to formulate the glimpse agent as a reinforcement learner with the reconstruction error as its reward.
+
+An interpretation of this objective is that the glimpse agent is surprise seeking, or curiosity driven. It is attending to parts of the state that are novel in the sense that they are difficult for the agent’s current model to predict. Another interpretation is that the glimpse agent is acting adversarially to the DMM’s model of the environment. At each step the glimpse agent tries to focus on parts of the state that are the most difficult for DMM to reconstruct. By doing so, it is indirectly creating a curriculum of increasingly difficult to model aspects of the environment.
+
+# 5.3 FULL TRAINING OBJECTIVE
+
+Let us look at a different expansion of the objective in eq. 1,
+
+$$
+I ( s _ { t + 1 } ; l _ { t } ) = \operatorname* { m a x } _ { l _ { t } } \mathcal { H } ( l _ { t } ) - \mathcal { H } ( l _ { t } | s _ { t + 1 } ) .
+$$
+
+The first term is $\mathcal { H } ( l _ { t } )$ , the entropy over the attention location, which is controlled by the policy of the glimpse agent. This term can easily be maximized by standard RL algorithms, such as A2C (Mnih et al., 2016; Wu et al., 2017), using weighted entropy maximization.
+
+Combining the first term from equation 5 and second term from equation 2, the final objective that we use to train the glimpse agent is
+
+$$
+\operatorname* { m a x } _ { l _ { t } } \mathcal { H } ( l _ { t } ) - \mathcal { H } ( s _ { t + 1 } | l _ { t } ) \equiv \alpha \mathcal { H } ( \pi _ { g l i m p s e } ) + \operatorname* { m a x } _ { l _ { t } } \sum _ { i \in I _ { t } } ( s _ { t } ^ { i } - \tau _ { t } ^ { i } ) ^ { 2 } / 2 ,
+$$
+
+where $\alpha$ is the entropy weighting set to 0.001 in our experiments.
+
+Note that this objective is not the mutual information from equation 1. It is possible to design a variational algorithm that maximizes the approximate mutual information based on the expansion in eq. 5 alone (instead of mixing terms from both the expansions). Empirically, however, we found that our objective performs better than the variational approximation to mutual information. See section A.4 for more detail.
+
+# 6 RESULTS
+
+We evaluate our approach in two environments: a gridworld environment (figure 5) and PhysEnv (figure 1). In both environments, the task is to navigate to a goal while avoiding obstacles and enemies. The agent’s actions are movement in the four discrete cardinal directions. In the gridworld, the state is a one-hot image of the objects in the environment (agent, walls, enemies and goal). The agent’s and enemies’ movements are restricted to the grid squares and the dynamics of the enemies are simple: moving in straight lines until an obstacle is hit in which case it reflects. In PhysEnv, observations are provided as RGB images and the motions of the objects are more varied.
+
+The glimpse agent $^ +$ DMM are first trained on offline data of state and action trajectories (no reward) collected from each environment. More details on data collection are provided in section A.1. We evaluate this part of the training by how accurately the full state is reconstructed by DMM at each step. Once the glimpse agent $+ \mathrm { D M M }$ have converged, i.e. the reconstruction loss has plateaued, we freeze their parameters and initialize an agent within the environment. The glimpse agent $^ +$ DMM serve as fixed components within the agent while it is learns to solve a task. The agent is trained using an off-the-shelf RL algorithm PPO (Schulman et al., 2017) and is evaluated by the total environment reward it collects during testing episodes.
+
+We compare our method against baselines inspired from related work. In the follow baseline the attention is always focused on the agent itself. This is a formulation of the partial observability as seen in Parisotto & Salakhutdinov (2017). Next, the environment baseline mimics the approach by Mnih et al. (2014), where the environment rewards are used to train the glimpse agent. Finally, the random baseline moves the attention randomly at each time step. For the environment task, we have an additional baseline, full, where the full state is provided to the agent to solve the task. This is an upper limit for how well any method that only receives partial observations can perform.
+
+# 6.1 STATE RECONSTRUCTION
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12 reward (ours)</td><td rowspan=1 colspan=1>random</td><td rowspan=1 colspan=1>follow</td><td rowspan=1 colspan=1>environment</td></tr><tr><td rowspan=1 colspan=1>Gridworld</td><td rowspan=1 colspan=1>0.00555</td><td rowspan=1 colspan=1>0.007666</td><td rowspan=1 colspan=1>0.01941</td><td rowspan=1 colspan=1>0.00827</td></tr><tr><td rowspan=1 colspan=1>PhysEnv</td><td rowspan=1 colspan=1>0.0521</td><td rowspan=1 colspan=1>0.0614</td><td rowspan=1 colspan=1>0.1186</td><td rowspan=1 colspan=1>0.0826</td></tr></table>
+
+Table 1: Per-pixel L2 loss between agent’s reconstruction $\tau$ and the ground truth full state $s$ . All methods only see partial observations, but must attempt to reconstruct the full state from internal representation. In both environments, our method has the lowest reconstruction error.
+
+We measure the performance of the glimpse agent $+ { \bf D M M }$ by the average reconstruction error to the true full (unobserved) state in unseen testing episodes. 25 consecutive observations from 6 unseen test episodes are passed through each model and error is averaged over reconstructions at each time step. This gauges the ability of our attention mechanism and memory to reconstruct the full state using only partial observations. Table 1 shows that using surprise driven reward (our approach) for learning attention control leads to a lower reconstruction error than the other baselines. See figures 5 through 9 in the appendix for visualizations of the reconstructions and training error. In particular, figure 7 and 8 show qualitative comparisons between our method with the baselines. Broadly speaking, our method learns to focus on dynamic objects within the environment and preserve them over many steps in the memory, whereas the baselines lose track of or blur objects as seen in their state reconstructions.
+
+![](images/a4007fb83f0203dda06cb9cef47af6d46205c4b567c68486399ad510a9b2b788.jpg)  
+Figure 3: (left) Object-wise breakdown of reconstruction error in gridworld. (right) $\%$ of frames in which the object occurs, i.e. the frequency with which the attention is focuses on a particular object.
+
+Figure 3 shows an analysis of what objects are under focus during training in the gridworld environment and their corresponding reconstruction error. The attention forms an interesting curriculum for training the DMM, where it learns to focus on different objects in the environment over time. It starts with walls, then moves to the agent itself, then goal and finally the enemies. Thus, it moves gradually from static to dynamic, harder to model parts of the state, allowing the DMM to learn how to accurately model each object. This very interesting curriculum-like behavior emerges naturally and is not pre-programmed into DMM or the attention agent. It is a consequence of maximizing surprise at every time step. See section A.2 for a more detailed discussion.
+
+# 6.2 REINFORCEMENT LEARNING
+
+![](images/ef6d6a60aa12e4e740937406379fe1c39ede3ddf9eb0d647ace2af3595c1d0ed.jpg)  
+Figure 4: Average testing reward for goal seeking task in PhysEnv (left) and gridworld (right). Comparison between agent with full observations (upper limit) and partial observations trained with surprise (ours), environment rewards and simply following the agent’s location.
+
+Figure 4 shows the average episodic reward over five testing episodes during the training of the agent. Our method approaches the performance of the upper bound agent that receives full environment states in the gridworld environment. The environment baseline performs similarly to our approach in gridworld, but does poorly in PhysEnv, where our approach performs the best. This may be because the attention policy of the environment baseline has high entropy and ends up exploring large parts of the state within an episode. But it does not learn to focus on the most unpredictable parts of the state as our method does. The follow baseline performs poorly in both environments, likely because it only focuses on the agent and does not explore the rest of the state.
+
+We also show the performance of our agent (l2 online) in the condition where the DMM and glimpse agent is trained online along with the policy, all components starting from random initialization. This agent did not have a pretraining step and was trained entirely on random exploratory data, therefore obviating the need of a pretraining dataset. The performance is slightly worse than the pretrained agent. This is an encouraging result in a particularly challenging setting of a non-stationary state space as the memory representation and glimpse policy are changing while the agent learns.
+
+# 7 CONCLUSIONS
+
+We have presented an approach of using mutual information to control a hard attention window in environments with dynamic objects. From partial observations under the attention window, our system is able to reconstruct the full state at every time step with the least error. The representation learned using our method enables RL agents to solve tasks within the environment, while the baseline methods are unable to learn useful representations on the same memory architecture. This demonstrates that attention control plays a large role in enabling task completion.
+
+Note that our attention control objective is independent of the task. It is solely concerned with gathering information from the environment where it is most unpredictable. This is similar to curiosity driven RL that seeks novel states or surprising transitions. Hence, attention control learned in this manner is a generic solution that can be applied to many downstream tasks. It may be able to be combined with task-specific attention control by fine-tuning on a task reward. A better foveal glimpse model may also be used instead of the fixed size hard attention used here. We made an assumption that only one glimpse is allowed per environment state. But the rate of collecting glimpses may be variable with respect to environment speed.
+
+Curiosity based approaches tend to suffer from the “noisy TV” problem (Savinov et al., 2018) where the agent tends to get distracted by and focus on random, unpredictable events in the environment. Orthogonal approaches have been developed to counter this common weakness by, for example, taking into account the agent’s own actions (Pathak et al., 2017). These approaches can be combined with our method as well. We leave the incorporation of this in attention control for future work.
+
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+
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+
+# A APPENDIX
+
+# A.1 DETAILS OF DATA COLLECTION
+
+For gridworld we used 20000 random steps in the environment to create this training set. In PhysEnv we used a previously trained agent to generate 500,000 steps of demonstrations. We experienced that using a random policy in PhysEnv led to quick deaths of the agent with insufficient episode lengths for learning. In both cases, the data are sequences of full states and agent actions from multiple episodes. In PhysEnv, the map size was $2 1 \times 2 1$ compared to the full state size of $8 4 \times 8 4$ and in gridworld both the map and state were $1 0 \times 1 0$ . The size of the attention window was fixed to be $3 \times 3$ in the gridworld $9 \%$ of the full state visible in an observation) and $2 1 \times 2 1$ in PhysEnv $( 6 . 2 5 \%$ of the state visible).
+
+# A.2 VISUALIZATIONS OF GLIMPSES DURING TRAINING
+
+We will discuss where the attention focuses and how that effects the reconstruction (and underlying belief) as the training progresses for the gridworld environment. Please refer to 3 for this discussion.
+
+Initially, the attention learns to focus on the wall objects and the corresponding reconstruction error for the walls decreases rapidly (3.b). This is because the walls are long objects in the environment, present at multiple pixels and side-by-side, providing the largest and most reliable source of error under the attention window initially. The error is provided as reward to the glimpse agent, hence it seeks more walls in the state.
+
+Soon, the error it receives from walls decreases as the agent’s DMM learns to persistently record the location of stationary walls over multiple time steps. Then around 10000 iters, the attention switches to focusing on the agent’s location itself. This is because the agent’s apparent erratic movements provide a larger reward due to their unpredictability than the stationary walls can anymore. This has dual consequences. First, since now the DMM is seeing a lot of examples of the agent’s movement, it learns to accurately model the agent based on an initial state and its actions and the reconstruction error corresponding to the agent falls. At its peak, over $70 \%$ of glimpses contain the agent within them (figure 3.b). Interestingly, the error corresponding to the wall and goal objects rises as some forgetting of the representation and dynamics of these objects occurs. A little before 20000 iters though, the agent has learnt to represent simultaneously all three objects and it switches attention to the enemies, the only remaining un-modeled objects in the environment. The enemies move predictably, bouncing off walls and other objects, but otherwise in straight lines. The agent does not have direct access to their actions and must infer their motion over multiple time steps using only glimpses.
+
+Figures 5 and 6 visualize the behavior of the glimpse agent during the initial stages of training. Initially, (100 iters), the attention mostly moves around randomly as can be seen by the heatmap being active throughout the state. The reconstruction does not correspond to anything in the real state since the agent has only just begun learning. The glimpse agent here is receiving high reward for pretty much every location in the state. By 2000 iters, the glimpse agent has learnt to mainly focus on the wall once it locates it. This can be seen in the high probabilities of the glimpse agent policy near the wall locations and the attention repeatedly fixating on the wall. The agent is simultaneously striving to learn to represent the walls as they are showing up a lot in its training data, and steadily reducing the reconstruction error (the glimpse agent’s reward) coming from walls.
+
+By iter 11000, the focus has shifted to tracking the agent as can be seen in the heatmap. The heatmap shows that the glimpse agent is singularly focused on the agent’s movements as the error reward it is getting from the walls has dropped off and the agent’s movements are not fully predicatable yet. In the figure for iter 19000, it can be seen that the agent can now track its own location very reliably with an initial position and the actions it is taking, without having to focus the attention on itself. In fact, after step 8 when the agent is discovered, the attention almost never moved back to it and yet the location in the reconstruction perfectly matches that of the unobserved full state. Hence the agent’s location based reward source has largely dried up. The heatmap of attention policy shows that the glimpse agent has become interested in the goals and walls again briefly as it learns to represent all three objects together. It can also be seen from the reconstructions here that the agent is unable to accurately track and model the enemies locations at this point.
+
+By iter 21000, the attention has shifted focus to the remaining object in the state that is producing a high reconstruction error, i.e. the enemy. Once the enemy is discovered on step 14, the glimpse agent’s policy puts all probability near its location so it can keep observing and model its behavior.
+
+The glimpse agent creates a natural curriculum for training the models within DMM, starting at large stationary objects that do not move, then moving to the agent’s followed by the objects with the most complex dynamics. We conjecture this automatically discovered curriculum is the reason our system is able to learn better reconstructions than the baselines.
+
+![](images/1b1d00dcc6be27103726a94bc00d44de1546ef78fc25ae6a395229173f0eb670.jpg)  
+Figure 5: Visualizations of glimpse behavior in gridworld environment. Each sub-figure is the same testing episode at different stages of training, with the rows corresponding to the time within the episode. In each subfigure, the first column shows the full (unobserved) state. The second column is a heatmap showing the glimpse policy, i.e. the likelihood the glimpse agent will select a location for placing attention. The location that was sampled is indicated in the first column by the yellow square. The final, third, column is the reconstruction $\tau$ based on the agent’s current current belief about the environment.
+
+![](images/58c94bc80bbdf079dc9466a7affe5cf5415435ae4e3ae90bbd022b6dbc4ed401.jpg)  
+Figure 6: Visualizations of glimpse behavior in PhysEnv environment.
+
+Next we show qualitative comparisons of the glimpse policy and state reconstruction between our method and the baselines. Each subfigure in figures 7 and 8 shows the same episode of length 20 played out with different glimpse agents. The first column is our method, the second shows a uniform random policy for the glimpse agent, the third shows a glimpse agent trained on environment (task) rewards, and lastly the fourth shows the glimpse following the agent. In both environments, it can be seen that our method learns to focus on dynamic objects within the environment and the resulting state reconstruction shows that these objects are preserved in the memory.
+
+Note that the glimpse policy learned by the environment rewards is significantly more entropic than the one learned by our method, despite having the same entropy weighting hyperparameter. We conjecture that this is because the task reward does not send a clear and consistent learning signal to control the attention window to create better reconstructions, or as seen in our RL results, to facilitate downstream tasks. As such, the reconstructions appear qualitatively similar to the random baseline.
+
+The random baseline quantitatively performs the closest to our approach, but it fails to focus on dynamic objects consistently like over time, and hence fails to track these objects over long periods in the episode. The reconstructions therefore have streaks and blurs where the objects were last spotted and the physics model has not learnt how to update their positions properly.
+
+![](images/cc103f12df50b384b3a935f94d0676dbedf84b35dc8f71f8d322648c777ca892.jpg)  
+Figure 7: Glimpse behavior for fully trained models of our method and the baselines in gridworld. Each subfigure is a different baseline, with columns within the subfigure being the full unobserved state, heatmap of attention policy, and reconstruction of full state respectively.
+
+![](images/5c07ecfa5d304f6dbceff467d21009d726361025cd0f9b88ca983615555186e2.jpg)  
+Figure 8: Glimpse behavior for fully trained models of our method and the baselines in PhysEnv.
+
+![](images/41edf80cf0f8258eb553d4de174a5a77bd347e291bb26bf204f84f910b503956.jpg)  
+Figure 9: (left) PhysEnv environment. (right) Gridworld environment. Reconstruction error (L2) from the full state during training of the $\mathrm { D M M } +$ glimpse agent. All approaches have converged by 200k iters, with our approach having the lowest error, i.e. the most faithful reconstruction of the full unobserved state.
+
+These training curves correspond to the results in table 1. The reconstruction error for each method has stabilized. Our method has the lowest reconstruction error for both environments.
+
+![](images/6112eec42e26da8bcef385499fbaeb3966e29deb24cf3bf17f83bb8b5e9aefa4.jpg)  
+Figure 10: (left) PhysEnv environment. (right) Gridworld environment. Episodic reward for glimpse agent during training (higher is better).
+
+Figure 10 shows the episodic reward for the glimpse agent on six testing episodes as the training progresses. This corresponds to the mutual information objective in equation 4, or the reconstruction eror under the attention window in the next step. The modules within DMM are constantly improving to reduce prediction error and hence the reward goes down as the training progresses. But the glimpse agent is simultaneously trying to maximize the objective by searching for areas within the state that still provide high error. Our method maximizes this objective in both environments. This leads to lower overall reconstruction error as the glimpse agent is learning to predict where the highest source of error is going to be at the next step. The environment baseline performs close to random and the follow baseline gathers the least reward in both environments.
+
+# A.4 VARIATIONAL APPROXIMATION TO MUTUAL INFORMATION MAXIMIZATION
+
+The mutual information between attention and environment state as expanded in eq. 5 is
+
+$$
+I ( s _ { t } ; l _ { t } ) = \mathcal { H } ( l _ { t } ) - \mathcal { H } ( l _ { t } | s _ { t } ) .
+$$
+
+In our algorithm, we combine the first term in this expansion with another term in eq. 2. This works well empirically. Theoretically, however, it is computationally possible to maximize both terms in eq. 6, hence maximizing the actual mutual information. However, as we shall show here, this does not work well practically.
+
+The first term can be maximized by using maximum entropy RL algorithms for the attention policy as discussed in the main text. Let us consider the second term.
+
+$$
+\begin{array} { r l } {  { - \mathcal { H } ( l _ { t } | s _ { t } ) = - \sum _ { s _ { t } } p ( s _ { t } ) \mathcal { H } ( l _ { t } | s _ { t } ) } } \\ & { = \sum _ { s _ { t } } p ( s _ { t } ) \sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \log p ( l _ { t } | s _ { t } ) } \\ & { = \frac { \mathbb { E } } { s _ { t } , l _ { t } \sim p ( s _ { t } , l _ { t } ) } [ \log p ( l _ { t } | s _ { t } ) ] } \end{array}
+$$
+
+While $\log p ( l _ { t } | s _ { t } )$ is not directly known, we can construct a variational function, $q ( l _ { t } | s _ { t } )$ , to approximate it.
+
+Since $K L ( p ( l _ { t } | s _ { t } ) | | q ( l _ { t } | s _ { t } ) ) \geq 0$ , we have,
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \log p ( l _ { t } | s _ { t } ) \geq \sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \log q ( l _ { t } | s _ { t } ) } \\ & { \displaystyle \sum _ { s _ { t } , l _ { t } \sim p ( s _ { t } , l _ { t } ) } [ \log p ( l _ { t } | s _ { t } ) ] \geq \underset { s _ { t } , l _ { t } \sim p ( s _ { t } , l _ { t } ) } { \mathbb { E } } [ \log q ( l _ { t } | s _ { t } ) ] } \end{array}
+$$
+
+Plugging this into eq. 6,
+
+$$
+I ( s _ { t } ; l _ { t } ) \geq \mathcal { H } ( l _ { t } ) + \underset { s _ { t } , l _ { t } \sim p ( s _ { t } , l _ { t } ) } { \mathbb { E } } [ \log q ( l _ { t } | s _ { t } ) ]
+$$
+
+Therefore, we can maximize the mutual information by maximizing this lower bound w.r.t. the policy parameters of the glimpse agent. This can be achieved by providing $\log q ( l _ { t } | s _ { t } )$ as a reward to the glimpse agent and using RL algorithms to maximize its long term value along with the policy entropy. As such, maximizing the mutual information.
+
+We can additionally condition $q$ on the internal state and action of the agent at previous time step. Since we do not have access to the full environment state, $s _ { t }$ , we substitute it with the agent’s internal state. The final variational function looks like $q ( l _ { t } | \mu _ { t } , \mu _ { t - 1 } , a _ { t - 1 } )$ . $q$ must be trained to match the true posterior $p ( l _ { t } | s _ { t } )$ to provide accurate rewards to the glimpse agent.
+
+The question now is how to learn $q \mathrm { ? }$ In this work, it is a neural network trained on samples. This is done by minimizing the KL divergence,
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { \substack { s _ { t } , l _ { t } \sim p ( s _ { t } , l _ { t } ) } } [ K L ( p ( l _ { t } | s _ { t } ) | | q ( l _ { t } | \mu _ { t } , \mu _ { t - 1 } , a _ { t - 1 } ) ) ] } \end{array}
+$$
+
+on samples from the environment and the glimpse agent’s policy. This will ensure $q$ is likely to assign high likelihood to actual locations conditioned on its inputs.
+
+Intuitively, $q$ will be high at locations of attention that are easy to identify by looking at the previous state, the resulting state, and the agent action. Otherwise, its output will be flat and hard to predict. Since $q$ is provided to the glimpse agent as reward, it will be rewarded for selecting easily identifiable locations and prefer such a policy. These can be locations that produce a high amount of localized surprising information in $\mu _ { t }$ as compared to $\mu _ { t - 1 }$ , as unexpected appearance of goal or enemies somewhere will signal that the attention was just moved there, hence producing a high likelihood in $q$ . Conversely, low reward locations will be ones that are hard to identify by $q$ as they do not impact any specific part of $\mu _ { t }$ , leaving it largely the same as $\mu _ { t - 1 }$ , or observe a location that was already well-modeled by the agent and hence do not leave an identifiable mark on $\mu _ { t }$ that is a tell-tale sign for q.
+
+![](images/81d5b51ffa8b2a07090878da948998cb61d625782c10e709f32794011c0322fe.jpg)  
+(b) Write error of observation under attention window.
+
+![](images/ba2c2fa373647194fb86f18cc133ed7112d12d0587957870325442344e3db85d.jpg)  
+Figure 11: State reconstruction and observation reconstruction errors in gridworld environment. Observation reconstruction error falls to almost zero for both methods while full state reconstruction remains high for variational approach.
+
+(a) Overall state reconstruction error per pixel.
+
+The above method does not work well in practice. Figure 11a shows the reconstruction error of the unobserved state with our proposed method and the variational approach. Figure 11b shows the observation reconstruction error, which is the error in reproducing just the observation under the attention immediately after it is made. It can be seen that the latter quantity goes to near zero for both methods, meaning that immediate observations are being faithfully recorded into memory and reconstructed. Yet, the overall state reconstruction error remains high for the variational approach, which means that the agent is not able to retain observations over time steps as the attention moves around the state and the environment model is not learning to predict the evolution of the state over time without direct observation.
+
+A reason why the variational approach fails in practice could be that we do not have direct access to $p ( l _ { t } | s _ { t } )$ in order to train $q$ . Instead we must rely on the agent’s internal estimate of the environment state, $\mu _ { t }$ , which itself is being trained alongside the glimpse agent. This results in noisy rewards to the glimpse agent as the $q$ function is not able to accurately predict where the attention is. As we have seen in section A.2, attention control creates a curriculum for training the agent’s state estimate. Hence this forms a loop, where attention control is dependent on agent’s state estimate which is in return dependent on the curriculum generated by the attention.
+
+Extensive hyperparameter tuning may also be required for this approach to work as the glimpse agent can easily default to chosing the same attention location at each time step as it ensures the highest $q ( l _ { t } | s _ { t } )$ .

parse/train/TV9INIrmtWN/TV9INIrmtWN_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "HARD ATTENTION CONTROL BY MUTUAL INFORMATION MAXIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Biological agents have adopted the principle of attention to limit the rate of incoming information from the environment. One question that arises is if an artificial agent has access to only a limited view of its surroundings, how can it control its attention to effectively solve tasks? We propose an approach for learning how to control a hard attention window by maximizing the mutual information between the environment state and the attention location at each step. The agent employs an internal world model to make predictions about its state and focuses attention towards where the predictions may be wrong. Attention is trained jointly with a dynamic memory architecture that stores partial observations and keeps track of the unobserved state. We demonstrate that our approach is effective in predicting the full state from a sequence of partial observations. We also show that the agent’s internal representation of the surroundings, a live mental map, can be used for control in two partially observable reinforcement learning tasks. Videos of the trained agent can be found at https://sites.google.com/view/hard-attention-control. ",
+        "bbox": [
+            233,
+            268,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            507,
+            336,
+            522
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) algorithms have successfully employed neural networks over the past few years, surpassing human level performance in many tasks (Mnih et al., 2015; Silver et al., 2017; Berner et al., 2019; Schulman et al., 2017). But a key difference in the way tasks are performed by humans versus RL algorithms is that humans have the ability to focus on parts of the state at a time, using attention to limit the amount of information gathered at every step. We actively control our attention to build an internal representation of our surroundings over multiple fixations (Fourtassi et al., 2017; Barrouillet et al., 2004; Yarbus, 2013; Itti, 2005). We also use memory and internal world models to predict motions of dynamic objects in the scene when they are not under direct observation (Bosco et al., 2012). By limiting the amount of input information in these two ways, i.e. directing attention only where needed and internally modeling the rest of the environment, we are able to be more efficient in terms of data that needs to be collected from the environment and processed at each time step. ",
+        "bbox": [
+            173,
+            540,
+            825,
+            707
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/d6707bda87abc67308d0b23dca6be6c3e5da4d8361981355bd723cb070b529c9.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the PhysEnv domain (Du & Narasimhan, 2019). Here, the agent (dark blue) needs to navigate to the goal (red) while avoiding enemies (light blue). Typically the full environment states (top) are provided to the RL algorithm to learn. We consider the case where only observations from under a hard attention window (bottom) are available. The attention’s location (yellow square) at each time step is controllable by the agent. Using only partial observations, the agent must learn to represent its surroundings and complete its task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            295,
+            722,
+            702,
+            825
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "By contrast, modern reinforcement learning methods often operate on the entire state. Observing the entire state simultaneously may be difficult in realistic environments. Consider an embodied agent that must actuate its camera to gather visual information about its surroundings learning how to cross a busy street. At every moment, there are different objects in the environment competing for its attention. The agent needs to learn to look left and right to store locations, heading, and speed of nearby vehicles, and perhaps other pedestrians. It must learn to create a live map of its surroundings, frequently checking back on moving objects to update their dynamics. In other words, the agent must learn to selectively move its attention so as to maximize the amount of information it collects from the environment at a time, while internally modeling motions of the other, more predictable parts of the state. Its internal representation, built using successive glimpses, must be sufficient to learn how to complete tasks in this partially observable environment. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            256
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider the problem of acting in an environment that is only partially observable through a controllable, fixed-size, hard attention window (see figure 1). Only the part of the state that is under the attention window is available to the agent as its observation. The rest must be inferred from previous observations and experience. We assume that the location of the attention window at every time step is under control of the agent and its size compared to the full environment state is known to the agent. We distinguish this task from that of learning soft attention (Vaswani et al., 2017), where the full state is attended to, weighted by a vector, and then fed into subsequent layers. Our system must learn to 1) decide where to place the attention in order to gather more information about its surroundings, 2) record the observation made into an internal memory and model the motion within parts of the state that were unobserved, and 3) use this internal representation to learn how to solve its task within the environment. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            415
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our approach for controlling attention uses RL to maximize an information theoretic objective closely related to the notion of surprise or novelty (Schmidhuber, 1991). It is unsupervised in terms of environment rewards, i.e. it can be trained on offline data (states and actions) without knowing the task or related rewards. We discuss this in more detail in section 5. Memory also plays a crucial role in allowing agents to solve tasks in partially observable environments. We pair our attention control mechanism with a memory architecture inspired largely by Du & Narasimhan (2019)’s SpatialNet, but modified to work in partially observable domains. This is described in section 4. Empirically, we show in section 6.1 that our system is able to reconstruct the full state image including dynamic objects at all time steps given only partial observations. Further, we show in section 6.2 that the internal representation built by our attention control mechanism and memory architecture is sufficient for the agent to learn to solve tasks in this challenging partially observable environment. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            598,
+            354,
+            614
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Using hard attention for image classification or object recognition is well studied in computer vision (Alexe et al., 2012; Butko & Movellan, 2009; Larochelle & Hinton, 2010; Paletta et al., 2005; Zoran et al., 2020; Welleck et al., 2017). Attention allows for processing only the salient or interesting parts of the image (Itti et al., 1998). Similarly, attention control has been applied to tracking objects within a video (Denil et al., 2012; Kamkar et al., 2020; Yu et al., 2020). Surprisingly, not a lot of recent work exists on the topic of hard attention control in reinforcement learning domains, where a sequential decision making task has to be solved by using partial observations from under the attention. ",
+        "bbox": [
+            173,
+            631,
+            825,
+            742
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Mnih et al. (2014) proposed a framework for hard attention control in the classification setting and a simple reinforcement learning task. Their approach consists of using environment rewards to train an attention control RL agent. Our approach differs mainly in that we train the attention control using our novel information theoretic objective as reward. Mnih et al. (2014)’s approach leads to a task specific policy for attention control, whereas our approach is unsupervised in terms of the task and can be applied generally to downstream tasks in the environment. Our approach also differs in that we use a memory architecture that is more suited to partially observable tasks with 2D images as input, compared to a RNN used by Mnih et al. (2014). ",
+        "bbox": [
+            174,
+            750,
+            825,
+            861
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There has been much prior work on memory and world models for reinforcement learning (Ha & Schmidhuber, 2018; Graves et al., 2016; Hausknecht & Stone, 2015; Khan et al., 2017). The work closest to our own is Du & Narasimhan (2019)’s SpatialNet, which attempts to learn a task-agnostic world model for multi-task settings. Our memory architecture is largely inspired by SpatialNet and adapted to work in the partially observable setting. We also use their PhysEnv environment to evaluate our approach. Closely related work is Neural Map (Parisotto & Salakhutdinov, 2017), which uses a structured 2D memory map to store information about the environment. Their approach also applies to partially observable RL tasks, but the attention is fixed to the agent. In contrast, we consider the problem of learning to control an attention window that can move independently of the agent location. Recently, Freeman et al. (2019) showed that world models can be learnt by simply limiting the agent’s ability to observe the environment. They apply observational dropout, where the output of the agent’s world model, rather than the environment state, is occasionally provided to the policy. We consider the related scenario where only a part of the true environment state is provided to the agent at each time step and the rest must be modeled using previous observations. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Finally, mutual information has been used to train self-supervised RL agents. This line of work originates in curiosity driven and intrinsically motivated RL (Schmidhuber, 1991; Pathak et al., 2017; Bellemare et al., 2016). Typically, some notion of predictive error or novelty about an aspect of the environment is optimized in lieu of environment rewards. Multiple papers have successfully used different formulations of mutual information to learn how to efficiently explore the environment without extrinsic rewards (Mohamed & Rezende, 2015; Houthooft et al., 2016; Achiam et al., 2018; Gregor et al., 2016; Eysenbach et al., 2018; Sharma et al., 2019). We apply the idea of using mutual information to the problem of curiosity driven attention control. ",
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+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
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+    {
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+        "text": "In this work, we are concerned with tasks solved using reinforcement learning (RL). RL is the study of agents acting in an environment to maximize some notion of long term utility. It is almost always formalized using the language of Markov decision problems (MDPs). States, actions, rewards and transitions form the components of an MDP, often represented as the tuple $\\langle S , A , R , T \\rangle$ . Maximizing the expected sum of (discounted) rewards over rollouts in the environment is usually set as the objective of learning. Once a problem is formulated as an MDP, such that the components $\\langle S , A , R , T \\rangle$ are well defined, one can apply a range of model-free RL algorithms to attempt to solve it (Schulman et al., 2017; Mnih et al., 2016; Wu et al., 2017). A solution to an MDP is typically sought as an optimal policy, $\\pi ^ { * } : S  A$ , which is a function that maps every state to an action that maximizes the long term expected rewards. In this section, we will attempt to formalize the components of the partially observable reinforcement learning problem under study. ",
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+    {
+        "type": "text",
+        "text": "3.1 INTERNAL REPRESENTATION ",
+        "text_level": 1,
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+        "img_path": "images/7d3892123220a2471b3b98f0b35abb357f6151636242067b3385ba55b94cce43.jpg",
+        "image_caption": [
+            "Figure 2: The agent starts with a representation $\\left( \\mu _ { t } \\right)$ from which it attempts a reconstruction $\\left( \\tau _ { t } \\right)$ of the likely current full (unobserved) state $s _ { t }$ . Then, it picks an attention location $\\left( l _ { t } \\right)$ and receives an observation of the state from under the attention $\\left( o _ { t } \\right)$ . We assume that the size of the attention window is known to the agent. The observation is written into $\\mu _ { t }$ to form $\\mu _ { t } ^ { w }$ . The entire map is then stepped through an internal world model to update the dynamical objects, forming $\\mu _ { t + 1 }$ , the representation for the next step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we give a brief description of how the agent stores observations in an internal representation of its surroundings (figure 2). An map $( \\mu )$ tracks the full environment state (s), which is never directly observed. The map is empty at the start of the episode and gets sequentially written into as observations are made. The agent also creates a reconstruction of the full state $( \\tau )$ at every time step based on its map. This will become useful later for training. For now, it suffices that $\\mu _ { t }$ is the map before an observation is made and $\\mu _ { t } ^ { w }$ is the one after the observation is made. $\\mu _ { t + 1 }$ is the map after the dynamics of the system for the next time step are taken into account. ",
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+    {
+        "type": "text",
+        "text": "3.2 THE TWO AGENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We formulate the solutions to controlling the attention at every time step and completing the environment task as two separate reinforcement learning agents. The attention location is controlled by the glimpse agent (the eye in figure 2), and actions within the environment are taken by the regular agent. These two agents have their own separate MDPs with their $\\langle S , A , R \\rangle$ tuples defined below. ",
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+    {
+        "type": "text",
+        "text": "The glimpse agent’s state at every timestep is $\\mu _ { t }$ . Its set of actions is all possible attention locations within the full state (Width $\\times$ Height actions). Its reward is based on the information theoretic objective discussed in section 5. Thus, the glimpse agent is provided the map before an observation is made and it must decide where the observation should be made from in order to optimize its reward. ",
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+    {
+        "type": "text",
+        "text": "The environment agent acting in the environment receives as input $\\mu _ { t } ^ { w }$ , i.e. the internal representation after the observation has been recorded. Its actions are the normal set of actions in the environment and its reward is the normal environment reward. We emphasize that neither agent has access to the full state at any time. They must both act based on the internal representation alone. They also cannot make multiple observations from the same environment state. Once an observation is made, an action must be selected that will change the environment state. In the next section, we will describe in detail how $\\mu _ { t } ^ { w }$ , $\\mu _ { t + 1 }$ and $\\tau _ { t }$ are formed and how the internal map is trained through a sequence of partial observations. ",
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+    {
+        "type": "text",
+        "text": "4 DYNAMIC MEMORY MAP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Memory plays a critical role in enabling agents to act in partially observable environments. It allows the agent to remember parts of the state that are not under direct observation. Along with memory, the agent must be able to model the dynamics of objects in its environment, as it may not receive observation of them for long periods. Finally, the effect of its own actions must be reflected in its belief about the environment. For this purpose, we design a special recurrent memory unit for visual environments called the Dynamic Memory Map (DMM). DMM is inspired largely by Du & Narasimhan (2019)’s work on SpatialNet, but modified to handle partial observations and work in tandem with the glimpse agent. ",
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+    {
+        "type": "text",
+        "text": "4.1 MEMORY MODULES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The DMM consists of three major modules: write, step and reconstruct. ",
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+    {
+        "type": "text",
+        "text": "Write. This module encodes an incoming observation, $o _ { t }$ , into the current memory representation, $\\mu _ { t }$ . The observation is first passed through a series of convolutional operations, $W$ , possibly downsampling it for a more efficient representation. Then it is blended with $\\mu$ using a series of convolutions, $B$ . Finally it is written into the memory but only in the locations where the observation was made, leaving the rest of the memory intact. The write operation can be written as: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d19e9f19fa2f2f3ff0fddbe2761782dd7c69925a2efab3a21527b2702ded923.jpg",
+        "text": "$$\n\\mu _ { t } ^ { w } = C _ { t } * B ( W ( o _ { t } ) , \\mu _ { t } ) + ( 1 - C _ { t } ) * \\mu _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $C _ { t }$ is a mask which is 1 under the attention window and 0 otherwise. ",
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+    {
+        "type": "text",
+        "text": "Step. This module is responsible for modeling the dynamics of the environment and updating the memory to track the full state from one time step to the next. Objects in the environment can be static or dynamic and may be affected by the agent’s own actions. This module is implemented as a residual network, $S$ , similar to SpatialNet (Du & Narasimhan, 2019). We additionally condition $S$ on the agent’s own action $a _ { t }$ . The entire map is updated at each time step to ensure that objects observed previously are also updated to new likely locations. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/025850cb3b1c13c59bbdeef22a71cf33f30fd083203f8770f05dd128826ac9e3.jpg",
+        "text": "$$\n\\mu _ { t + 1 } = \\mu _ { t } + S ( \\mu _ { t } ^ { w } , a _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Reconstruct. This module converts the memory representation to a reconstruction of the full state using a series of deconvolutions $( R )$ . ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/cd3dfe03a748a43fa6a17791860e4d0c23b0f8e53d2d44e6e08e8fbb699f8e8e.jpg",
+        "text": "$$\n\\tau _ { t } = R ( \\mu _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A reconstruction can also be made immediately after the observation, i.e. $\\tau _ { t } ^ { w } = R ( \\mu _ { t } ^ { w } )$ . Reconstruction is essential for training the other two modules of DMM as we explain in section 4.2. The reconstruction error, or the discrepancy between the observation and the reconstruction under the attention window can be back-propagated through the write and step layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 TRAINING LOSS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The write loss, $L _ { w }$ is incurred under the current attention window immediately after the observation is made into $\\mu _ { t } ^ { w }$ . This ensures that the immediate observation is correctly encoded into the memory and trains the write and reconstruction modules. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1795d46a458f7322fd16e568682a80663baac6759c58840b3dee5a908b81c1d5.jpg",
+        "text": "$$\nL _ { w } = C _ { t } * \\lvert \\lvert \\tau _ { t } ^ { w } - o _ { t } \\rvert \\rvert _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The step loss, $L _ { s }$ is incurred after stepping the map and under the attention window in the next step. ",
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+    {
+        "type": "equation",
+        "img_path": "images/9e00058991d9eff6949c2983ee3da034de45b7fbf10ed0cd372fc6e58b93fc3e.jpg",
+        "text": "$$\nL _ { s } = C _ { t + 1 } * \\lvert \\lvert \\tau _ { t + 1 } - o _ { t + 1 } \\rvert \\rvert _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This trains the step module, $S$ , to accurately model the motion of objects in the entire state, so that a faithful representation of the state can be guessed before the next observation is taken. Once the next observation is made, we can calculate the difference between what the agent’s model expected to be at the glimpse location and what actually was there. In figure 2 this is the difference between the images under the yellow rectangle in $\\tau _ { t }$ and $o _ { t }$ . This quantity is the amount of surprise in the dynamics model and we will use this again when training the glimpse agent. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition to the reconstruction loss, the absolute value of the output by the write and step modules is also penalized in order to regularize the contents of the DMM. The total loss for a single step within the rollout is (where $\\alpha = \\beta = 0 . 0 1$ here): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/360dc0754258b57b1e040e6c30feeb483ced9791a514795c88b70bd5839d24e8.jpg",
+        "text": "$$\nL _ { t } = L _ { w } + L _ { s } + \\alpha * C _ { t } * | \\mu _ { t } ^ { w } | + \\beta * | S ( \\mu _ { t } ^ { w } , a _ { t } ) | .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We sum this loss over the entire rollout and jointly minimize it over all steps. The reconstruction error at every location can be back-propagated to the step an observation was recorded there. ",
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+    {
+        "type": "text",
+        "text": "5 MAXIMIZING MUTUAL INFORMATION TO CONTROL ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The intuition behind our approach is that attention should be used to gather information from the environment when and where it is required. The DMM is constantly modeling its environment and attention should be directed to parts of the world where the model is uncertain. This directly informs the agent about difficult to model parts of the environment, thus reducing uncertainty in the state as a whole. Secondly, by focusing on areas with hard to predict dynamics, more data can be collected from those areas, updating the model for future predictions. ",
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+    {
+        "type": "text",
+        "text": "5.1 MUTUAL INFORMATION OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The idea of reducing uncertainty about the environment can be captured in the language of information theory by using mutual information. Specifically, we propose selecting the location of the attention such that its mutual information with the state at the following step is maximized, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d81359222e0e62809fab29d390eccc0312d26db3d0aebcb34bf1c86e4c8a80bc.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { l _ { t } } I ( s _ { t + 1 } ; l _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Where $l _ { t }$ is the location of the attention window at time $t$ . Eq. 1 can be expanded as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/374898378a7127e04be4ba6b5f735f5dafc82543964db0d077d51cd2a10b76fb.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { l _ { t } } \\mathcal { H } ( s _ { t + 1 } ) - \\mathcal { H } ( s _ { t + 1 } | l _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { H }$ denotes the entropy. This expansion brings out a very intuitive explanation of the objective. We would like to pick an attention location that maximizes the reduction in entropy (uncertainty) of the environment state after an observation is made. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 CURIOSITY DRIVEN ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In this section, we show how maximizing the amount of surprise can lead to maximizing the mutual information. Let us begin by further expanding eq. 2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bcb292ab5f7ebff21a3ce5878ad819b663aedfb3eab13521cd24926ffd181234.jpg",
+        "text": "$$\n\\begin{array} { r l } & { I ( s _ { t + 1 } ; l _ { t } ) = \\mathcal { H } ( s _ { t + 1 } ) - \\mathcal { H } ( s _ { t + 1 } | l _ { t } ) } \\\\ & { \\qquad = - \\displaystyle \\sum _ { s _ { t + 1 } } p ( s _ { t + 1 } ) \\log p ( s _ { t + 1 } ) + \\displaystyle \\sum _ { l _ { t } } p ( l _ { t } ) \\sum _ { s _ { t + 1 } } p ( s _ { t + 1 } | l _ { t } ) \\log p ( s _ { t + 1 } | l _ { t } ) } \\\\ & { \\qquad = - \\displaystyle \\sum _ { s _ { t + 1 } \\sim p ( s _ { t + 1 } ) } [ \\log p ( s _ { t + 1 } ) ] + \\displaystyle \\sum _ { l _ { t } } \\sum _ { s _ { t + 1 } } p ( s _ { t + 1 } , l _ { t } ) \\log p ( s _ { t + 1 } | l _ { t } ) } \\\\ & { \\qquad = - \\displaystyle \\operatorname* { l i m } _ { s _ { t + 1 } \\sim p ( s _ { t + 1 } ) } [ \\log p ( s _ { t + 1 } ) ] + \\displaystyle \\sum _ { s _ { t + 1 } , l _ { t } \\sim p ( s _ { t + 1 } , l _ { t } ) } [ \\log p ( s _ { t + 1 } | l _ { t } ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "So, maximizing mutual information between the attention location and the environment state is equivalent to minimizing $\\log p ( s _ { t + 1 } )$ and maximizing $\\log p ( s _ { t + 1 } | l _ { t } )$ under expectation. The first term is the log-likelihood of the next state prior to selecting $l _ { t }$ . Computing it requires marginalization over all possible $l _ { t }$ , which is prohibited by our environment as only a single partial observation from the full state can be provided to the agent. The second term, $\\log p ( s _ { t + 1 } | l _ { t } )$ , is computable at a single attention location from a single state and hence we will focus on maximizing this term. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Now, assume the agent’s belief over the true environment state at the next step, $s _ { t + 1 }$ , is represented by a Gaussian with unit variance around the reconstruction, $\\tau _ { t + 1 }$ . So, $s _ { t + 1 } \\sim \\mathcal { N } ( \\tau _ { t + 1 } , \\mathcal { T } )$ and ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61f2658369b8ff1bcd855e190e8495473f2662450e8c7087c17d64b6a37a8eb1.jpg",
+        "text": "$$\n\\begin{array} { l } { \\log p ( s _ { t + 1 } | \\tau _ { t + 1 } ) \\propto \\log e x p [ - ( s _ { t + 1 } - \\tau _ { t + 1 } ) ^ { T } ( s _ { t + 1 } - \\tau _ { t + 1 } ) / 2 ] } \\\\ { \\quad \\quad \\quad = - \\displaystyle \\sum _ { i } ( s _ { t + 1 } ^ { i } - \\tau _ { t + 1 } ^ { i } ) ^ { 2 } / 2 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "So, minimizing the squared difference between $s _ { t + 1 }$ and $\\tau _ { t + 1 }$ leads to maximizing $\\log p ( s _ { t + 1 } | \\tau _ { t + 1 } )$ . Maximizing $\\log p ( s _ { t + 1 } \\vert \\tau _ { t + 1 } )$ in turn means maximizing $\\log p ( s _ { t + 1 } | l _ { t } )$ , since $\\tau _ { t + 1 }$ is constructed using the deterministic functions $W$ , $S$ , and $R$ once $l _ { t }$ has been picked (see section 4). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to minimize the squared difference between $s _ { t + 1 }$ and $\\tau _ { t + 1 }$ , the agent needs to pick a location $l _ { t }$ that observes the maximum error between $s _ { t }$ and its reconstruction $\\tau _ { t }$ . This is because at step $t$ , if the agent observes the location with the highest reconstruction error, i.e. the least likelihood $p ( s _ { t } | \\tau _ { t } )$ , it will ensure that at the next time step $t + 1$ the agent has the most recent information on that region and can make a good reconstruction of it. Hence, maximizing $\\log p ( s _ { t + 1 } | l _ { t } )$ is equivalent to picking an $l _ { t }$ that will lead to the highest reconstruction error at step $t$ . ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e83e27e473d3926180a3d33c43fbc5eedbc7e396ddfc6200913ab6e24796efcb.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { l _ { t } } \\log p ( s _ { t + 1 } | l _ { t } ) \\implies \\operatorname* { m a x } _ { l _ { t } } \\sum _ { i \\in I _ { t } } ( s _ { t } ^ { i } - \\tau _ { t } ^ { i } ) ^ { 2 } / 2\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            627,
+            656,
+            662
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $I _ { t }$ forms the set of indices under the window at location $l _ { t }$ . Directly optimizing this quantity with respect to $l _ { t }$ is not possible as it can only be computed once an observation has been made and only at a single location. Hence, the glimpse agent must learn to predict the location that is most likely to result in high reconstruction error before an observation is made. Our approach is to formulate the glimpse agent as a reinforcement learner with the reconstruction error as its reward. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An interpretation of this objective is that the glimpse agent is surprise seeking, or curiosity driven. It is attending to parts of the state that are novel in the sense that they are difficult for the agent’s current model to predict. Another interpretation is that the glimpse agent is acting adversarially to the DMM’s model of the environment. At each step the glimpse agent tries to focus on parts of the state that are the most difficult for DMM to reconstruct. By doing so, it is indirectly creating a curriculum of increasingly difficult to model aspects of the environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 FULL TRAINING OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Let us look at a different expansion of the objective in eq. 1, ",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9c2af5cae878e02e50bb475d94d672a21c7cabf73fe6eb3a99862fa3d4d8e51.jpg",
+        "text": "$$\nI ( s _ { t + 1 } ; l _ { t } ) = \\operatorname* { m a x } _ { l _ { t } } \\mathcal { H } ( l _ { t } ) - \\mathcal { H } ( l _ { t } | s _ { t + 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first term is $\\mathcal { H } ( l _ { t } )$ , the entropy over the attention location, which is controlled by the policy of the glimpse agent. This term can easily be maximized by standard RL algorithms, such as A2C (Mnih et al., 2016; Wu et al., 2017), using weighted entropy maximization. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Combining the first term from equation 5 and second term from equation 2, the final objective that we use to train the glimpse agent is ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/348340b7ddc057670d8bcea9cd2515f9b471e7e3d0dfa2638ffe1f2d16205dd3.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { l _ { t } } \\mathcal { H } ( l _ { t } ) - \\mathcal { H } ( s _ { t + 1 } | l _ { t } ) \\equiv \\alpha \\mathcal { H } ( \\pi _ { g l i m p s e } ) + \\operatorname* { m a x } _ { l _ { t } } \\sum _ { i \\in I _ { t } } ( s _ { t } ^ { i } - \\tau _ { t } ^ { i } ) ^ { 2 } / 2 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            188,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ is the entropy weighting set to 0.001 in our experiments. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that this objective is not the mutual information from equation 1. It is possible to design a variational algorithm that maximizes the approximate mutual information based on the expansion in eq. 5 alone (instead of mixing terms from both the expansions). Empirically, however, we found that our objective performs better than the variational approximation to mutual information. See section A.4 for more detail. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate our approach in two environments: a gridworld environment (figure 5) and PhysEnv (figure 1). In both environments, the task is to navigate to a goal while avoiding obstacles and enemies. The agent’s actions are movement in the four discrete cardinal directions. In the gridworld, the state is a one-hot image of the objects in the environment (agent, walls, enemies and goal). The agent’s and enemies’ movements are restricted to the grid squares and the dynamics of the enemies are simple: moving in straight lines until an obstacle is hit in which case it reflects. In PhysEnv, observations are provided as RGB images and the motions of the objects are more varied. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The glimpse agent $^ +$ DMM are first trained on offline data of state and action trajectories (no reward) collected from each environment. More details on data collection are provided in section A.1. We evaluate this part of the training by how accurately the full state is reconstructed by DMM at each step. Once the glimpse agent $+ \\mathrm { D M M }$ have converged, i.e. the reconstruction loss has plateaued, we freeze their parameters and initialize an agent within the environment. The glimpse agent $^ +$ DMM serve as fixed components within the agent while it is learns to solve a task. The agent is trained using an off-the-shelf RL algorithm PPO (Schulman et al., 2017) and is evaluated by the total environment reward it collects during testing episodes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare our method against baselines inspired from related work. In the follow baseline the attention is always focused on the agent itself. This is a formulation of the partial observability as seen in Parisotto & Salakhutdinov (2017). Next, the environment baseline mimics the approach by Mnih et al. (2014), where the environment rewards are used to train the glimpse agent. Finally, the random baseline moves the attention randomly at each time step. For the environment task, we have an additional baseline, full, where the full state is provided to the agent to solve the task. This is an upper limit for how well any method that only receives partial observations can perform. ",
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+    },
+    {
+        "type": "text",
+        "text": "6.1 STATE RECONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/71092c719b65f81742005a021994a55980797aaa67ea2dd72ad222f3683f81a9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12 reward (ours)</td><td rowspan=1 colspan=1>random</td><td rowspan=1 colspan=1>follow</td><td rowspan=1 colspan=1>environment</td></tr><tr><td rowspan=1 colspan=1>Gridworld</td><td rowspan=1 colspan=1>0.00555</td><td rowspan=1 colspan=1>0.007666</td><td rowspan=1 colspan=1>0.01941</td><td rowspan=1 colspan=1>0.00827</td></tr><tr><td rowspan=1 colspan=1>PhysEnv</td><td rowspan=1 colspan=1>0.0521</td><td rowspan=1 colspan=1>0.0614</td><td rowspan=1 colspan=1>0.1186</td><td rowspan=1 colspan=1>0.0826</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Per-pixel L2 loss between agent’s reconstruction $\\tau$ and the ground truth full state $s$ . All methods only see partial observations, but must attempt to reconstruct the full state from internal representation. In both environments, our method has the lowest reconstruction error. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We measure the performance of the glimpse agent $+ { \\bf D M M }$ by the average reconstruction error to the true full (unobserved) state in unseen testing episodes. 25 consecutive observations from 6 unseen test episodes are passed through each model and error is averaged over reconstructions at each time step. This gauges the ability of our attention mechanism and memory to reconstruct the full state using only partial observations. Table 1 shows that using surprise driven reward (our approach) for learning attention control leads to a lower reconstruction error than the other baselines. See figures 5 through 9 in the appendix for visualizations of the reconstructions and training error. In particular, figure 7 and 8 show qualitative comparisons between our method with the baselines. Broadly speaking, our method learns to focus on dynamic objects within the environment and preserve them over many steps in the memory, whereas the baselines lose track of or blur objects as seen in their state reconstructions. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a4007fb83f0203dda06cb9cef47af6d46205c4b567c68486399ad510a9b2b788.jpg",
+        "image_caption": [
+            "Figure 3: (left) Object-wise breakdown of reconstruction error in gridworld. (right) $\\%$ of frames in which the object occurs, i.e. the frequency with which the attention is focuses on a particular object. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows an analysis of what objects are under focus during training in the gridworld environment and their corresponding reconstruction error. The attention forms an interesting curriculum for training the DMM, where it learns to focus on different objects in the environment over time. It starts with walls, then moves to the agent itself, then goal and finally the enemies. Thus, it moves gradually from static to dynamic, harder to model parts of the state, allowing the DMM to learn how to accurately model each object. This very interesting curriculum-like behavior emerges naturally and is not pre-programmed into DMM or the attention agent. It is a consequence of maximizing surprise at every time step. See section A.2 for a more detailed discussion. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6.2 REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "image",
+        "img_path": "images/ef6d6a60aa12e4e740937406379fe1c39ede3ddf9eb0d647ace2af3595c1d0ed.jpg",
+        "image_caption": [
+            "Figure 4: Average testing reward for goal seeking task in PhysEnv (left) and gridworld (right). Comparison between agent with full observations (upper limit) and partial observations trained with surprise (ours), environment rewards and simply following the agent’s location. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 shows the average episodic reward over five testing episodes during the training of the agent. Our method approaches the performance of the upper bound agent that receives full environment states in the gridworld environment. The environment baseline performs similarly to our approach in gridworld, but does poorly in PhysEnv, where our approach performs the best. This may be because the attention policy of the environment baseline has high entropy and ends up exploring large parts of the state within an episode. But it does not learn to focus on the most unpredictable parts of the state as our method does. The follow baseline performs poorly in both environments, likely because it only focuses on the agent and does not explore the rest of the state. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also show the performance of our agent (l2 online) in the condition where the DMM and glimpse agent is trained online along with the policy, all components starting from random initialization. This agent did not have a pretraining step and was trained entirely on random exploratory data, therefore obviating the need of a pretraining dataset. The performance is slightly worse than the pretrained agent. This is an encouraging result in a particularly challenging setting of a non-stationary state space as the memory representation and glimpse policy are changing while the agent learns. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented an approach of using mutual information to control a hard attention window in environments with dynamic objects. From partial observations under the attention window, our system is able to reconstruct the full state at every time step with the least error. The representation learned using our method enables RL agents to solve tasks within the environment, while the baseline methods are unable to learn useful representations on the same memory architecture. This demonstrates that attention control plays a large role in enabling task completion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that our attention control objective is independent of the task. It is solely concerned with gathering information from the environment where it is most unpredictable. This is similar to curiosity driven RL that seeks novel states or surprising transitions. Hence, attention control learned in this manner is a generic solution that can be applied to many downstream tasks. It may be able to be combined with task-specific attention control by fine-tuning on a task reward. A better foveal glimpse model may also be used instead of the fixed size hard attention used here. We made an assumption that only one glimpse is allowed per environment state. But the rate of collecting glimpses may be variable with respect to environment speed. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Curiosity based approaches tend to suffer from the “noisy TV” problem (Savinov et al., 2018) where the agent tends to get distracted by and focus on random, unpredictable events in the environment. Orthogonal approaches have been developed to counter this common weakness by, for example, taking into account the agent’s own actions (Pathak et al., 2017). These approaches can be combined with our method as well. We leave the incorporation of this in attention control for future work. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DETAILS OF DATA COLLECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            434,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For gridworld we used 20000 random steps in the environment to create this training set. In PhysEnv we used a previously trained agent to generate 500,000 steps of demonstrations. We experienced that using a random policy in PhysEnv led to quick deaths of the agent with insufficient episode lengths for learning. In both cases, the data are sequences of full states and agent actions from multiple episodes. In PhysEnv, the map size was $2 1 \\times 2 1$ compared to the full state size of $8 4 \\times 8 4$ and in gridworld both the map and state were $1 0 \\times 1 0$ . The size of the attention window was fixed to be $3 \\times 3$ in the gridworld $9 \\%$ of the full state visible in an observation) and $2 1 \\times 2 1$ in PhysEnv $( 6 . 2 5 \\%$ of the state visible). ",
+        "bbox": [
+            174,
+            165,
+            825,
+            275
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 VISUALIZATIONS OF GLIMPSES DURING TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            297,
+            573,
+            311
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We will discuss where the attention focuses and how that effects the reconstruction (and underlying belief) as the training progresses for the gridworld environment. Please refer to 3 for this discussion. ",
+        "bbox": [
+            174,
+            325,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Initially, the attention learns to focus on the wall objects and the corresponding reconstruction error for the walls decreases rapidly (3.b). This is because the walls are long objects in the environment, present at multiple pixels and side-by-side, providing the largest and most reliable source of error under the attention window initially. The error is provided as reward to the glimpse agent, hence it seeks more walls in the state. ",
+        "bbox": [
+            174,
+            361,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Soon, the error it receives from walls decreases as the agent’s DMM learns to persistently record the location of stationary walls over multiple time steps. Then around 10000 iters, the attention switches to focusing on the agent’s location itself. This is because the agent’s apparent erratic movements provide a larger reward due to their unpredictability than the stationary walls can anymore. This has dual consequences. First, since now the DMM is seeing a lot of examples of the agent’s movement, it learns to accurately model the agent based on an initial state and its actions and the reconstruction error corresponding to the agent falls. At its peak, over $70 \\%$ of glimpses contain the agent within them (figure 3.b). Interestingly, the error corresponding to the wall and goal objects rises as some forgetting of the representation and dynamics of these objects occurs. A little before 20000 iters though, the agent has learnt to represent simultaneously all three objects and it switches attention to the enemies, the only remaining un-modeled objects in the environment. The enemies move predictably, bouncing off walls and other objects, but otherwise in straight lines. The agent does not have direct access to their actions and must infer their motion over multiple time steps using only glimpses. ",
+        "bbox": [
+            174,
+            438,
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+            631
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figures 5 and 6 visualize the behavior of the glimpse agent during the initial stages of training. Initially, (100 iters), the attention mostly moves around randomly as can be seen by the heatmap being active throughout the state. The reconstruction does not correspond to anything in the real state since the agent has only just begun learning. The glimpse agent here is receiving high reward for pretty much every location in the state. By 2000 iters, the glimpse agent has learnt to mainly focus on the wall once it locates it. This can be seen in the high probabilities of the glimpse agent policy near the wall locations and the attention repeatedly fixating on the wall. The agent is simultaneously striving to learn to represent the walls as they are showing up a lot in its training data, and steadily reducing the reconstruction error (the glimpse agent’s reward) coming from walls. ",
+        "bbox": [
+            174,
+            638,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "By iter 11000, the focus has shifted to tracking the agent as can be seen in the heatmap. The heatmap shows that the glimpse agent is singularly focused on the agent’s movements as the error reward it is getting from the walls has dropped off and the agent’s movements are not fully predicatable yet. In the figure for iter 19000, it can be seen that the agent can now track its own location very reliably with an initial position and the actions it is taking, without having to focus the attention on itself. In fact, after step 8 when the agent is discovered, the attention almost never moved back to it and yet the location in the reconstruction perfectly matches that of the unobserved full state. Hence the agent’s location based reward source has largely dried up. The heatmap of attention policy shows that the glimpse agent has become interested in the goals and walls again briefly as it learns to represent all three objects together. It can also be seen from the reconstructions here that the agent is unable to accurately track and model the enemies locations at this point. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "By iter 21000, the attention has shifted focus to the remaining object in the state that is producing a high reconstruction error, i.e. the enemy. Once the enemy is discovered on step 14, the glimpse agent’s policy puts all probability near its location so it can keep observing and model its behavior. ",
+        "bbox": [
+            173,
+            103,
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+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The glimpse agent creates a natural curriculum for training the models within DMM, starting at large stationary objects that do not move, then moving to the agent’s followed by the objects with the most complex dynamics. We conjecture this automatically discovered curriculum is the reason our system is able to learn better reconstructions than the baselines. ",
+        "bbox": [
+            174,
+            152,
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+            208
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b1d00dcc6be27103726a94bc00d44de1546ef78fc25ae6a395229173f0eb670.jpg",
+        "image_caption": [
+            "Figure 5: Visualizations of glimpse behavior in gridworld environment. Each sub-figure is the same testing episode at different stages of training, with the rows corresponding to the time within the episode. In each subfigure, the first column shows the full (unobserved) state. The second column is a heatmap showing the glimpse policy, i.e. the likelihood the glimpse agent will select a location for placing attention. The location that was sampled is indicated in the first column by the yellow square. The final, third, column is the reconstruction $\\tau$ based on the agent’s current current belief about the environment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            101,
+            818,
+            849
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/58c94bc80bbdf079dc9466a7affe5cf5415435ae4e3ae90bbd022b6dbc4ed401.jpg",
+        "image_caption": [
+            "Figure 6: Visualizations of glimpse behavior in PhysEnv environment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            172,
+            751,
+            827
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Next we show qualitative comparisons of the glimpse policy and state reconstruction between our method and the baselines. Each subfigure in figures 7 and 8 shows the same episode of length 20 played out with different glimpse agents. The first column is our method, the second shows a uniform random policy for the glimpse agent, the third shows a glimpse agent trained on environment (task) rewards, and lastly the fourth shows the glimpse following the agent. In both environments, it can be seen that our method learns to focus on dynamic objects within the environment and the resulting state reconstruction shows that these objects are preserved in the memory. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Note that the glimpse policy learned by the environment rewards is significantly more entropic than the one learned by our method, despite having the same entropy weighting hyperparameter. We conjecture that this is because the task reward does not send a clear and consistent learning signal to control the attention window to create better reconstructions, or as seen in our RL results, to facilitate downstream tasks. As such, the reconstructions appear qualitatively similar to the random baseline. ",
+        "bbox": [
+            174,
+            208,
+            823,
+            291
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The random baseline quantitatively performs the closest to our approach, but it fails to focus on dynamic objects consistently like over time, and hence fails to track these objects over long periods in the episode. The reconstructions therefore have streaks and blurs where the objects were last spotted and the physics model has not learnt how to update their positions properly. ",
+        "bbox": [
+            174,
+            299,
+            823,
+            354
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/cc103f12df50b384b3a935f94d0676dbedf84b35dc8f71f8d322648c777ca892.jpg",
+        "image_caption": [
+            "Figure 7: Glimpse behavior for fully trained models of our method and the baselines in gridworld. Each subfigure is a different baseline, with columns within the subfigure being the full unobserved state, heatmap of attention policy, and reconstruction of full state respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            157,
+            753,
+            818
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c07ecfa5d304f6dbceff467d21009d726361025cd0f9b88ca983615555186e2.jpg",
+        "image_caption": [
+            "Figure 8: Glimpse behavior for fully trained models of our method and the baselines in PhysEnv. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            172,
+            751,
+            829
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/41edf80cf0f8258eb553d4de174a5a77bd347e291bb26bf204f84f910b503956.jpg",
+        "image_caption": [
+            "Figure 9: (left) PhysEnv environment. (right) Gridworld environment. Reconstruction error (L2) from the full state during training of the $\\mathrm { D M M } +$ glimpse agent. All approaches have converged by 200k iters, with our approach having the lowest error, i.e. the most faithful reconstruction of the full unobserved state. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            138,
+            803,
+            310
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "These training curves correspond to the results in table 1. The reconstruction error for each method has stabilized. Our method has the lowest reconstruction error for both environments. ",
+        "bbox": [
+            173,
+            404,
+            823,
+            433
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/6112eec42e26da8bcef385499fbaeb3966e29deb24cf3bf17f83bb8b5e9aefa4.jpg",
+        "image_caption": [
+            "Figure 10: (left) PhysEnv environment. (right) Gridworld environment. Episodic reward for glimpse agent during training (higher is better). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            449,
+            803,
+            621
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 shows the episodic reward for the glimpse agent on six testing episodes as the training progresses. This corresponds to the mutual information objective in equation 4, or the reconstruction eror under the attention window in the next step. The modules within DMM are constantly improving to reduce prediction error and hence the reward goes down as the training progresses. But the glimpse agent is simultaneously trying to maximize the objective by searching for areas within the state that still provide high error. Our method maximizes this objective in both environments. This leads to lower overall reconstruction error as the glimpse agent is learning to predict where the highest source of error is going to be at the next step. The environment baseline performs close to random and the follow baseline gathers the least reward in both environments. ",
+        "bbox": [
+            173,
+            685,
+            825,
+            811
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "A.4 VARIATIONAL APPROXIMATION TO MUTUAL INFORMATION MAXIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            746,
+            118
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The mutual information between attention and environment state as expanded in eq. 5 is ",
+        "bbox": [
+            171,
+            128,
+            751,
+            145
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/565c89db39c89d399b722573af72e2b31040e600a420cd04fc19b4a56696ff0d.jpg",
+        "text": "$$\nI ( s _ { t } ; l _ { t } ) = \\mathcal { H } ( l _ { t } ) - \\mathcal { H } ( l _ { t } | s _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            172,
+            596,
+            191
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In our algorithm, we combine the first term in this expansion with another term in eq. 2. This works well empirically. Theoretically, however, it is computationally possible to maximize both terms in eq. 6, hence maximizing the actual mutual information. However, as we shall show here, this does not work well practically. ",
+        "bbox": [
+            173,
+            205,
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+            262
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The first term can be maximized by using maximum entropy RL algorithms for the attention policy as discussed in the main text. Let us consider the second term. ",
+        "bbox": [
+            173,
+            268,
+            823,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1602998e6a55ec5aba7f6b39ffddc07b317ce04d937030e3793cdd0a39be7e7e.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { - \\mathcal { H } ( l _ { t } | s _ { t } ) = - \\sum _ { s _ { t } } p ( s _ { t } ) \\mathcal { H } ( l _ { t } | s _ { t } ) } } \\\\ & { = \\sum _ { s _ { t } } p ( s _ { t } ) \\sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \\log p ( l _ { t } | s _ { t } ) } \\\\ & { = \\frac { \\mathbb { E } } { s _ { t } , l _ { t } \\sim p ( s _ { t } , l _ { t } ) } [ \\log p ( l _ { t } | s _ { t } ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            321,
+            655,
+            420
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "While $\\log p ( l _ { t } | s _ { t } )$ is not directly known, we can construct a variational function, $q ( l _ { t } | s _ { t } )$ , to approximate it. ",
+        "bbox": [
+            176,
+            435,
+            823,
+            464
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Since $K L ( p ( l _ { t } | s _ { t } ) | | q ( l _ { t } | s _ { t } ) ) \\geq 0$ , we have, ",
+        "bbox": [
+            174,
+            469,
+            452,
+            487
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a90228b032286a24abe506f7c57610a4bdac5111a924f48f351741d791078295.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \\log p ( l _ { t } | s _ { t } ) \\geq \\sum _ { l _ { t } } p ( l _ { t } | s _ { t } ) \\log q ( l _ { t } | s _ { t } ) } \\\\ & { \\displaystyle \\sum _ { s _ { t } , l _ { t } \\sim p ( s _ { t } , l _ { t } ) } [ \\log p ( l _ { t } | s _ { t } ) ] \\geq \\underset { s _ { t } , l _ { t } \\sim p ( s _ { t } , l _ { t } ) } { \\mathbb { E } } [ \\log q ( l _ { t } | s _ { t } ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            511,
+            673,
+            574
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Plugging this into eq. 6, ",
+        "bbox": [
+            174,
+            587,
+            333,
+            602
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/541aeccf8fcb114fbdd66e5161d377e1b58e189b3c1b92e8342eb2954bfa4801.jpg",
+        "text": "$$\nI ( s _ { t } ; l _ { t } ) \\geq \\mathcal { H } ( l _ { t } ) + \\underset { s _ { t } , l _ { t } \\sim p ( s _ { t } , l _ { t } ) } { \\mathbb { E } } [ \\log q ( l _ { t } | s _ { t } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            631,
+            647,
+            656
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we can maximize the mutual information by maximizing this lower bound w.r.t. the policy parameters of the glimpse agent. This can be achieved by providing $\\log q ( l _ { t } | s _ { t } )$ as a reward to the glimpse agent and using RL algorithms to maximize its long term value along with the policy entropy. As such, maximizing the mutual information. ",
+        "bbox": [
+            173,
+            671,
+            825,
+            728
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "We can additionally condition $q$ on the internal state and action of the agent at previous time step. Since we do not have access to the full environment state, $s _ { t }$ , we substitute it with the agent’s internal state. The final variational function looks like $q ( l _ { t } | \\mu _ { t } , \\mu _ { t - 1 } , a _ { t - 1 } )$ . $q$ must be trained to match the true posterior $p ( l _ { t } | s _ { t } )$ to provide accurate rewards to the glimpse agent. ",
+        "bbox": [
+            173,
+            734,
+            825,
+            791
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The question now is how to learn $q \\mathrm { ? }$ In this work, it is a neural network trained on samples. This is done by minimizing the KL divergence, ",
+        "bbox": [
+            173,
+            796,
+            825,
+            827
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e5c05949696b53e40bb5759607212fa4f3253416ca9af982adcfe16f62a88e1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatorname* { m i n } _ { \\substack { s _ { t } , l _ { t } \\sim p ( s _ { t } , l _ { t } ) } } [ K L ( p ( l _ { t } | s _ { t } ) | | q ( l _ { t } | \\mu _ { t } , \\mu _ { t - 1 } , a _ { t - 1 } ) ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            854,
+            668,
+            881
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "on samples from the environment and the glimpse agent’s policy. This will ensure $q$ is likely to assign high likelihood to actual locations conditioned on its inputs. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, $q$ will be high at locations of attention that are easy to identify by looking at the previous state, the resulting state, and the agent action. Otherwise, its output will be flat and hard to predict. Since $q$ is provided to the glimpse agent as reward, it will be rewarded for selecting easily identifiable locations and prefer such a policy. These can be locations that produce a high amount of localized surprising information in $\\mu _ { t }$ as compared to $\\mu _ { t - 1 }$ , as unexpected appearance of goal or enemies somewhere will signal that the attention was just moved there, hence producing a high likelihood in $q$ . Conversely, low reward locations will be ones that are hard to identify by $q$ as they do not impact any specific part of $\\mu _ { t }$ , leaving it largely the same as $\\mu _ { t - 1 }$ , or observe a location that was already well-modeled by the agent and hence do not leave an identifiable mark on $\\mu _ { t }$ that is a tell-tale sign for q. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            242
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/81d5b51ffa8b2a07090878da948998cb61d625782c10e709f32794011c0322fe.jpg",
+        "image_caption": [
+            "(b) Write error of observation under attention window. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            262,
+            490,
+            435
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba2c2fa373647194fb86f18cc133ed7112d12d0587957870325442344e3db85d.jpg",
+        "image_caption": [
+            "Figure 11: State reconstruction and observation reconstruction errors in gridworld environment. Observation reconstruction error falls to almost zero for both methods while full state reconstruction remains high for variational approach. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            508,
+            263,
+            808,
+            434
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "(a) Overall state reconstruction error per pixel. ",
+        "bbox": [
+            200,
+            446,
+            477,
+            460
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "The above method does not work well in practice. Figure 11a shows the reconstruction error of the unobserved state with our proposed method and the variational approach. Figure 11b shows the observation reconstruction error, which is the error in reproducing just the observation under the attention immediately after it is made. It can be seen that the latter quantity goes to near zero for both methods, meaning that immediate observations are being faithfully recorded into memory and reconstructed. Yet, the overall state reconstruction error remains high for the variational approach, which means that the agent is not able to retain observations over time steps as the attention moves around the state and the environment model is not learning to predict the evolution of the state over time without direct observation. ",
+        "bbox": [
+            174,
+            530,
+            825,
+            655
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "A reason why the variational approach fails in practice could be that we do not have direct access to $p ( l _ { t } | s _ { t } )$ in order to train $q$ . Instead we must rely on the agent’s internal estimate of the environment state, $\\mu _ { t }$ , which itself is being trained alongside the glimpse agent. This results in noisy rewards to the glimpse agent as the $q$ function is not able to accurately predict where the attention is. As we have seen in section A.2, attention control creates a curriculum for training the agent’s state estimate. Hence this forms a loop, where attention control is dependent on agent’s state estimate which is in return dependent on the curriculum generated by the attention. ",
+        "bbox": [
+            174,
+            662,
+            825,
+            760
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Extensive hyperparameter tuning may also be required for this approach to work as the glimpse agent can easily default to chosing the same attention location at each time step as it ensures the highest $q ( l _ { t } | s _ { t } )$ . ",
+        "bbox": [
+            174,
+            767,
+            823,
+            810
+        ],
+        "page_idx": 20
+    }
+]

parse/train/jnkE5c5f9m/jnkE5c5f9m.md

@@ -0,0 +1,341 @@
+# Canonical Capsules: Self-Supervised Capsules in Canonical Pose
+
+# Weiwei $\mathbf { S u n ^ { 1 , 4 , * } }$
+
+Andrea Tagliasacchi2,3,⇤ Boyang Deng3 Sara Sabour2,3
+
+Soroosh Yazdani3 Geoffrey Hinton2,3 Kwang Moo Yi1,4
+
+1University of British Columbia, 2University of Toronto, 3Google Research, 4University of Victoria, ⇤equal contributions
+
+https://canonical-capsules.github.io
+
+# Abstract
+
+We propose a self-supervised capsule architecture for 3D point clouds. We compute capsule decompositions of objects through permutation-equivariant attention, and self-supervise the process by training with pairs of randomly rotated objects. Our key idea is to aggregate the attention masks into semantic keypoints, and use these to supervise a decomposition that satisfies the capsule invariance/equivariance properties. This not only enables the training of a semantically consistent decomposition, but also allows us to learn a canonicalization operation that enables object-centric reasoning. To train our neural network we require neither classification labels nor manually-aligned training datasets. Yet, by learning an object-centric representation in a self-supervised manner, our method outperforms the state-of-the-art on 3D point cloud reconstruction, canonicalization, and unsupervised classification.
+
+# 1 Introduction
+
+Understanding objects is one of the core problems of computer vision $[ \sqrt { 3 2 } , \sqrt { 1 4 } , \sqrt { 3 8 } ]$ . While this task has traditionally relied on large annotated datasets $[ \hat { \lVert { \boldsymbol { 4 } } \hat { 2 } } , \hat { \lVert { \boldsymbol { 2 } } \hat { 2 } } ] ]$ , unsupervised approaches that utilize self-supervision $ { \mathbb { I } }$ have emerged to remove the need for labels. Recently, researchers have attempted to extend these methods to work on 3D point clouds $\mathbb { \lVert 5 9 \rVert }$ , but the field of unsupervised 3D learning remains relatively uncharted. Conversely, researchers have been extensively investigating 3D deep representations for shape auto-encoding1 [61, 19, 33, 16], making one wonder whether these discoveries can now benefit from unsupervised learning for tasks other than auto-encoding.
+
+Importantly, these recent methods for 3D deep representation learning are not entirely unsupervised. Whether using point clouds $\mathbb { \left[ 6 2 \right] }$ , meshes $\bar { \| 1 9 \| }$ , or implicits $\mathbb { \left[ \left. 3 3 \right] \right. }$ , they owe much of their success to the bias within the dataset that was used for training. Specifically, all 3D models in the popular ShapeNet $\pmb { \mathbb { B } } \|$ dataset are “object-centric” – they are pre-canonicalized to a unit bounding box, and, even more importantly, with an orientation that synchronizes object semantics to Euclidean frame axes (e.g. airplane cockpit is always along $+ y$ , car wheels always touch $z = 0$ ). Differentiable 3D decoders are heavily affected by the consistent alignment of their output with an Euclidean frame $\boxed { 8 } \boxed { 1 6 }$ as local-to-global transformations cannot be easily learnt by fully connected layers. As we will show in Section $^ { 4 . 2 , }$ these methods fail in the absence of pre-alignment, even when data augmentation is used. A concurrent work $\lVert \rVert \bigstar \bigstar \rVert$ also recognizes this problem and proposes a separate learnt canonicalizer, which is shown helpful in downstream classification tasks.
+
+In this work, we leverage the modeling paradigm of capsule networks $[ \left[ 2 3 \right] ]$ . In capsule networks, a scene is perceived via its decomposition into part hierarchies, and each part is represented with a (pose, descriptor) pair: $\textcircled{1}$ The capsule pose specifies the frame of reference of a part, and hence should be transformation equivariant; $\textcircled{2}$ The capsule descriptor specifies the appearance of a part, and hence should be transformation invariant. Thus, one does not have to worry how the data is oriented or translated, as these changes can be encoded within the capsule representation.
+
+We introduce Canonical Capsules, a novel capsule architecture to compute a K-part decomposition of a point cloud. We train our network by feeding pairs of a randomly rotated/translated copies of the same shape (i.e. siamese training) hence removing the requirement of pre-aligned training datasets. We then decompose the point cloud by assigning each point into one of the K parts via attention, which we aggregate into K keypoints. Equivariance is then enforced by requiring the two keypoint sets to only differ by the known (relative) transformation (i.e. a form of self-supervision). For invariance, we simply ask that the descriptors of each keypoint of the two instances match.
+
+With Canonical Capsules, we exploit our decomposition to recover a canonical frame that allows unsupervised “object-centric” learning of 3D deep representations without requiring a semantically aligned dataset. We achieve this task by regressing canonical capsule poses from capsule descriptors via a deep network, and computing a canonicalizing transformation by solving a shape-matching problem $\bar { \textregistered 4 4 }$ . This not only allows more effective shape auto-encoding, but our experiments confirm this results in a latent representation that is more effective in unsupervised classification tasks. Note that, like our decomposition, our canonicalizing transformations are also learnt in a self-supervised fashion, by only training on randomly transformed point clouds.
+
+Contributions. In summary, in this paper we:
+
+• propose an architecture for 3D self-supervised learning based on capsule decomposition; • enable object-centric unsupervised learning by introducing a learned canonical frame of reference; • achieve state-of-the-art performance 3D point cloud auto-encoding/reconstruction, canonicalization, and unsupervised classification.
+
+# 2 Related works
+
+Convolutional Neural Networks lack equivariance to rigid transformations, despite their pivotal role in describing the structure of the 3D scene behind a 2D image. One promising approach to overcome this shortcoming is to add equivariance under a group action in each layer [49, 9]. In our work, we remove the need for a global $\mathbf { S E } ( 3 )$ -equivariant network by canonicalizing the input.
+
+Capsule Networks. Capsule Networks $\mathbb { \left[ \left. 2 3 \right] \right. }$ have been proposed to overcome this issue towards a relational and hierarchical understanding of natural images. Of particular relevance to our work, are methods that apply capsule networks to 3D input data [64, 65, 46], but note these methods are not unsupervised, as they either rely on classification supervision [65], or on datasets that present a significant inductive bias in the form of pre-alignment $\pmb { \| 6 4 \| }$ . In this paper, we take inspiration from the recent Stacked Capsule Auto-Encoders $\bar { \| 3 0 \| }$ , which shows how capsule-style reasoning can be effective as long as the primary capsules can be trained in a self-supervised fashion (i.e. via reconstruction losses). The natural question, which we answer in this paper, is “How can we engineer networks that generate 3D primary capsules in an unsupervised fashion?”
+
+Deep 3D representations. Reconstructing 3D objects requires effective inductive biases about 3D vision and 3D geometry. When the input is images, the core challenge is how to encode $3 D$ projective geometry concepts into the model. This can be achieved by explicitly modeling multi-view geometry $\bar { \lVert 2 8 \rVert }$ , by attempting to learn it $\mathbb { \lVert 1 3 \rVert }$ , or by hybrid solutions $\dot { \left[ 6 0 \right] }$ . But even when input is 3D, there are still significant challenges. It is still not clear which is the 3D representation that is most amenable to deep learning. Researchers proposed the use of meshes $\lVert \boldsymbol { \bar { 5 } 3 } \rVert \textcircled { 3 1 }$ , voxels $\mathbb { 1 } 5 4 , \lvert 5 5 \rvert$ , surface patches [19, 12, 10], and implicit functions $\pm \pm \infty \pm \infty \pm 1$ . Unfortunately, the importance of geometric structures (i.e. part-to-whole relationships) is often overlooked. Recent works have tried to close this gap by using part decomposition consisting of oriented boxes $ { \mathbb { I } } ^ {  { \left[ 5 0 \right] } }$ , ellipsoids $\mathbb { I B Z } , \mathbb { I G } ]$ , convex polytopes $\mathbf { \widehat { \mathbb { m } } }$ , and grids [2]. However, as previously discussed, most of these still heavily rely on a pre-aligned training dataset; our paper attempts to bridge this gap, allowing learning of structured 3D representations without requiring pre-aligned data.
+
+![](images/b214db76fe7d07e923b4ee2f128eac4da2df13f56c046f7d9cb4366137ea8671.jpg)  
+Figure 1: Framework – We learn a capsule encoder for 3D point clouds by relating the decomposition result of two random rigid transformations $\mathbf { T } ^ { a }$ and $\mathbf { T } ^ { b }$ , of a given point cloud, i.e., a Siamese training setup. We learn the parameters of an encoder $\mathcal { E }$ , a per-capsule decoder $\mathcal { D } _ { k }$ , as well as a network that represents a learnt canonical frame $\kappa$ . For illustrative purposes, we shade-out the outputs that do not flow forward, and with $\Sigma$ summarize the aggregations in $\overset { \cdot } { ( 2 ) }$ .
+
+Canonicalization. One way to circumvent the requirement of pre-aligned datasets is to rely on methods capable of registering a point cloud into a canonical frame. The recently proposed CaSPR [39] fulfills this premise, but requires ground-truth canonical point clouds in the form of normalized object coordinate spaces $[ \bar { 1 } \bar { 5 } \bar { 2 } ]$ for supervision. Similarly, $\bar { \| 2 0 \| }$ regresses each view’s pose relative to the canonical pose, but still requires weak annotations in the form of multiple partial views. C3DPO $\textcircled { 1 3 4 } \textcircled { 1 }$ learns a canonical 3D frame based on 2D input keypoints. In contrast to these methods, our solution is completely self-supervised. The concurrent Compass [45] also learns to canonicalize in a self-supervised fashion, but as the process is not end-to-end, this results in a worse performance than ours, as it will be shown in Section 4.3.
+
+Registration. Besides canonicalization, many pairwise registration techniques based on deep learning have been proposed (e.g. [56, $\boxed { 6 3 }$ ), even using semantic keypoints and symmetry to perform the task $\mathbb { \lVert \boldsymbol { 1 5 } \rVert }$ . These methods typically register a pair of instances from the same class, but lack the ability to jointly and consistently register all instances to a shared canonical frame.
+
+# 3 Method
+
+Our network trains on unaligned point clouds as illustrated in Figure $\boldsymbol { 1 } \colon$ we train a network that decomposes point clouds into parts, and enforce invariance/equivariance through a Siamese training setup $\dot { \left[ \left| 4 8 \right| \right] }$ . We then canonicalize the point cloud to a learnt frame of reference, and perform autoencoding in this coordinate space. The losses employed to train $\varepsilon , \kappa$ , and $\mathcal { D }$ , will be covered in Section $^ { 3 . 1 , }$ while the details of their architecture are in Section $3 . 2 .$
+
+Decomposition. In more detail, given a point cloud $\mathbf { P } \in \mathbb { R } ^ { P \times D }$ of $P$ points in $D$ dimensions (in our case $D { = } 3$ ), we perturb it with two random transformations $\mathbf { T } ^ { a }$ ${ { \mathbf { \Omega } } ^ { a } } , { { \mathbf { T } } ^ { b } } { \dot { \mathbf { \Omega } } } \in { \mathbf { S } } \mathbf { E } ( D )$ to produce point clouds $\mathbf { P } ^ { a } , \mathbf { P } ^ { b }$ . We then use a shared permutation-equivariant capsule encoder $\mathcal { E }$ to compute a $K$ -fold attention map $\mathbf { A } \in \mathbb { R } ^ { P \times K }$ for $K$ capsules, as well as per-point feature map $\mathbf { F } \in \overset { \bullet } { \mathbb { R } ^ { P \times C } }$ with $C$ channels:
+
+$$
+\mathbf { A } , \mathbf { F } = { \mathcal { E } } ( \mathbf { P } ) \ ,
+$$
+
+where we drop the superscript indexing the Siamese branch for simplicity. From these attention masks, we then compute, for the $k$ –th capsule its pose $\theta _ { k } \in \mathbb { R } ^ { 3 }$ parameterized by its location in 3D space, and the corresponding capsule descriptor $\beta _ { k } \in \mathbb { R } ^ { C }$ :
+
+$$
+\theta _ { k } = \frac { \sum _ { p } \mathbf { A } _ { p , k } \mathbf { P } _ { p } } { \sum _ { p } \mathbf { A } _ { p , k } } , \qquad \beta _ { k } = \frac { \sum _ { p } \mathbf { A } _ { p , k } \mathbf { F } _ { p } } { \sum _ { p } \mathbf { A } _ { p , k } } .
+$$
+
+Hence, as long as $\mathcal { E }$ is invariant w.r.t. rigid transformations of $\mathbf { P }$ , the pose $\pmb { \theta } _ { k }$ will be transformation equivariant, and the descriptor $\beta _ { k }$ will be transformation invariant. Note that this simplifies the design (and training) of the encoder $\mathcal { E }$ , which only needs to be invariant, rather than equivariant $\textcircled { 1 4 9 } , \textcircled { 4 6 } $ .
+
+Canonicalization. Simply enforcing invariance and equivariance with the above framework is not enough to learn 3D representations that are object-centric, as we lack an (unsupervised) mechanism to bring information into a shared “object-centric” reference frame. Furthermore, the “right” canonical frame is nothing but a convention, thus we need a mechanism that allows the network to make a choice – a choice, however, that must then be consistent across all objects. For example, a learnt canonical frame where the cockpit of airplanes is consistently positioned along $+ z$ is just as good as a canonical frame where it is positioned along the $+ y$ axis. To address this, we propose to link the capsule descriptors to the capsule poses in canonical space, that is, we ask that objects with similar appearance to be located in similar Euclidean neighborhoods in canonical space. We achieve this by regressing canonical capsules poses (i.e. canonical keypoints) $\bar { \pmb { \theta } } \in \mathbb { R } ^ { K \times 3 }$ using the descriptors ${ \bar { \beta } } \in { \breve { \mathbb { R } } } ^ { K \times C }$ via a fully connected deep network $\kappa$ :
+
+$$
+\bar { \pmb { \theta } } = \mathcal { K } \left( \beta \right)
+$$
+
+Because fully connected layers are biased towards learning low-frequency representations $ { \mathbb { \left[ \left[ 2 7 \right] \right] } }$ , this regressor also acts as a regularizer that enforces semantic locality.
+
+Auto-encoding. Finally, in the learnt canonical frame of reference, to train the capsule descriptors via auto-encoding, we reconstruct the point clouds with per-capsule decoders $\mathcal { D } _ { k }$ :
+
+$$
+\tilde { \bf P } = \cup _ { k } \left\{ \mathcal { D } _ { k } ( \bar { \bf R } \pmb { \theta } _ { k } + \bar { \bf t } , \beta _ { k } ) \right\} ,
+$$
+
+where $\cup$ denotes the union operator. The canonicalizing transformation $\bar { \bf T } = ( \bar { \bf R } , \bar { \bf t } )$ can be readily computed by solving a shape-matching problem $\pm \ddagger { 4 } \rVert$ , thanks to the property that our capsule poses and regressed keypoints are in one-to-one correspondence:
+
+$$
+\bar { \bf R } , \bar { \bf t } = \underset { { \bf R } , \mathbf { t } } { \arg \operatorname* { m i n } } \frac { 1 } { K } \sum _ { k } \| ( { \bf R } \pmb { \theta } _ { k } + { \bf t } ) - \bar { \pmb { \theta } } _ { k } \| _ { 2 } ^ { 2 } .
+$$
+
+While the reconstruction in $\textcircled{4}$ is in canonical frame, note it is trivial to transform the point cloud back to the original coordinate system after reconstruction, as the transformation $\bar { \mathbf { T } } ^ { - 1 }$ is available.
+
+# 3.1 Losses
+
+As common in unsupervised methods, our framework relies on a number of losses that control the different characteristics we seek to obtain in our representation. Note none of these losses require ground truth labels. We organize the losses according to the portion of the network they supervise: decomposition, canonicalization, and reconstruction.
+
+Decomposition. While a transformation invariant encoder architecture should be sufficient to achieve equivariance/invariance, this does not prevent the encoder from producing trivial solutions/decompositions once trained. As capsule poses should be transformation equivariant, the poses of the two rotation augmentations $\pmb { \theta } _ { k } ^ { a }$ and $\theta _ { k } ^ { b }$ should only differ by the (known) relative transformation:
+
+$$
+\mathcal { L } _ { \mathrm { e q u i v a r i a n c e } } = \frac { 1 } { K } \sum _ { k } \| \pmb { \theta } _ { k } ^ { a } - ( \mathbf { T } ^ { a } ) ( \mathbf { T } ^ { b } ) ^ { - 1 } \pmb { \theta } _ { k } ^ { b } \| _ { 2 } ^ { 2 } .
+$$
+
+Conversely, capsule descriptors should be transformation invariant, and as the two input points clouds are of the same object, the corresponding capsule descriptors $\beta$ should be identical:
+
+$$
+\mathcal { L } _ { \mathrm { i n v a r i a n c e } } = \frac { 1 } { K } \sum _ { k } \| \beta _ { k } ^ { a } - \beta _ { k } ^ { b } \| _ { 2 } ^ { 2 } .
+$$
+
+We further regularize the capsule decomposition to ensure each of the $K$ heads roughly represent the same “amount” of the input point cloud, hence preventing degenerate (zero attention) capsules. This is achieved by penalizing the attention variance:
+
+$$
+\mathcal { L } _ { \mathrm { e q u i l i b r i u m } } = \frac { 1 } { K } \sum _ { k } \| a _ { k } - \frac { 1 } { K } \Sigma _ { k } a _ { k } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $a _ { k } = \Sigma _ { p } ( \mathbf { A } _ { p , k } )$ denotes the total attention exerted by the $\mathbf { k }$ -th head on the point cloud.
+
+Finally, to facilitate the training process, we ask for capsules to learn a localized representation of geometry. We express the spatial extent of a capsule by computing first-order moments of the represented points with respect to the capsule pose $\pmb { \theta } _ { k }$ :
+
+$$
+\mathcal { L } _ { \mathrm { l o c a l i z a t i o n } } = \frac { 1 } { K } \sum _ { k } \frac { 1 } { a _ { k } } \sum _ { p } { \bf A } _ { p , k } \| \pmb { \theta } _ { k } - { \bf P } _ { p } \| _ { 2 } ^ { 2 } .
+$$
+
+Canonicalization. To train our canonicalizer $\kappa$ , we relate the predicted capsule poses to regressed canonical capsule poses via the optimal rigid transformation from $\textcircled{5}$ :
+
+$$
+\mathcal { L } _ { \mathrm { c a n o n i c a l } } = \frac { 1 } { K } \sum _ { k } \| ( \bar { \bf R } \pmb { \theta } _ { k } + \bar { \bf t } ) - \bar { \pmb { \theta } } _ { k } \| _ { 2 } ^ { 2 } .
+$$
+
+Recall that $\bar { \mathbf { R } }$ and $\bar { \mathbf T }$ are obtained through a differentiable process. Thus, this loss is forcing the aggregated pose $\pmb { \theta } _ { k }$ to agree with the one that goes through the regression path, $\bar { \pmb { \theta } } _ { k }$ . Now, since $\bar { \pmb { \theta } } _ { k }$ is regressed solely from the set of capsule descriptors, similar shapes will result in similar canonical keypoints, and the coordinate system of $\bar { \pmb { \theta } } _ { k }$ is one that employs Euclidean space to encode semantics.
+
+Reconstruction. To learn canonical capsule descriptors in an unsupervised fashion, we rely on an auto-encoding task. We train the decoders $\{ \mathcal { D } _ { k } \}$ by minimizing the Chamfer Distance (CD) between the (canonicalized) input point cloud and the reconstructed one, as in [61, 19]:
+
+$$
+\mathcal { L } _ { \mathrm { r e c o n } } = \mathrm { C D } \left( \bar { \mathbf { R } } \mathbf { P } + \bar { \mathbf { t } } , \tilde { \mathbf { P } } \right) .
+$$
+
+# 3.2 Network Architectures
+
+We briefly summarize our implementation details, including the network architecture; for further details, please refer to the supplementary material.
+
+Encoder – $\mathcal { E }$ . Our architecture is based on the one suggested in $\textcircled { \lVert { 4 7 } \rVert }$ : a pointnet-like architecture with residual connections and attentive context normalization. We utilize Batch Normalization [26] instead of the Group Normalization $\lVert \boldsymbol { \bar { 5 } } 8 \rVert$ , which trained faster in our experiments. We further extend their method to have multiple attention maps, where each attention map corresponds to a capsule.
+
+Decoder – $\mathcal { D }$ . The decoder from $\textcircled{4}$ operates on a per-capsule basis. Our decoder architecture is similar to AtlasNetV2 $\mathbb { [ [  2 ]  }$ (with trainable grids). The difference is that we translate the per-capsule decoded point cloud by the corresponding capsule pose.
+
+Regressor – $\kappa$ . We concatenate the descriptors and apply a series of fully connected layers with ReLU activation to regress the $P$ capsule locations. At the output layer we use a linear activation and subtract the mean of the outputs to make our regressed locations zero-centered in the canonical frame.
+
+Canonicalizing the descriptors. As our descriptors are only approximately rotation invariant (via $\mathcal { L } _ { \mathrm { i n v a r i a n c e } } )$ , we found it useful to re-extract the capsule descriptors $\beta _ { k }$ after canonicalization. Specifically, we compute $\bar { \bf F }$ with the same encoder setup, but with $\bar { \bf P } { = } \bar { \bf R } \bar { \bf P } { + } \bar { \bf T }$ instead of $\mathbf { P }$ and use it to compute $\bar { \boldsymbol { \beta } } _ { k }$ ; we validate this empirically in the supplementary material.
+
+# 4 Results
+
+We first discuss the experimental setup, and then validate our method on a variety of tasks: autoencoding, canonicalization, and unsupervised classification. While the task differs, our learning process remains the same: we learn capsules by reconstructing objects in a learnt canonical frame. We also provide an ablation study, which is expanded in detail in the supplementary material, where we provide additional qualitative results.
+
+# 4.1 Experimental setup
+
+To evaluate our method, we rely on the ShapeNet (Core) dataset $\pmb { \bigtriangledown } ^ { 2 } .$ We follow the category choices from AtlasNetV2 [12], using the airplane and chair classes for single-category experiments, while for multi-category experiments we use all 13 classes: airplane, bench, cabinet, car, chair, monitor, lamp, speaker, firearm, couch, table, cellphone, and watercraft. To make our results most compatible with those reported in the literature, we also use the same splits as in AtlasNetV2 [12]: 31747 shapes in the train, and 7943 shapes in the test set. Unless otherwise noted, we randomly sample 1024 points from the object surface for each shape to create our 3D point clouds.
+
+De-canonicalizing the dataset. As discussed in the introduction, ShapeNet (Core) contains substantial inductive bias in the form of consistent semantic alignment. To remove this bias, we create random
+
+SE(3) transformations, and apply them to each point cloud. We first generate uniformly sampled random rotations, and add uniformly sampled random translations within the range $[ - 0 . 2 , 0 . 2 ]$ , where the bounding volume of the shape ranges in $[ - 1 , + 1 ]$ . Note the relatively limited translation range is chosen to give state-of-the-art methods a chance to compete with our solution. We then use the relative transformation between the point clouds extracted from this ground-truth transformation to evaluate our methods. We refer to this unaligned version of the ShapeNet Core dataset as the unaligned setup, and using the vanilla ShapeNet Core dataset as the aligned setup. For the aligned setup, as there is no need for equivariance adaptation, we simply train our method without the random transformations, and so $\mathcal { L } _ { \mathrm { e q u i v a r i a n c e } }$ and $\mathcal { L } _ { \mathrm { i n v a r i a n c e } }$ are not used. This setup is to simply demonstrate how Canonical Capsules would perform in the presence of a dataset bias.
+
+We emphasize here that a proper generation of random rotation is important. While some existing works have generated them by uniformly sampling the degrees of freedom of an Euler-angle representation, this is known to be an incorrect way to sample random rotations $\pmb { \mathbb { H } }$ , leading to biases in the generated dataset; see the supplementary material.
+
+Implementation details. For all our experiments we use the Adam optimizer $\mathbb { \left[ \left[ 2 9 \right] \right] }$ with an initial learning rate of 0.001 and decay rate of 0.1. We train for 325 epochs for the aligned setup to match the AtlasNetV2 $\mathbb { \lVert 1 2 \rVert }$ original setup. For the unaligned setting, as the problem is harder, we train for a longer number of 450 epochs. We use a batch size of 16. The training rate is ${ \sim } 2 . 5$ iters/sec. We train each model on a single NVidia V100 GPU. Unless stated otherwise, we use $k { = } 1 0$ capsules and capsule descriptors of dimension $C { = } 1 2 8$ . We train three models with our method: two that are single-category (i.e., for airplane and chairs), and one that is multi-category (i.e., all 13 classes). To set the weights for each loss term, we rely on the reconstruction performance (CD) in the training set. We set weights to be one for all terms except for $\mathcal { L } _ { \mathrm { e q u i v a r i a n c e } }$ (5) and $\mathcal { L } _ { \mathrm { e q u i l i b r i u m } } ( 1 0 ^ { - 3 } )$ . In the aligned case, because $\mathcal { L } _ { \mathrm { e q u i v a r i a n c e } }$ and $\mathcal { L } _ { \mathrm { i n v a r i a n c e } }$ are not needed (always zero), we reduce the weights for the other decomposition losses by $1 0 ^ { 3 }$ ; $ { \mathcal { L } } _ { \mathrm { l o c a l i z a t i o n } }$ to $1 0 ^ { - 3 }$ and $\mathcal { L } _ { \mathrm { { e q u i l i b r i u m } } }$ to $1 0 ^ { - 6 }$ .
+
+# 4.2 Auto-encoding – Figure 2 and Table 1
+
+Table 1: Auto-encoding / quantitative – Performance in terms of Chamfer distance with 1024 points per point cloud – metric is multiplied by $\mathrm { 1 0 ^ { 3 } }$ as in $\pmb { \mathbb { I } }$ .   
+
+<table><tr><td></td><td>Method</td><td>Airplane</td><td>Chair</td><td>Multi</td></tr><tr><td>Pgia</td><td>3D-PointCapsNet [64]</td><td>1.94</td><td>3.30</td><td>2.49</td></tr><tr><td></td><td>AtlasNetV212</td><td>1.28</td><td>2.36</td><td>2.14</td></tr><tr><td></td><td>Our method</td><td>0.96</td><td>1.99</td><td>1.76</td></tr><tr><td>nrtgren</td><td>3D-PointCapsNet [64</td><td>5.58</td><td>7.57</td><td>4.66</td></tr><tr><td></td><td>AtlasNetV2 国</td><td>2.80</td><td>3.98</td><td>3.08</td></tr><tr><td></td><td>AtlasNetV2 园 w/ STN</td><td>1.90</td><td>2.99</td><td>2.60</td></tr><tr><td></td><td>Our method</td><td>1.11</td><td>2.58</td><td>2.22</td></tr></table>
+
+We evaluate the performance of our method for the task that was used to train the network – reconstruction / auto-encoding – against three baselines (trained in both single-category and multi-category variants): $\textcircled{1}$ 3D-PointCapsNet $\dot { \left[ 6 4 \right] }$ , an auto-encoder for 3D point clouds that utilize a capsule architecture; $\textcircled{2}$ AtlasNetV2 $\mathbb { [ [  2 ]  }$ , a state-of-the-art auto-encoder which utilizes a multi-head patch-based decoder; $\textcircled{3}$ AtlasNetV2 $\mathbb { \lVert 1 2 \rVert }$ with a spatial-transformer network (STN) aligner from PointNet $[ \beta 6 ]$ , a baseline with canonicalization. We do not compare against $\lVert \overline { { 4 6 } } \rVert$ , as unfortunately source code is not publicly available.
+
+Quantitative analysis – Table 1. We achieve state-of-the-art performance in both the aligned and unaligned settings. The wider margin in the unaligned setup indicates tackling this more challenging scenario damages the performance of AtlasNetV2 [12] and 3D-PointCapsNet $\textcircled { 6 4 } \textcircled { 1 6 }$ much more than our method3. We also include a variant of AtlasNetV2 for which a STN (Spatial Transformer Network) is used to pre-align the point clouds [36], demonstrating how the simplest form of prealigner/canonicalizer is not sufficient.
+
+Qualitative analysis – Figure 2. We illustrate our decomposition-based reconstruction of 3D point clouds, as well as the reconstructions of 3D-PointCapsNet [64] and AtlasNetV2 [12]. As shown, even in the unaligned setup, our method is able to provide semantically consistent capsule decompositions – e.g. the wings of the airplane have consistent colours, and when aligned in the canonical frame, the different airplane instances are well-aligned. Compared to AtlasNetV2 $\mathbb { \lVert \rVert }$ and 3D-PointCapsNet $\pmb { \mathbb { E 4 } }$ , the reconstruction quality is also visibly improved: we better preserve details along the engines of the airplane, or the thin structures of the bench; note also that the decompositions are semantically consistent in our examples. Results are better appreciated in our supplementary material, where we visualize the performance as we continuously traverse $\mathbf { S E } ( 3 )$ .
+
+![](images/d9254b3576e6f0b73d8a67dbdef09b1f14b32c94cc04c89a3c04cfdc66f4a905.jpg)  
+Figure 2: Auto-encoding / qualitative – Example decomposition and reconstruction results setup using Canonical Capsules on several unaligned point cloud instances from the test set. We color each Canonical Capsule with a unique colour, and similarly color “patches” from the reconstruction heads of 3D-PointCapsNet [64] and AtlasNetV2 [12]. Canonical Capsules provide semantically consistent decomposition that is aligned in canonical frame, leading to improved reconstruction quality.
+
+# 4.3 Canonicalization – Table 2
+
+We compare against three baselines: $\textcircled{1}$ Deep Closest Points $\left[ \left[ 5 6 \right] \right]$ , a deep learning-based pairwise point cloud registration method; $\textcircled{2}$ DeepGMR $\begin{array} { r l } {  { \bigl [ \bigl | 6 3 \bigr | \bigr ] } } \end{array}$ a state-of-the-art pairwise registration method that decomposes clouds into Gaussian mixtures and utilizes Rigorously Rotation-Invariant (RRI) features [4]; $\textcircled{3}$ Compass $[ | 4 5 | ]$ a concurrent work on learnt alignment/canonicalization. For all compared methods we use the official implementation. For DeepGMR we use both RRI and the typical XYZ coordinates as input. We also try our method with the RRI features, following DeepGMR’s training protocol and train for 100 epochs.
+
+Metrics. To evaluate the canonicalization performance, we look into the stability of the canonicalization – the shakiness shown in the videos in our supplementary material– represented as the mean standard deviation of the rotations (mStd):
+
+$$
+\mathrm { m S t d } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sqrt { \frac { 1 } { m } \sum _ { j = 1 } ^ { m } ( \angle ( \mathbf { R } ^ { i j } , \mathbf { R } _ { m e a n } ^ { i } ) ) ^ { 2 } } ~ ,
+$$
+
+where $\angle$ is the angular distance between two rotation matrices $[ \overline { { 1 8 } } , \overline { { 6 2 } } , \overline { { 6 3 } } ]$ , $\mathbf { R } ^ { i j }$ is the rotation matrix of the $j$ -th instance of the $i$ -th object in canonical frame, and ${ \bf R } _ { m e a n } ^ { i }$ is the mean rotation $\left[ \left[ 2 1 \right] \right]$ of the $i$ -th object. Note that with mStd we measure the stability of canonicalization with respect to rotations to accommodate for methods that do not deal with translation $[ | 4 5 | ]$ . To allow for comparisons with pairwise registration methods, we also measure performance in terms of the RMSE metric $\textcircled { 1 6 3 } , \textcircled { 6 6 } \textcircled { 1 }$ .
+
+Quantitative analysis – Table $\bigtriangledown$ Compared to Compass $\boldsymbol { \left[ \left[ 4 5 \right] \right] }$ , our method provides improved stability in canonicalization. This also provides an advantage in pairwise registration, delivering state-of-the-art results when XYZ-coordinates are used. Note that while pairwise methods can align two sets of given point clouds, creating a canonical frame that simultaneously registers all point clouds is a non-trivial extension to the problem.
+
+Table 2: Canonicalization – Quantitative evaluation for canonicalization, where we highlight significantly better performance than the concurrent work Compass [45]. While pairwise registration is not our main objective, the global alignment frame created by our method still allows for effective registration (on par, or better) than the state-of-the-art.   
+
+<table><tr><td></td><td></td><td colspan="3">Canonicalization (mStd)↓</td><td colspan="3">Pairwise registration (RMSE)↓</td></tr><tr><td></td><td>Method</td><td>Airplane</td><td>Chair</td><td>Multi</td><td>Airplane</td><td>Chair</td><td>Multi</td></tr><tr><td></td><td>Deep Closest Points [56]</td><td>1</td><td>1</td><td>1</td><td>0.318</td><td>0.160</td><td>0.131</td></tr><tr><td></td><td>DeepGMR [63]</td><td></td><td></td><td>1</td><td>0.079</td><td>0.082</td><td>0.077</td></tr><tr><td>P.JCOo-zX</td><td>Compass 4</td><td>19.105</td><td>19.508</td><td>51.811</td><td>0.412</td><td>0.482</td><td>0.515</td></tr><tr><td></td><td>Our method</td><td>8.278</td><td>5.730</td><td>21.210</td><td>0.022</td><td>0.027</td><td>0.074</td></tr><tr><td>R</td><td>DeepGMR [63]</td><td></td><td></td><td></td><td>0.0001</td><td>0.0001</td><td>0.0001</td></tr><tr><td></td><td>Our method</td><td>(unstable)</td><td>(unstable)</td><td>(unstable)</td><td>0.0006</td><td>0.0009</td><td>0.0016</td></tr></table>
+
+When RRI is used as input, our method is on par with DeepGMR $\pmb { \mathbb { \left\| 6 3 \right\| } }$ , up to a level where registration is near perfect – alignment differences when errors are in the $1 0 ^ { - 4 }$ ballpark are indiscernible. We note that the performance of Deep Closest Points $\begin{array} { r l } {  { \bigl \| \boldsymbol { 5 } 6 \bigr \| } } & { { } } \end{array}$ is not as good as reported in the original paper, as we uniformly draw rotations from SO(3). When a sub-portion of $\mathbf { S O } ( 3 )$ is used, e.g. a quarter of what we are using, DCP performs relatively well (0.008 in the multi-class experiment). While curriculum learning could be used to enhance the performance of DCP, our technique does not need to rely on these more complex training techniques.
+
+We further note that, while RRI delivers good registration performance, using RRI features cause the learnt canonicalization to fail – training becomes unstable. This hints that RRI features may be throwing away too much information to achieve transformation invariance. Our method using raw XYZ coordinates as input, on the other hand, provides comparable registration performance, and is able to do significantly more than just registration (i.e. classification, reconstruction).
+
+# 4.4 Unsupervised classification – Table 3
+
+Beyond reconstruction and canonicalization, we evaluate the usefulness of our method via a classification task that is not related in any way to the losses used for training. We compute the features from the auto-encoding methods from Section $4 . 2$ against those from our method (where we build features by combining pose with descriptors) to perform 13-way classification with two different techniques:
+
+We train a supervised linear Support Vector Machine (SVM) on the extracted features [1, Ch. 7]; We perform unsupervised K-Means clustering [1, Ch. 9] and then label each cluster via bipartite matching with the actual labels through the Hungarian algorithm.
+
+Note the former provides an upper bound for unsupervised classification, while better performance on the latter implies that the learnt features are able to separate the classes into clusters that are compact (in an Euclidean sense).
+
+Table 3: Classification – Top-1 accuracy $\% )$   
+
+<table><tr><td>Method</td><td></td><td>SVM</td><td>K-Means</td></tr><tr><td>Pse</td><td>3D-PointCapsNet [64]</td><td>93.81</td><td>65.87</td></tr><tr><td></td><td>AtlasNetV2[12]</td><td>94.07</td><td>61.66</td></tr><tr><td></td><td>Our method</td><td>94.21</td><td>69.82</td></tr><tr><td>aggen</td><td>3D-PointCapsNet [64]</td><td>71.13</td><td>14.59</td></tr><tr><td></td><td>AtlasNetV2 国</td><td>64.85</td><td>17.12</td></tr><tr><td></td><td>AtlasNetV2 圆 w/ STN</td><td>78.55</td><td>20.03</td></tr><tr><td></td><td>Our method</td><td>87.33</td><td>43.04</td></tr></table>
+
+Analysis of results – SVM. Note how our method provides best results in all cases, and when the dataset is not unaligned the difference is significant. This shows that, while 3D-PointCapsNet and AtlasNetV2 (with and without STN) are able to somewhat auto-encode point clouds in the unaligned setup, what they learn does not translate well to classification. However, the features learned with Canonical Capsules are more related to the semantics of the object, which helps classification.
+
+Analysis of results – K-Means. The performance gap becomes wider when K-Means is used – even in the aligned case. This could mean that the features extracted by Canonical Capsules are better suited for other unsupervised tasks, having a feature space that is close to being Euclidean in terms of semantics. The difference is striking in the unaligned setup. We argue that these results emphasize the importance of the capsule framework – jointly learning the invariances and equivariances in the data – is cardinal to unsupervised learning [25, $\checkmark$ .
+
+Table 4: Effect of losses – Reconstruction performance, and canonicalization performance when loss terms are removed.   
+
+<table><tr><td></td><td>Full</td><td>-Linvar</td><td>-Lequiv</td><td>-Lcanonical</td><td>-Llocalization</td><td>-Leqilibrium</td></tr><tr><td>CD</td><td>1.11</td><td>1.12</td><td>1.16</td><td>1.12</td><td>1.44</td><td>1.60</td></tr><tr><td>Std</td><td>8.278</td><td>9.983</td><td>110.174</td><td>8.421</td><td>113.204</td><td>92.970</td></tr></table>
+
+Table 5: Backbone – Auto-encoding performance (Chamfer distance) when we use various permutation-invariant backbones.   
+
+<table><tr><td></td><td>PointNet</td><td>PointNet++</td><td>DGCNN</td><td>ACNe</td></tr><tr><td>CD</td><td>1.28</td><td>1.34</td><td>1.21</td><td>1.11</td></tr></table>
+
+# 4.5 Ablation study
+
+To make the computational cost manageable, we perform all ablations with the airplane category (the category with most instances), and in the unaligned setup (unless otherwise noted). Please also see supplementary material for more ablation studies.
+
+Losses – Table 4. We analyze the importance of each loss term, with the exception of $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n } }$ which is necessary for training. All losses beneficially contribute to reconstruction performance, but note how $\mathcal { L } _ { \mathrm { e q u i v } }$ , $ { \mathcal { L } } _ { \mathrm { l o c a l i z a t i o n } }$ and $\mathcal { L } _ { \mathrm { { e q u i l i b r i u m } } }$ affect it to a larger degree. By considering our canonicalization metric, we can motivate this outcome by observing that the method fails to perform canonicalization when these losses are not employed (i.e. training collapses).
+
+Encoder architecture – Table $\boxed { \pmb { 5 } }$ Our method can be used with any backbone, as our main contribution lies in the self-supervised canonicalization architecture. For completeness, we explore variants of our method using different back-bones: PointNet $\textcircled { \lvert 3 6 \rvert }$ , PointNet+ $^ { - + }$ [37], DGCNN $\boxed { \sqrt { 5 7 } }$ , and ACNe $\mathbb { H }$ . Among these, the ACNe backbone performs best; note that all the variants in Table 5, regardless of backbone choice, significantly outperforms all other methods reported in Table 1.
+
+# 5 Conclusions
+
+In this paper, we provide a self-supervised framework to train primary capsule decompositions for 3D point clouds. We rely on a Siamese setup that allows self-supervision and auto-encoding in canonical space, circumventing the customary need to train on pre-aligned datasets. Despite being trained in a self-supervised fashion, our representation achieves state-of-the-art performance across auto-encoding, canonicalization and classification tasks. These results are made possible by allowing the network to learn a canonical frame of reference. We interpret this result as giving our neural networks a mechanism to construct a “mental picture” of a given 3D object – so that downstream tasks are executed within an object-centric coordinate frame.
+
+Future work. As many objects have natural symmetries that we do not consider at the moment $\mathbb { \left[ 1 2 \right] }$ providing our canonicalizer a way to encode such a prior is likely to further improve the representation. We perform decomposition at a single level, and it would be interesting to investigate how to effectively engineer multi-level decompositions $\mathbb { \left[ \left[ 5 \right] \right] }$ ; one way could be to over-decompose the input in a redundant fashion (with large $K$ ), and use a downstream layers that “selects” the decomposition heads to be used $\textcircled { 6 }$ . We would also like to extend our results to more “in-the-wild” 3D computer vision and understand whether learning object-centric representations is possible when incomplete (i.e., single view $\pmb { \Vert 3 5 \Vert }$ or occluded) data is given in input, when an entire scene with potentially multiple objects is given $\mathbb { \lVert \boldsymbol { 4 3 } \rVert }$ , or where our measurement of the 3D world is a single 2D image [48], or by exploiting the persistence of objects in video [40].
+
+Broader impact. While our work is exclusively on 3D shapes, and thus not immediately subject to any societal concerns, it enhances how Artificial Intelligence (AI) can understand and model 3D geometry. Thus, similar to how image recognition could be misused, one should be careful when extending the use of our method. In addition, while not subject to how the data itself is aligned, the learnt canonical frame of our method is still data-driven, thus subject to any data collection biases that may exist – canonicalization will favour shapes that appear more often. This should also be taken into account with care to prevent any biased decisions when utilizing our method within a decision making AI platform.
+
+# Acknowledgements
+
+This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, NSERC Collaborative Research and Development Grant, Google, Compute Canada, and Advanced Research Computing at the University of British Columbia.
+
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+[40] Sara Sabour, Andrea Tagliasacchi, Soroosh Yazdani, Geoffrey E. Hinton, and David J. Fleet. Unsupervised part representation by flow capsules, 2021.   
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Abstract and Section 1.   
+(b) Did you describe the limitations of your work? [Yes] See Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] There are no direct ethics concerns for the paper.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] In addition to the public code release that we will do, we have included the code in the supplementary   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Due to the large amount of compute requirement for repeated experiments we have not been able to include them in the submission. However, the results do not vary much from one run to the other in our experience, and we will release code to reproduce the results.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4.1. Note that to preserve anonymity we have decided not to include exact detail.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 4.1.   
+(b) Did you mention the license of the assets? [Yes] See Section 4.1.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/jnkE5c5f9m/jnkE5c5f9m_content_list.json

@@ -0,0 +1,1430 @@
+[
+    {
+        "type": "text",
+        "text": "Canonical Capsules: Self-Supervised Capsules in Canonical Pose ",
+        "text_level": 1,
+        "bbox": [
+            236,
+            122,
+            763,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Weiwei $\\mathbf { S u n ^ { 1 , 4 , * } }$ ",
+        "text_level": 1,
+        "bbox": [
+            202,
+            224,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andrea Tagliasacchi2,3,⇤ Boyang Deng3 Sara Sabour2,3 ",
+        "bbox": [
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+            224,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Soroosh Yazdani3 Geoffrey Hinton2,3 Kwang Moo Yi1,4 ",
+        "bbox": [
+            269,
+            260,
+            728,
+            276
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of British Columbia, 2University of Toronto, 3Google Research, 4University of Victoria, ⇤equal contributions ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "https://canonical-capsules.github.io ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
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+            535,
+            407
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a self-supervised capsule architecture for 3D point clouds. We compute capsule decompositions of objects through permutation-equivariant attention, and self-supervise the process by training with pairs of randomly rotated objects. Our key idea is to aggregate the attention masks into semantic keypoints, and use these to supervise a decomposition that satisfies the capsule invariance/equivariance properties. This not only enables the training of a semantically consistent decomposition, but also allows us to learn a canonicalization operation that enables object-centric reasoning. To train our neural network we require neither classification labels nor manually-aligned training datasets. Yet, by learning an object-centric representation in a self-supervised manner, our method outperforms the state-of-the-art on 3D point cloud reconstruction, canonicalization, and unsupervised classification. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding objects is one of the core problems of computer vision $[ \\sqrt { 3 2 } , \\sqrt { 1 4 } , \\sqrt { 3 8 } ]$ . While this task has traditionally relied on large annotated datasets $[ \\hat { \\lVert { \\boldsymbol { 4 } } \\hat { 2 } } , \\hat { \\lVert { \\boldsymbol { 2 } } \\hat { 2 } } ] ]$ , unsupervised approaches that utilize self-supervision $ { \\mathbb { I } }$ have emerged to remove the need for labels. Recently, researchers have attempted to extend these methods to work on 3D point clouds $\\mathbb { \\lVert 5 9 \\rVert }$ , but the field of unsupervised 3D learning remains relatively uncharted. Conversely, researchers have been extensively investigating 3D deep representations for shape auto-encoding1 [61, 19, 33, 16], making one wonder whether these discoveries can now benefit from unsupervised learning for tasks other than auto-encoding. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Importantly, these recent methods for 3D deep representation learning are not entirely unsupervised. Whether using point clouds $\\mathbb { \\left[ 6 2 \\right] }$ , meshes $\\bar { \\| 1 9 \\| }$ , or implicits $\\mathbb { \\left[ \\left. 3 3 \\right] \\right. }$ , they owe much of their success to the bias within the dataset that was used for training. Specifically, all 3D models in the popular ShapeNet $\\pmb { \\mathbb { B } } \\|$ dataset are “object-centric” – they are pre-canonicalized to a unit bounding box, and, even more importantly, with an orientation that synchronizes object semantics to Euclidean frame axes (e.g. airplane cockpit is always along $+ y$ , car wheels always touch $z = 0$ ). Differentiable 3D decoders are heavily affected by the consistent alignment of their output with an Euclidean frame $\\boxed { 8 } \\boxed { 1 6 }$ as local-to-global transformations cannot be easily learnt by fully connected layers. As we will show in Section $^ { 4 . 2 , }$ these methods fail in the absence of pre-alignment, even when data augmentation is used. A concurrent work $\\lVert \\rVert \\bigstar \\bigstar \\rVert$ also recognizes this problem and proposes a separate learnt canonicalizer, which is shown helpful in downstream classification tasks. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we leverage the modeling paradigm of capsule networks $[ \\left[ 2 3 \\right] ]$ . In capsule networks, a scene is perceived via its decomposition into part hierarchies, and each part is represented with a (pose, descriptor) pair: $\\textcircled{1}$ The capsule pose specifies the frame of reference of a part, and hence should be transformation equivariant; $\\textcircled{2}$ The capsule descriptor specifies the appearance of a part, and hence should be transformation invariant. Thus, one does not have to worry how the data is oriented or translated, as these changes can be encoded within the capsule representation. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We introduce Canonical Capsules, a novel capsule architecture to compute a K-part decomposition of a point cloud. We train our network by feeding pairs of a randomly rotated/translated copies of the same shape (i.e. siamese training) hence removing the requirement of pre-aligned training datasets. We then decompose the point cloud by assigning each point into one of the K parts via attention, which we aggregate into K keypoints. Equivariance is then enforced by requiring the two keypoint sets to only differ by the known (relative) transformation (i.e. a form of self-supervision). For invariance, we simply ask that the descriptors of each keypoint of the two instances match. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With Canonical Capsules, we exploit our decomposition to recover a canonical frame that allows unsupervised “object-centric” learning of 3D deep representations without requiring a semantically aligned dataset. We achieve this task by regressing canonical capsule poses from capsule descriptors via a deep network, and computing a canonicalizing transformation by solving a shape-matching problem $\\bar { \\textregistered 4 4 }$ . This not only allows more effective shape auto-encoding, but our experiments confirm this results in a latent representation that is more effective in unsupervised classification tasks. Note that, like our decomposition, our canonicalizing transformations are also learnt in a self-supervised fashion, by only training on randomly transformed point clouds. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions. In summary, in this paper we: ",
+        "bbox": [
+            176,
+            435,
+            475,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• propose an architecture for 3D self-supervised learning based on capsule decomposition; • enable object-centric unsupervised learning by introducing a learned canonical frame of reference; • achieve state-of-the-art performance 3D point cloud auto-encoding/reconstruction, canonicalization, and unsupervised classification. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 Related works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            544,
+            325,
+            560
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks lack equivariance to rigid transformations, despite their pivotal role in describing the structure of the 3D scene behind a 2D image. One promising approach to overcome this shortcoming is to add equivariance under a group action in each layer [49, 9]. In our work, we remove the need for a global $\\mathbf { S E } ( 3 )$ -equivariant network by canonicalizing the input. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Capsule Networks. Capsule Networks $\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }$ have been proposed to overcome this issue towards a relational and hierarchical understanding of natural images. Of particular relevance to our work, are methods that apply capsule networks to 3D input data [64, 65, 46], but note these methods are not unsupervised, as they either rely on classification supervision [65], or on datasets that present a significant inductive bias in the form of pre-alignment $\\pmb { \\| 6 4 \\| }$ . In this paper, we take inspiration from the recent Stacked Capsule Auto-Encoders $\\bar { \\| 3 0 \\| }$ , which shows how capsule-style reasoning can be effective as long as the primary capsules can be trained in a self-supervised fashion (i.e. via reconstruction losses). The natural question, which we answer in this paper, is “How can we engineer networks that generate 3D primary capsules in an unsupervised fashion?” ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep 3D representations. Reconstructing 3D objects requires effective inductive biases about 3D vision and 3D geometry. When the input is images, the core challenge is how to encode $3 D$ projective geometry concepts into the model. This can be achieved by explicitly modeling multi-view geometry $\\bar { \\lVert 2 8 \\rVert }$ , by attempting to learn it $\\mathbb { \\lVert 1 3 \\rVert }$ , or by hybrid solutions $\\dot { \\left[ 6 0 \\right] }$ . But even when input is 3D, there are still significant challenges. It is still not clear which is the 3D representation that is most amenable to deep learning. Researchers proposed the use of meshes $\\lVert \\boldsymbol { \\bar { 5 } 3 } \\rVert \\textcircled { 3 1 }$ , voxels $\\mathbb { 1 } 5 4 , \\lvert 5 5 \\rvert$ , surface patches [19, 12, 10], and implicit functions $\\pm \\pm \\infty \\pm \\infty \\pm 1$ . Unfortunately, the importance of geometric structures (i.e. part-to-whole relationships) is often overlooked. Recent works have tried to close this gap by using part decomposition consisting of oriented boxes $ { \\mathbb { I } } ^ {  { \\left[ 5 0 \\right] } }$ , ellipsoids $\\mathbb { I B Z } , \\mathbb { I G } ]$ , convex polytopes $\\mathbf { \\widehat { \\mathbb { m } } }$ , and grids [2]. However, as previously discussed, most of these still heavily rely on a pre-aligned training dataset; our paper attempts to bridge this gap, allowing learning of structured 3D representations without requiring pre-aligned data. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/b214db76fe7d07e923b4ee2f128eac4da2df13f56c046f7d9cb4366137ea8671.jpg",
+        "image_caption": [
+            "Figure 1: Framework – We learn a capsule encoder for 3D point clouds by relating the decomposition result of two random rigid transformations $\\mathbf { T } ^ { a }$ and $\\mathbf { T } ^ { b }$ , of a given point cloud, i.e., a Siamese training setup. We learn the parameters of an encoder $\\mathcal { E }$ , a per-capsule decoder $\\mathcal { D } _ { k }$ , as well as a network that represents a learnt canonical frame $\\kappa$ . For illustrative purposes, we shade-out the outputs that do not flow forward, and with $\\Sigma$ summarize the aggregations in $\\overset { \\cdot } { ( 2 ) }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            90,
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+            181
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            285,
+            821,
+            314
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Canonicalization. One way to circumvent the requirement of pre-aligned datasets is to rely on methods capable of registering a point cloud into a canonical frame. The recently proposed CaSPR [39] fulfills this premise, but requires ground-truth canonical point clouds in the form of normalized object coordinate spaces $[ \\bar { 1 } \\bar { 5 } \\bar { 2 } ]$ for supervision. Similarly, $\\bar { \\| 2 0 \\| }$ regresses each view’s pose relative to the canonical pose, but still requires weak annotations in the form of multiple partial views. C3DPO $\\textcircled { 1 3 4 } \\textcircled { 1 }$ learns a canonical 3D frame based on 2D input keypoints. In contrast to these methods, our solution is completely self-supervised. The concurrent Compass [45] also learns to canonicalize in a self-supervised fashion, but as the process is not end-to-end, this results in a worse performance than ours, as it will be shown in Section 4.3. ",
+        "bbox": [
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+            319,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Registration. Besides canonicalization, many pairwise registration techniques based on deep learning have been proposed (e.g. [56, $\\boxed { 6 3 }$ ), even using semantic keypoints and symmetry to perform the task $\\mathbb { \\lVert \\boldsymbol { 1 5 } \\rVert }$ . These methods typically register a pair of instances from the same class, but lack the ability to jointly and consistently register all instances to a shared canonical frame. ",
+        "bbox": [
+            173,
+            450,
+            825,
+            507
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            526,
+            271,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our network trains on unaligned point clouds as illustrated in Figure $\\boldsymbol { 1 } \\colon$ we train a network that decomposes point clouds into parts, and enforce invariance/equivariance through a Siamese training setup $\\dot { \\left[ \\left| 4 8 \\right| \\right] }$ . We then canonicalize the point cloud to a learnt frame of reference, and perform autoencoding in this coordinate space. The losses employed to train $\\varepsilon , \\kappa$ , and $\\mathcal { D }$ , will be covered in Section $^ { 3 . 1 , }$ while the details of their architecture are in Section $3 . 2 .$ ",
+        "bbox": [
+            173,
+            558,
+            825,
+            628
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Decomposition. In more detail, given a point cloud $\\mathbf { P } \\in \\mathbb { R } ^ { P \\times D }$ of $P$ points in $D$ dimensions (in our case $D { = } 3$ ), we perturb it with two random transformations $\\mathbf { T } ^ { a }$ ${ { \\mathbf { \\Omega } } ^ { a } } , { { \\mathbf { T } } ^ { b } } { \\dot { \\mathbf { \\Omega } } } \\in { \\mathbf { S } } \\mathbf { E } ( D )$ to produce point clouds $\\mathbf { P } ^ { a } , \\mathbf { P } ^ { b }$ . We then use a shared permutation-equivariant capsule encoder $\\mathcal { E }$ to compute a $K$ -fold attention map $\\mathbf { A } \\in \\mathbb { R } ^ { P \\times K }$ for $K$ capsules, as well as per-point feature map $\\mathbf { F } \\in \\overset { \\bullet } { \\mathbb { R } ^ { P \\times C } }$ with $C$ channels: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dab0d03a2759dc4b53aada43e1d65d535a911d1d5e2a0e5dcd19d7343bd86bcc.jpg",
+        "text": "$$\n\\mathbf { A } , \\mathbf { F } = { \\mathcal { E } } ( \\mathbf { P } ) \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            704,
+            549,
+            722
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we drop the superscript indexing the Siamese branch for simplicity. From these attention masks, we then compute, for the $k$ –th capsule its pose $\\theta _ { k } \\in \\mathbb { R } ^ { 3 }$ parameterized by its location in 3D space, and the corresponding capsule descriptor $\\beta _ { k } \\in \\mathbb { R } ^ { C }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49607a0225e78028587dcb1226b801ea35dde2b3f3e30e41288cc282b94ec101.jpg",
+        "text": "$$\n\\theta _ { k } = \\frac { \\sum _ { p } \\mathbf { A } _ { p , k } \\mathbf { P } _ { p } } { \\sum _ { p } \\mathbf { A } _ { p , k } } , \\qquad \\beta _ { k } = \\frac { \\sum _ { p } \\mathbf { A } _ { p , k } \\mathbf { F } _ { p } } { \\sum _ { p } \\mathbf { A } _ { p , k } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hence, as long as $\\mathcal { E }$ is invariant w.r.t. rigid transformations of $\\mathbf { P }$ , the pose $\\pmb { \\theta } _ { k }$ will be transformation equivariant, and the descriptor $\\beta _ { k }$ will be transformation invariant. Note that this simplifies the design (and training) of the encoder $\\mathcal { E }$ , which only needs to be invariant, rather than equivariant $\\textcircled { 1 4 9 } , \\textcircled { 4 6 } $ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Canonicalization. Simply enforcing invariance and equivariance with the above framework is not enough to learn 3D representations that are object-centric, as we lack an (unsupervised) mechanism to bring information into a shared “object-centric” reference frame. Furthermore, the “right” canonical frame is nothing but a convention, thus we need a mechanism that allows the network to make a choice – a choice, however, that must then be consistent across all objects. For example, a learnt canonical frame where the cockpit of airplanes is consistently positioned along $+ z$ is just as good as a canonical frame where it is positioned along the $+ y$ axis. To address this, we propose to link the capsule descriptors to the capsule poses in canonical space, that is, we ask that objects with similar appearance to be located in similar Euclidean neighborhoods in canonical space. We achieve this by regressing canonical capsules poses (i.e. canonical keypoints) $\\bar { \\pmb { \\theta } } \\in \\mathbb { R } ^ { K \\times 3 }$ using the descriptors ${ \\bar { \\beta } } \\in { \\breve { \\mathbb { R } } } ^ { K \\times C }$ via a fully connected deep network $\\kappa$ : ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86dbb1174784cf1174c4710c073ad549e2d239cf15d052cb7f25111df9ff9a48.jpg",
+        "text": "$$\n\\bar { \\pmb { \\theta } } = \\mathcal { K } \\left( \\beta \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Because fully connected layers are biased towards learning low-frequency representations $ { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }$ , this regressor also acts as a regularizer that enforces semantic locality. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Auto-encoding. Finally, in the learnt canonical frame of reference, to train the capsule descriptors via auto-encoding, we reconstruct the point clouds with per-capsule decoders $\\mathcal { D } _ { k }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3d74c1c82daad71c8200df564e1b61ce5cdfb4224cda6dd9cec2aacc5cbcf5a.jpg",
+        "text": "$$\n\\tilde { \\bf P } = \\cup _ { k } \\left\\{ \\mathcal { D } _ { k } ( \\bar { \\bf R } \\pmb { \\theta } _ { k } + \\bar { \\bf t } , \\beta _ { k } ) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\cup$ denotes the union operator. The canonicalizing transformation $\\bar { \\bf T } = ( \\bar { \\bf R } , \\bar { \\bf t } )$ can be readily computed by solving a shape-matching problem $\\pm \\ddagger { 4 } \\rVert$ , thanks to the property that our capsule poses and regressed keypoints are in one-to-one correspondence: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3099658afddc18c7961e0749d3bfd39af458ea3f92a5177ce5abc7cd421c6f3.jpg",
+        "text": "$$\n\\bar { \\bf R } , \\bar { \\bf t } = \\underset { { \\bf R } , \\mathbf { t } } { \\arg \\operatorname* { m i n } } \\frac { 1 } { K } \\sum _ { k } \\| ( { \\bf R } \\pmb { \\theta } _ { k } + { \\bf t } ) - \\bar { \\pmb { \\theta } } _ { k } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While the reconstruction in $\\textcircled{4}$ is in canonical frame, note it is trivial to transform the point cloud back to the original coordinate system after reconstruction, as the transformation $\\bar { \\mathbf { T } } ^ { - 1 }$ is available. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 Losses ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As common in unsupervised methods, our framework relies on a number of losses that control the different characteristics we seek to obtain in our representation. Note none of these losses require ground truth labels. We organize the losses according to the portion of the network they supervise: decomposition, canonicalization, and reconstruction. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Decomposition. While a transformation invariant encoder architecture should be sufficient to achieve equivariance/invariance, this does not prevent the encoder from producing trivial solutions/decompositions once trained. As capsule poses should be transformation equivariant, the poses of the two rotation augmentations $\\pmb { \\theta } _ { k } ^ { a }$ and $\\theta _ { k } ^ { b }$ should only differ by the (known) relative transformation: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0afa6c000eab8b6e4d3503f540555ca133c53b51ce615118b57c33e1a20d6ce7.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { e q u i v a r i a n c e } } = \\frac { 1 } { K } \\sum _ { k } \\| \\pmb { \\theta } _ { k } ^ { a } - ( \\mathbf { T } ^ { a } ) ( \\mathbf { T } ^ { b } ) ^ { - 1 } \\pmb { \\theta } _ { k } ^ { b } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Conversely, capsule descriptors should be transformation invariant, and as the two input points clouds are of the same object, the corresponding capsule descriptors $\\beta$ should be identical: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/288a883ecaa5d9617c22046404f07d6548e88f9f6bfc406b8d57075849694b1d.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { i n v a r i a n c e } } = \\frac { 1 } { K } \\sum _ { k } \\| \\beta _ { k } ^ { a } - \\beta _ { k } ^ { b } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We further regularize the capsule decomposition to ensure each of the $K$ heads roughly represent the same “amount” of the input point cloud, hence preventing degenerate (zero attention) capsules. This is achieved by penalizing the attention variance: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/040046403298e0c6b00e7677687c7b0dc66bc76a8cfbe7fbeda32f9304ac51c5.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { e q u i l i b r i u m } } = \\frac { 1 } { K } \\sum _ { k } \\| a _ { k } - \\frac { 1 } { K } \\Sigma _ { k } a _ { k } \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $a _ { k } = \\Sigma _ { p } ( \\mathbf { A } _ { p , k } )$ denotes the total attention exerted by the $\\mathbf { k }$ -th head on the point cloud. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, to facilitate the training process, we ask for capsules to learn a localized representation of geometry. We express the spatial extent of a capsule by computing first-order moments of the represented points with respect to the capsule pose $\\pmb { \\theta } _ { k }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/33899b27f6c1965758e9be0d4a3c431892190eae84079ad6d8bc3d7af67fd187.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { l o c a l i z a t i o n } } = \\frac { 1 } { K } \\sum _ { k } \\frac { 1 } { a _ { k } } \\sum _ { p } { \\bf A } _ { p , k } \\| \\pmb { \\theta } _ { k } - { \\bf P } _ { p } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Canonicalization. To train our canonicalizer $\\kappa$ , we relate the predicted capsule poses to regressed canonical capsule poses via the optimal rigid transformation from $\\textcircled{5}$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef942ee0e9f007beda15d3aaf6518f042ad7e7ef0410a4dd195762b54e31b27f.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c a n o n i c a l } } = \\frac { 1 } { K } \\sum _ { k } \\| ( \\bar { \\bf R } \\pmb { \\theta } _ { k } + \\bar { \\bf t } ) - \\bar { \\pmb { \\theta } } _ { k } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            125,
+            635,
+            162
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recall that $\\bar { \\mathbf { R } }$ and $\\bar { \\mathbf T }$ are obtained through a differentiable process. Thus, this loss is forcing the aggregated pose $\\pmb { \\theta } _ { k }$ to agree with the one that goes through the regression path, $\\bar { \\pmb { \\theta } } _ { k }$ . Now, since $\\bar { \\pmb { \\theta } } _ { k }$ is regressed solely from the set of capsule descriptors, similar shapes will result in similar canonical keypoints, and the coordinate system of $\\bar { \\pmb { \\theta } } _ { k }$ is one that employs Euclidean space to encode semantics. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Reconstruction. To learn canonical capsule descriptors in an unsupervised fashion, we rely on an auto-encoding task. We train the decoders $\\{ \\mathcal { D } _ { k } \\}$ by minimizing the Chamfer Distance (CD) between the (canonicalized) input point cloud and the reconstructed one, as in [61, 19]: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbfd84cbd9c491d1a04fb6ab29c0f266cbdf4c8557d11614056d4cedd6c45ac1.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { r e c o n } } = \\mathrm { C D } \\left( \\bar { \\mathbf { R } } \\mathbf { P } + \\bar { \\mathbf { t } } , \\tilde { \\mathbf { P } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Network Architectures ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            320,
+            372,
+            335
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We briefly summarize our implementation details, including the network architecture; for further details, please refer to the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Encoder – $\\mathcal { E }$ . Our architecture is based on the one suggested in $\\textcircled { \\lVert { 4 7 } \\rVert }$ : a pointnet-like architecture with residual connections and attentive context normalization. We utilize Batch Normalization [26] instead of the Group Normalization $\\lVert \\boldsymbol { \\bar { 5 } } 8 \\rVert$ , which trained faster in our experiments. We further extend their method to have multiple attention maps, where each attention map corresponds to a capsule. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Decoder – $\\mathcal { D }$ . The decoder from $\\textcircled{4}$ operates on a per-capsule basis. Our decoder architecture is similar to AtlasNetV2 $\\mathbb { [ [  2 ]  }$ (with trainable grids). The difference is that we translate the per-capsule decoded point cloud by the corresponding capsule pose. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Regressor – $\\kappa$ . We concatenate the descriptors and apply a series of fully connected layers with ReLU activation to regress the $P$ capsule locations. At the output layer we use a linear activation and subtract the mean of the outputs to make our regressed locations zero-centered in the canonical frame. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Canonicalizing the descriptors. As our descriptors are only approximately rotation invariant (via $\\mathcal { L } _ { \\mathrm { i n v a r i a n c e } } )$ , we found it useful to re-extract the capsule descriptors $\\beta _ { k }$ after canonicalization. Specifically, we compute $\\bar { \\bf F }$ with the same encoder setup, but with $\\bar { \\bf P } { = } \\bar { \\bf R } \\bar { \\bf P } { + } \\bar { \\bf T }$ instead of $\\mathbf { P }$ and use it to compute $\\bar { \\boldsymbol { \\beta } } _ { k }$ ; we validate this empirically in the supplementary material. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We first discuss the experimental setup, and then validate our method on a variety of tasks: autoencoding, canonicalization, and unsupervised classification. While the task differs, our learning process remains the same: we learn capsules by reconstructing objects in a learnt canonical frame. We also provide an ablation study, which is expanded in detail in the supplementary material, where we provide additional qualitative results. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental setup ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To evaluate our method, we rely on the ShapeNet (Core) dataset $\\pmb { \\bigtriangledown } ^ { 2 } .$ We follow the category choices from AtlasNetV2 [12], using the airplane and chair classes for single-category experiments, while for multi-category experiments we use all 13 classes: airplane, bench, cabinet, car, chair, monitor, lamp, speaker, firearm, couch, table, cellphone, and watercraft. To make our results most compatible with those reported in the literature, we also use the same splits as in AtlasNetV2 [12]: 31747 shapes in the train, and 7943 shapes in the test set. Unless otherwise noted, we randomly sample 1024 points from the object surface for each shape to create our 3D point clouds. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "De-canonicalizing the dataset. As discussed in the introduction, ShapeNet (Core) contains substantial inductive bias in the form of consistent semantic alignment. To remove this bias, we create random ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SE(3) transformations, and apply them to each point cloud. We first generate uniformly sampled random rotations, and add uniformly sampled random translations within the range $[ - 0 . 2 , 0 . 2 ]$ , where the bounding volume of the shape ranges in $[ - 1 , + 1 ]$ . Note the relatively limited translation range is chosen to give state-of-the-art methods a chance to compete with our solution. We then use the relative transformation between the point clouds extracted from this ground-truth transformation to evaluate our methods. We refer to this unaligned version of the ShapeNet Core dataset as the unaligned setup, and using the vanilla ShapeNet Core dataset as the aligned setup. For the aligned setup, as there is no need for equivariance adaptation, we simply train our method without the random transformations, and so $\\mathcal { L } _ { \\mathrm { e q u i v a r i a n c e } }$ and $\\mathcal { L } _ { \\mathrm { i n v a r i a n c e } }$ are not used. This setup is to simply demonstrate how Canonical Capsules would perform in the presence of a dataset bias. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We emphasize here that a proper generation of random rotation is important. While some existing works have generated them by uniformly sampling the degrees of freedom of an Euler-angle representation, this is known to be an incorrect way to sample random rotations $\\pmb { \\mathbb { H } }$ , leading to biases in the generated dataset; see the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation details. For all our experiments we use the Adam optimizer $\\mathbb { \\left[ \\left[ 2 9 \\right] \\right] }$ with an initial learning rate of 0.001 and decay rate of 0.1. We train for 325 epochs for the aligned setup to match the AtlasNetV2 $\\mathbb { \\lVert 1 2 \\rVert }$ original setup. For the unaligned setting, as the problem is harder, we train for a longer number of 450 epochs. We use a batch size of 16. The training rate is ${ \\sim } 2 . 5$ iters/sec. We train each model on a single NVidia V100 GPU. Unless stated otherwise, we use $k { = } 1 0$ capsules and capsule descriptors of dimension $C { = } 1 2 8$ . We train three models with our method: two that are single-category (i.e., for airplane and chairs), and one that is multi-category (i.e., all 13 classes). To set the weights for each loss term, we rely on the reconstruction performance (CD) in the training set. We set weights to be one for all terms except for $\\mathcal { L } _ { \\mathrm { e q u i v a r i a n c e } }$ (5) and $\\mathcal { L } _ { \\mathrm { e q u i l i b r i u m } } ( 1 0 ^ { - 3 } )$ . In the aligned case, because $\\mathcal { L } _ { \\mathrm { e q u i v a r i a n c e } }$ and $\\mathcal { L } _ { \\mathrm { i n v a r i a n c e } }$ are not needed (always zero), we reduce the weights for the other decomposition losses by $1 0 ^ { 3 }$ ; $ { \\mathcal { L } } _ { \\mathrm { l o c a l i z a t i o n } }$ to $1 0 ^ { - 3 }$ and $\\mathcal { L } _ { \\mathrm { { e q u i l i b r i u m } } }$ to $1 0 ^ { - 6 }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Auto-encoding – Figure 2 and Table 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            470,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/97b1c8cfd7a1bb4b282c45bf2256f7fc52841b1afb987bf691a39d1ce9e43f73.jpg",
+        "table_caption": [
+            "Table 1: Auto-encoding / quantitative – Performance in terms of Chamfer distance with 1024 points per point cloud – metric is multiplied by $\\mathrm { 1 0 ^ { 3 } }$ as in $\\pmb { \\mathbb { I } }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Method</td><td>Airplane</td><td>Chair</td><td>Multi</td></tr><tr><td>Pgia</td><td>3D-PointCapsNet [64]</td><td>1.94</td><td>3.30</td><td>2.49</td></tr><tr><td></td><td>AtlasNetV212</td><td>1.28</td><td>2.36</td><td>2.14</td></tr><tr><td></td><td>Our method</td><td>0.96</td><td>1.99</td><td>1.76</td></tr><tr><td>nrtgren</td><td>3D-PointCapsNet [64</td><td>5.58</td><td>7.57</td><td>4.66</td></tr><tr><td></td><td>AtlasNetV2 国</td><td>2.80</td><td>3.98</td><td>3.08</td></tr><tr><td></td><td>AtlasNetV2 园 w/ STN</td><td>1.90</td><td>2.99</td><td>2.60</td></tr><tr><td></td><td>Our method</td><td>1.11</td><td>2.58</td><td>2.22</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the performance of our method for the task that was used to train the network – reconstruction / auto-encoding – against three baselines (trained in both single-category and multi-category variants): $\\textcircled{1}$ 3D-PointCapsNet $\\dot { \\left[ 6 4 \\right] }$ , an auto-encoder for 3D point clouds that utilize a capsule architecture; $\\textcircled{2}$ AtlasNetV2 $\\mathbb { [ [  2 ]  }$ , a state-of-the-art auto-encoder which utilizes a multi-head patch-based decoder; $\\textcircled{3}$ AtlasNetV2 $\\mathbb { \\lVert 1 2 \\rVert }$ with a spatial-transformer network (STN) aligner from PointNet $[ \\beta 6 ]$ , a baseline with canonicalization. We do not compare against $\\lVert \\overline { { 4 6 } } \\rVert$ , as unfortunately source code is not publicly available. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Quantitative analysis – Table 1. We achieve state-of-the-art performance in both the aligned and unaligned settings. The wider margin in the unaligned setup indicates tackling this more challenging scenario damages the performance of AtlasNetV2 [12] and 3D-PointCapsNet $\\textcircled { 6 4 } \\textcircled { 1 6 }$ much more than our method3. We also include a variant of AtlasNetV2 for which a STN (Spatial Transformer Network) is used to pre-align the point clouds [36], demonstrating how the simplest form of prealigner/canonicalizer is not sufficient. ",
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+        "type": "text",
+        "text": "Qualitative analysis – Figure 2. We illustrate our decomposition-based reconstruction of 3D point clouds, as well as the reconstructions of 3D-PointCapsNet [64] and AtlasNetV2 [12]. As shown, even in the unaligned setup, our method is able to provide semantically consistent capsule decompositions – e.g. the wings of the airplane have consistent colours, and when aligned in the canonical frame, the different airplane instances are well-aligned. Compared to AtlasNetV2 $\\mathbb { \\lVert \\rVert }$ and 3D-PointCapsNet $\\pmb { \\mathbb { E 4 } }$ , the reconstruction quality is also visibly improved: we better preserve details along the engines of the airplane, or the thin structures of the bench; note also that the decompositions are semantically consistent in our examples. Results are better appreciated in our supplementary material, where we visualize the performance as we continuously traverse $\\mathbf { S E } ( 3 )$ . ",
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+        "image_caption": [
+            "Figure 2: Auto-encoding / qualitative – Example decomposition and reconstruction results setup using Canonical Capsules on several unaligned point cloud instances from the test set. We color each Canonical Capsule with a unique colour, and similarly color “patches” from the reconstruction heads of 3D-PointCapsNet [64] and AtlasNetV2 [12]. Canonical Capsules provide semantically consistent decomposition that is aligned in canonical frame, leading to improved reconstruction quality. "
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+        "text": "4.3 Canonicalization – Table 2 ",
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+        "text": "We compare against three baselines: $\\textcircled{1}$ Deep Closest Points $\\left[ \\left[ 5 6 \\right] \\right]$ , a deep learning-based pairwise point cloud registration method; $\\textcircled{2}$ DeepGMR $\\begin{array} { r l } {  { \\bigl [ \\bigl | 6 3 \\bigr | \\bigr ] } } \\end{array}$ a state-of-the-art pairwise registration method that decomposes clouds into Gaussian mixtures and utilizes Rigorously Rotation-Invariant (RRI) features [4]; $\\textcircled{3}$ Compass $[ | 4 5 | ]$ a concurrent work on learnt alignment/canonicalization. For all compared methods we use the official implementation. For DeepGMR we use both RRI and the typical XYZ coordinates as input. We also try our method with the RRI features, following DeepGMR’s training protocol and train for 100 epochs. ",
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+        "type": "text",
+        "text": "Metrics. To evaluate the canonicalization performance, we look into the stability of the canonicalization – the shakiness shown in the videos in our supplementary material– represented as the mean standard deviation of the rotations (mStd): ",
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+        "img_path": "images/b615b52e549bd7a6969ee1a88d5340ee5796dbfbbd2c26d5bf5e03f0c4ac4bdb.jpg",
+        "text": "$$\n\\mathrm { m S t d } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sqrt { \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } ( \\angle ( \\mathbf { R } ^ { i j } , \\mathbf { R } _ { m e a n } ^ { i } ) ) ^ { 2 } } ~ ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\angle$ is the angular distance between two rotation matrices $[ \\overline { { 1 8 } } , \\overline { { 6 2 } } , \\overline { { 6 3 } } ]$ , $\\mathbf { R } ^ { i j }$ is the rotation matrix of the $j$ -th instance of the $i$ -th object in canonical frame, and ${ \\bf R } _ { m e a n } ^ { i }$ is the mean rotation $\\left[ \\left[ 2 1 \\right] \\right]$ of the $i$ -th object. Note that with mStd we measure the stability of canonicalization with respect to rotations to accommodate for methods that do not deal with translation $[ | 4 5 | ]$ . To allow for comparisons with pairwise registration methods, we also measure performance in terms of the RMSE metric $\\textcircled { 1 6 3 } , \\textcircled { 6 6 } \\textcircled { 1 }$ . ",
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+        "type": "text",
+        "text": "Quantitative analysis – Table $\\bigtriangledown$ Compared to Compass $\\boldsymbol { \\left[ \\left[ 4 5 \\right] \\right] }$ , our method provides improved stability in canonicalization. This also provides an advantage in pairwise registration, delivering state-of-the-art results when XYZ-coordinates are used. Note that while pairwise methods can align two sets of given point clouds, creating a canonical frame that simultaneously registers all point clouds is a non-trivial extension to the problem. ",
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+        "table_caption": [
+            "Table 2: Canonicalization – Quantitative evaluation for canonicalization, where we highlight significantly better performance than the concurrent work Compass [45]. While pairwise registration is not our main objective, the global alignment frame created by our method still allows for effective registration (on par, or better) than the state-of-the-art. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"3\">Canonicalization (mStd)↓</td><td colspan=\"3\">Pairwise registration (RMSE)↓</td></tr><tr><td></td><td>Method</td><td>Airplane</td><td>Chair</td><td>Multi</td><td>Airplane</td><td>Chair</td><td>Multi</td></tr><tr><td></td><td>Deep Closest Points [56]</td><td>1</td><td>1</td><td>1</td><td>0.318</td><td>0.160</td><td>0.131</td></tr><tr><td></td><td>DeepGMR [63]</td><td></td><td></td><td>1</td><td>0.079</td><td>0.082</td><td>0.077</td></tr><tr><td>P.JCOo-zX</td><td>Compass 4</td><td>19.105</td><td>19.508</td><td>51.811</td><td>0.412</td><td>0.482</td><td>0.515</td></tr><tr><td></td><td>Our method</td><td>8.278</td><td>5.730</td><td>21.210</td><td>0.022</td><td>0.027</td><td>0.074</td></tr><tr><td>R</td><td>DeepGMR [63]</td><td></td><td></td><td></td><td>0.0001</td><td>0.0001</td><td>0.0001</td></tr><tr><td></td><td>Our method</td><td>(unstable)</td><td>(unstable)</td><td>(unstable)</td><td>0.0006</td><td>0.0009</td><td>0.0016</td></tr></table>",
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+        "type": "text",
+        "text": "When RRI is used as input, our method is on par with DeepGMR $\\pmb { \\mathbb { \\left\\| 6 3 \\right\\| } }$ , up to a level where registration is near perfect – alignment differences when errors are in the $1 0 ^ { - 4 }$ ballpark are indiscernible. We note that the performance of Deep Closest Points $\\begin{array} { r l } {  { \\bigl \\| \\boldsymbol { 5 } 6 \\bigr \\| } } & { { } } \\end{array}$ is not as good as reported in the original paper, as we uniformly draw rotations from SO(3). When a sub-portion of $\\mathbf { S O } ( 3 )$ is used, e.g. a quarter of what we are using, DCP performs relatively well (0.008 in the multi-class experiment). While curriculum learning could be used to enhance the performance of DCP, our technique does not need to rely on these more complex training techniques. ",
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+        "type": "text",
+        "text": "We further note that, while RRI delivers good registration performance, using RRI features cause the learnt canonicalization to fail – training becomes unstable. This hints that RRI features may be throwing away too much information to achieve transformation invariance. Our method using raw XYZ coordinates as input, on the other hand, provides comparable registration performance, and is able to do significantly more than just registration (i.e. classification, reconstruction). ",
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+    {
+        "type": "text",
+        "text": "4.4 Unsupervised classification – Table 3 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Beyond reconstruction and canonicalization, we evaluate the usefulness of our method via a classification task that is not related in any way to the losses used for training. We compute the features from the auto-encoding methods from Section $4 . 2$ against those from our method (where we build features by combining pose with descriptors) to perform 13-way classification with two different techniques: ",
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+    {
+        "type": "text",
+        "text": "We train a supervised linear Support Vector Machine (SVM) on the extracted features [1, Ch. 7]; We perform unsupervised K-Means clustering [1, Ch. 9] and then label each cluster via bipartite matching with the actual labels through the Hungarian algorithm. ",
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+    {
+        "type": "text",
+        "text": "Note the former provides an upper bound for unsupervised classification, while better performance on the latter implies that the learnt features are able to separate the classes into clusters that are compact (in an Euclidean sense). ",
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+        "table_caption": [
+            "Table 3: Classification – Top-1 accuracy $\\% )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td></td><td>SVM</td><td>K-Means</td></tr><tr><td>Pse</td><td>3D-PointCapsNet [64]</td><td>93.81</td><td>65.87</td></tr><tr><td></td><td>AtlasNetV2[12]</td><td>94.07</td><td>61.66</td></tr><tr><td></td><td>Our method</td><td>94.21</td><td>69.82</td></tr><tr><td>aggen</td><td>3D-PointCapsNet [64]</td><td>71.13</td><td>14.59</td></tr><tr><td></td><td>AtlasNetV2 国</td><td>64.85</td><td>17.12</td></tr><tr><td></td><td>AtlasNetV2 圆 w/ STN</td><td>78.55</td><td>20.03</td></tr><tr><td></td><td>Our method</td><td>87.33</td><td>43.04</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Analysis of results – SVM. Note how our method provides best results in all cases, and when the dataset is not unaligned the difference is significant. This shows that, while 3D-PointCapsNet and AtlasNetV2 (with and without STN) are able to somewhat auto-encode point clouds in the unaligned setup, what they learn does not translate well to classification. However, the features learned with Canonical Capsules are more related to the semantics of the object, which helps classification. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis of results – K-Means. The performance gap becomes wider when K-Means is used – even in the aligned case. This could mean that the features extracted by Canonical Capsules are better suited for other unsupervised tasks, having a feature space that is close to being Euclidean in terms of semantics. The difference is striking in the unaligned setup. We argue that these results emphasize the importance of the capsule framework – jointly learning the invariances and equivariances in the data – is cardinal to unsupervised learning [25, $\\checkmark$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/bf847e6a330665babe28c2ca91a615e7675f6bbcd987e53d280aaaf980ca5722.jpg",
+        "table_caption": [
+            "Table 4: Effect of losses – Reconstruction performance, and canonicalization performance when loss terms are removed. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Full</td><td>-Linvar</td><td>-Lequiv</td><td>-Lcanonical</td><td>-Llocalization</td><td>-Leqilibrium</td></tr><tr><td>CD</td><td>1.11</td><td>1.12</td><td>1.16</td><td>1.12</td><td>1.44</td><td>1.60</td></tr><tr><td>Std</td><td>8.278</td><td>9.983</td><td>110.174</td><td>8.421</td><td>113.204</td><td>92.970</td></tr></table>",
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+    {
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+        "table_caption": [
+            "Table 5: Backbone – Auto-encoding performance (Chamfer distance) when we use various permutation-invariant backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PointNet</td><td>PointNet++</td><td>DGCNN</td><td>ACNe</td></tr><tr><td>CD</td><td>1.28</td><td>1.34</td><td>1.21</td><td>1.11</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.5 Ablation study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To make the computational cost manageable, we perform all ablations with the airplane category (the category with most instances), and in the unaligned setup (unless otherwise noted). Please also see supplementary material for more ablation studies. ",
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+    {
+        "type": "text",
+        "text": "Losses – Table 4. We analyze the importance of each loss term, with the exception of $\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n } }$ which is necessary for training. All losses beneficially contribute to reconstruction performance, but note how $\\mathcal { L } _ { \\mathrm { e q u i v } }$ , $ { \\mathcal { L } } _ { \\mathrm { l o c a l i z a t i o n } }$ and $\\mathcal { L } _ { \\mathrm { { e q u i l i b r i u m } } }$ affect it to a larger degree. By considering our canonicalization metric, we can motivate this outcome by observing that the method fails to perform canonicalization when these losses are not employed (i.e. training collapses). ",
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+    {
+        "type": "text",
+        "text": "Encoder architecture – Table $\\boxed { \\pmb { 5 } }$ Our method can be used with any backbone, as our main contribution lies in the self-supervised canonicalization architecture. For completeness, we explore variants of our method using different back-bones: PointNet $\\textcircled { \\lvert 3 6 \\rvert }$ , PointNet+ $^ { - + }$ [37], DGCNN $\\boxed { \\sqrt { 5 7 } }$ , and ACNe $\\mathbb { H }$ . Among these, the ACNe backbone performs best; note that all the variants in Table 5, regardless of backbone choice, significantly outperforms all other methods reported in Table 1. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we provide a self-supervised framework to train primary capsule decompositions for 3D point clouds. We rely on a Siamese setup that allows self-supervision and auto-encoding in canonical space, circumventing the customary need to train on pre-aligned datasets. Despite being trained in a self-supervised fashion, our representation achieves state-of-the-art performance across auto-encoding, canonicalization and classification tasks. These results are made possible by allowing the network to learn a canonical frame of reference. We interpret this result as giving our neural networks a mechanism to construct a “mental picture” of a given 3D object – so that downstream tasks are executed within an object-centric coordinate frame. ",
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+    {
+        "type": "text",
+        "text": "Future work. As many objects have natural symmetries that we do not consider at the moment $\\mathbb { \\left[ 1 2 \\right] }$ providing our canonicalizer a way to encode such a prior is likely to further improve the representation. We perform decomposition at a single level, and it would be interesting to investigate how to effectively engineer multi-level decompositions $\\mathbb { \\left[ \\left[ 5 \\right] \\right] }$ ; one way could be to over-decompose the input in a redundant fashion (with large $K$ ), and use a downstream layers that “selects” the decomposition heads to be used $\\textcircled { 6 }$ . We would also like to extend our results to more “in-the-wild” 3D computer vision and understand whether learning object-centric representations is possible when incomplete (i.e., single view $\\pmb { \\Vert 3 5 \\Vert }$ or occluded) data is given in input, when an entire scene with potentially multiple objects is given $\\mathbb { \\lVert \\boldsymbol { 4 3 } \\rVert }$ , or where our measurement of the 3D world is a single 2D image [48], or by exploiting the persistence of objects in video [40]. ",
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+    {
+        "type": "text",
+        "text": "Broader impact. While our work is exclusively on 3D shapes, and thus not immediately subject to any societal concerns, it enhances how Artificial Intelligence (AI) can understand and model 3D geometry. Thus, similar to how image recognition could be misused, one should be careful when extending the use of our method. In addition, while not subject to how the data itself is aligned, the learnt canonical frame of our method is still data-driven, thus subject to any data collection biases that may exist – canonicalization will favour shapes that appear more often. This should also be taken into account with care to prevent any biased decisions when utilizing our method within a decision making AI platform. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, NSERC Collaborative Research and Development Grant, Google, Compute Canada, and Advanced Research Computing at the University of British Columbia. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "[1] Christopher M Bishop. Pattern Recognition and Machine Learning. springer, 2006.   \n[2] Rohan Chabra, Jan Eric Lenssen, Eddy Ilg, Tanner Schmidt, Julian Straub, Steven Lovegrove, and Richard Newcombe. Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction. In European Conference on Computer Vision, 2020.   \n[3] Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al. Shapenet: An Information-Rich 3D Model Repository. arXiv Preprint, 2015.   \n[4] Chao Chen, Guanbin Li, Ruijia Xu, Tianshui Chen, Meng Wang, and Liang Lin. Clusternet: Deep Hierarchical Cluster Network with Rigorously Rotation-Invariant Representation for Point Cloud Analysis. In Conference on Computer Vision and Pattern Recognition, 2019.   \n[5] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A Simple Framework for Contrastive Learning of Visual Representations. 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ACM Transactions on Graphics, 2019.   \n[58] Yuxin Wu and Kaiming He. Group Normalization. In European Conference on Computer Vision, 2018.   \n[59] Saining Xie, Jiatao Gu, Demi Guo, Charles R. Qi, Leonidas J. Guibas, and Or Litany. PointContrast: Unsupervised Pre-training for 3D Point Cloud Understanding. In European Conference on Computer Vision, 2020.   \n[60] Xinchen Yan, Jimei Yang, Ersin Yumer, Yijie Guo, and Honglak Lee. Perspective Transformer Nets: Learning Single-View 3D Object Reconstruction without 3D Supervision. In Advances in Neural Information Processing Systems, 2016.   \n[61] Yaoqing Yang, Chen Feng, Yiru Shen, and Dong Tian. FoldingNet: Point Cloud Auto-Encoder via Deep Grid Deformation. In Conference on Computer Vision and Pattern Recognition, 2018.   \n[62] Zi Jian Yew and Gim Hee Lee. Rpm-net: Robust point matching using learned features. 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parse/train/qZzy5urZw9/qZzy5urZw9.md

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+# ROBUST OVERFITTING MAY BE MITIGATED BY PROPERLY LEARNED SMOOTHENING
+
+Tianlong Chen1\*, Zhenyu Zhang2\*, Sijia $\mathbf { L i u } ^ { 3 , 4 }$ , Shiyu Chang4, Zhangyang Wang1 1University of Texas at Austin,2University of Science and Technology of China 3Michigan State University, 4MIT-IBM Watson AI Lab, IBM Research {tianlong.chen,atlaswang}@utexas.edu, zzy19969@mail.ustc.edu.cn liusiji5@msu.edu, shiyu.chang@ibm.com
+
+# ABSTRACT
+
+A recent study (Rice et al., 2020) revealed overfitting to be a dominant phenomenon in adversarially robust training of deep networks, and that appropriate early-stopping of adversarial training (AT) could match the performance gains of most recent algorithmic improvements. This intriguing problem of robust overfitting motivates us to seek more remedies. As a pilot study, this paper investigates two empirical means to inject more learned smoothening during AT: one leveraging knowledge distillation and self-training to smooth the logits, the other performing stochastic weight averaging (Izmailov et al., 2018) to smooth the weights. Despite the embarrassing simplicity, the two approaches are surprisingly effective and hassle-free in mitigating robust overfitting. Experiments demonstrate that by plugging in them to AT, we can simultaneously boost the standard accuracy by $3 . 7 \bar { 2 } \% \sim 6 . 6 8 \%$ and robust accuracy by $0 . 2 2 \% \sim 2 . 0 3 \%$ , across multiple datasets (STL-10, SVHN, CIFAR-10, CIFAR-100, and Tiny ImageNet), perturbation types ( $\ell _ { \infty }$ and $\ell _ { 2 }$ ), and robustified methods (PGD, TRADES, and FSGM), establishing the new state-of-the-art bar in AT. We present systematic visualizations and analyses to dive into their possible working mechanisms. We also carefully exclude the possibility of gradient masking by evaluating our models’ robustness against transfer attacks. Codes are available at https: //github.com/VITA-Group/Alleviate-Robust-Overfitting.
+
+# 1 INTRODUCTION
+
+Adversarial training (AT) (Madry et al., 2018), i.e., training a deep network to minimize the worst-case training loss under input perturbations, is recognized as the current best defense method to adversarial attacks. However, one of its pitfalls was exposed by a recent work (Rice et al., 2020): in contrast to the commonly-held belief that overparameterized deep networks hardly overfit in standard training (Zhang et al., 2016; Neyshabur et al., 2017; Belkin et al., 2019), overfitting turns out to be a dominant phenomenon in adversarially robust training of deep networks. After a certain point in AT, e.g., immediately after the first learning rate decay, the robust test errors will only continue to substantially increase with further training (see Figure 1 bottom for example).
+
+That surprising phenomenon, termed as “robust overfitting”, has been prevalent on many datasets and models. As Rice et al. (2020) pointed out, it poses serious challenges to assess recent algorithmic advances upon AT: by just using an earlier checkpoint, the performance of AT be drastically boosted to match the more recently reported state-of-the-arts (Yang et al., 2019b; Zhang et al., 2019b). Even worse, Rice et al. (2020) tested several other implicit and explicit regularization methods, including weight decay, data augmentation and semi-supervised learning; they reported that none of those alternatives seem to combat robust overfitting (stably) better than simple early stopping. The authors thus advocated using the validation set to select a stopping point, although the manual picking would inevitably trade off between selecting either the peak point of robust test accuracy or that of standard accuracy, which often do not coincide (Chen et al., 2020a).
+
+![](images/919765c12a9a65b56db3bb040c8cd4527924e9d7d93e08f8f85af7d5f8f992c3.jpg)  
+Figure 1: The standard (top) and robust (bottom) test error rate comparison, between the vanilla PGD-AT baseline (Madry et al., 2018), and PGDAT with our proposed weight/label smoothening techniques applied, on CIFAR-10 with ResNet18. Our methods effectively mitigates the robust overfitting (Rice et al., 2020) even when trained to 200 epochs, while maintaining the same high standard/robust accuracies compared to the best early-stop checkpoint of the baseline.
+
+Does there exist more principled, hands-off, and hassle-free mitigation for this robust overfitting, for us to further unleash the competency of AT? This paper explores two options along the way, that draw two more sophisticated ideas from enhancing standard deep models’ generalization. Both could be viewed as certain types of learned smoothening, and are directly plugged into AT:
+
+• Our first approach is to smooth the logits in AT via self-training, using knowledge distillation with the same model pre-trained as a self-teacher. The idea is inspired by two facts: (1) label smoothening (Szegedy et al., 2016) can calibrate the notorious overconfidence of deep networks (Hein et al., 2019), and that was found to improve their standard generalization; (2) label smoothening can be viewed as a special case of knowledge distillation (Yuan et al., 2020), and self-training can produce more semantic-aware and discriminative soft label “self-teachers” than naive label smoothening (Chen et al., 2020b; Tang et al., 2020). Our second approach is to smooth the weights in AT via stochastic weight averaging (SWA) (Izmailov et al., 2018), a popular training technique that leads to better standard generalization than SGD, with almost no computational overhead. While SWA has not yet be applied to AT, it is known to find flatter minima which are widely believed to indicate stronger robustness (Hein & Andriushchenko, 2017; Wu et al., 2020a). Meanwhile, SWA could also be interpreted as a temporal model ensemble, and therefore might bring the extra robustness of ensemble defense (Tramer et al., 2018; Grefenstette et al., 2018) with the convenience \` of a single model. Those suggest that applying SWA is natural and promising for AT.
+
+To be clear, neither knowledge-distillation/self-training nor SWA was invented by this paper: they have been utilized in standard training to alleviate (standard) overfitting and improve generalization, by fixing over-confidence and by finding flatter solutions, respectively. By introducing and adapting them to AT, our aim is to complement the existing study, demonstrating that while simpler regularizations were unable to fix robustness overfitting as Rice et al. (2020) found, our learned logit/weight smoothening could effectively regularize and mitigate it, without needing early stopping.
+
+Experiments demonstrate that by plugging in the two techniques to AT, we can simultaneously boost the standard accuracy by $3 . 7 2 \% \sim 6 . 6 8 \%$ and robust accuracy by $0 . 2 2 \% \sim 2 . 0 3 \%$ , across multiple datasets (STL-10, SVHN, CIFAR-10, CIFAR-100, and Tiny ImageNet), perturbation types ( $\ell _ { \infty }$ and $\ell _ { 2 , }$ ), and robustified methods (PGD, TRADES, and FSGM), establishing the new state-of-the-art in AT. As shown in Figure 1 example, our method eliminates the robust overfitting phenomenon in AT, even when training up to 200 epochs. Our results imply that although robustness overfitting is more challenging than standard overfitting, its mitigation is still feasible with properly-chosen, advanced regularizations that were developed for the latter. Overall, our findings join (Rice et al., 2020) in re-establishing the competitiveness of the simplest AT baseline.
+
+# 1.1 BACKGROUND WORK
+
+Deep networks are easily fooled by imperceivable adversarial samples. To tackle this vulnerability, numerous defense methods were proposed (Goodfellow et al., 2015; Kurakin et al., 2016; Madry et al., 2018), yet many of them (Liao et al., 2018; Guo et al., 2018; Xu et al., 2017; Dziugaite et al., 2016; Dhillon et al., 2018; Xie et al., 2018; Jiang et al., 2020) were later found to result from training artifacts, such as obfuscated gradients (Athalye et al., 2018) caused by input transformation or randomization. Among them, adversarial training (AT) (Madry et al., 2018) remains one of the most competitive options. Recently more improved defenses have been reported (Dong et al., 2018; Yang et al., 2019b; Mosbach et al., 2018; Hu et al., 2020; Wang et al., 2020a; Dong et al., 2020; Zhang et al., 2020a;b), with some of them also being variants of AT, e.g. TRADES (Zhang et al., 2019b) and AT with metric learning regularizers (Mao et al., 2019; Pang et al., 2019; 2020).
+
+While overfitting has become less a practical concern in training deep networks nowadays, it was not yet noticed nor addressed in the adversarial defense field until lately. An overfitting phenomenon was first observed in a few fast adversarial training methods (Zhang et al., 2019a; Shafahi et al., 2019b; Wong et al., 2020) based on FGSM (Goodfellow et al., 2015), e.g., sometimes the robust accuracy against a PGD adversary suddenly drop to nearly zero after some training. (Andriushchenko & Flammarion, 2020) suggested it to be rooted in those methods’ local linearization assumptions of the loss landscape in those “fast” AT. The recently reported robust overfitting (Rice et al., 2020) seems to raise a completely new challenge for the classical AT (not fast): the model starts to irreversibly lose robustness after training with AT for a period, even the double-descent generalization curves still seemed to hold (Belkin et al., 2019; Nakkiran et al., 2019). Among various options tried in Rice et al. (2020), early-stopping was so far the only effective remedy found.
+
+# 2 METHODOLOGY
+
+# 2.1 LEARNING TO SMOOTH LOGITS IN AT
+
+Rationale: Why AT enforces models robust against adversarial attacks of a specific type and certain magnitudes. However, it has been shown to “overfit” the threat model “seen” during training (Kang et al., 2019; Maini et al., 2019; Stutz et al., 2020), and its gained robustness does not extrapolate to larger perturbations nor unseen attack types. Stutz et al. (2020) hypothesized this to be an unwanted consequence of enforcing high-confidence predictions on adversarial examples since high-confidence predictions are difficult to extrapolate to arbitrary regions beyond the seen examples during training. We generalize this observation: during AT, the attacks generated at every iteration can be naturally considered as continuously varying/evolving, along with the model training. Therefore, we hypothesize one source of robust overfitting might lie in that the model “overfits” the attacks generated in the early stage of AT and fails to generalize or adapt to the attacks in the late stage.
+
+To alleviate the overconfidence problem, we adapt the label smoothening (LS) technique in standard training (Szegedy et al., 2016). LS creates uncertainty in the one-hot labels, by computing crossentropy not with the “hard” targets from the dataset, but with a weighted mixture of these one-hot targets with the uniform distribution. This uncertainty helps to tackle alleviate the overconfidence problem Hein et al. (2019) and improves the standard generalization. The idea of LS was previously investigated in other defense methods (Shafahi et al., 2019a; Goibert & Dohmatob, 2019), but much of the observed robustness gains were later attributed to obfuscated gradients (Athalye et al., 2018). Two recent works (Stutz et al., 2020; Cheng et al., 2020) have integrated LS with AT to inject label uncertainty: Stutz et al. (2020) used a convex combination of uniform and one-hot distributions as target for the cross-entropy loss in AT, which resembles the LS regularizer, while Cheng et al. (2020) concurrently used an LS regularizer for AT.
+
+However, there is one pitfall of the naive LS in (Szegedy et al., 2016): over-smoothening labels in a data-blind way could cause loss of information in the logits, and hence weakened discriminative power of the trained models (Muller et al., 2019). That calls for a careful and adaptive balance be- ¨ tween discriminative capability and confidence calibration of the model. In the context of AT, Stutz et al. (2020) crafted a perturbation-dependent parameter, to explicitly control the transition from one-hot to the uniform distribution when the attack magnitude grows from small to large. To identify more automated and principled means, we notice another recent work (Yuan et al., 2020), who explicitly connected knowledge distillation (KD) (Hinton et al., 2015) to LS. The authors pointed out that LS equals a special case of KD using a virtual and hand-crafted teacher; on the contrary, the conventional KD provides data-driven soften labels rather than simply mixing one-shot and uniform vectors. Together with many others (Furlanello et al., 2018; Chen et al., 2020b), these works demonstrated that using model-based and learned soft labels supplies much superior confidence calibration and logit geometry compared to the naive LS (Tang et al., 2020).
+
+Furthermore, (Furlanello et al., 2018; Chen et al., 2020b; Yuan et al., 2020) unanimously revealed that another strong teacher model with extra privileged information is NOT critical to the success of KD. Yuan et al. (2020) shows that even a poorly-trained teacher with much lower accuracy can still improve the student. Moreover, Chen et al. (2020b); Yuan et al. (2020) find self-teacher to be sufficiently effective for KD, that is, using soft-logit outputs from the student or designed manually as the KD regularization to train itself (also called teacher-free KD (Tf-KD) in (Yuan et al., 2020)). These observations make the main cornerstone for our learned logit smoothening approach next.
+
+![](images/c8f1ee63e85dc8ebd65971757acdf9fbb76829204da7e6f27e67727ca772fcd8.jpg)  
+Figure 2: Comparing the logit distribution of different LS/KD means on CIFAR-10 using ResNet-18 (C7 is the correct label).
+
+Approach: How We follow (Chen et al., 2020b; Yuan et al., 2020) to use self-training with the same model, but introduce one specific modification. The one model could be trained with at least two different ways: standard training, or robust training (AT or other cheaper ways; see ablation experiments). That can yield two self-teachers. We assume both to be available; and let $\mathbf { x }$ be the input, $y$ the one-hot ground truth label, $\delta$ the adversarial perturbation bounded by $\ell _ { p }$ norm ball with radius $\epsilon$ , and $\theta _ { r } / \theta _ { s }$ the weights of the robust-/standard-trained self-teachers, respectively. Note the two self-teachers share the identical network architecture and training data with our target model. Our self-training smoothed loss function is expressed below $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are two hyperparameters):
+
+$$
+\begin{array} { r l } { \underset { \theta } { \mathop { \operatorname* { m i n } } } } & { \mathbb { E } _ { ( \mathbf { x } , y ) \in \mathcal { D } } \left\{ \left( 1 - \lambda _ { 1 } - \lambda _ { 2 } \right) \cdot \underset { \delta \in \mathrm { B } _ { \mathrm { c } } ( \mathbf { x } ) } { \operatorname* { m a x } } \mathcal { L } _ { \mathrm { X E } } ( f ( \theta , \mathbf { x } + \delta ) , y ) + \right. } \\ & { \left. \lambda _ { 1 } \cdot K \mathcal { D } _ { \mathrm { a d v } } \big ( f ( \theta , \mathbf { x } + \delta ) , f ( \theta _ { r } , \mathbf { x } + \delta ) \big ) + \lambda _ { 2 } \cdot K \mathcal { D } _ { \mathrm { s t d } } \big ( f ( \theta , \mathbf { x } + \delta ) , f ( \theta _ { s } , \mathbf { x } + \delta ) \big ) \right\} , } \end{array}
+$$
+
+where $\mathcal { L } _ { \mathrm { X E } }$ is robustified cross-entropy loss adopted in the original AT; $\kappa D _ { \mathrm { a d v } }$ and $\kappa \mathcal { D } _ { \mathrm { s t d } }$ are the Kullback–Leibler divergence loss with the robust-trained and standard-trained self-teachers, respectively. $\lambda _ { 1 } = 0 . 5$ and $\lambda _ { 2 } = 0 . 2 5$ are default in all experiments. More details are in Appendix A2.1.
+
+Figure 2 visualizes an example of logit distributions, generated by naive LS (Szegedy et al., 2016), the Tf-KD regularizer using manually-designed self-teacher in (Yuan et al., 2020), as well our standard- and robust-trained teachers, respectively. We observe both standard and robust selfteachers are more discriminative than the other two baseline smoothenings, while the robust selfteacher is relatively more conservative as one shall expect.
+
+# 2.2 LEARNING TO SMOOTH WEIGHTS IN AT
+
+Rationale: Why Another measure that is often believed to indicate the standard generalization is the flatness: the loss surface at the final learned weights for well-generalizing models is relatively “flat”. Similarly, Wu et al. (2020a) advocated that a flatter adversarial loss landscape shrinks the robustness generalization gap. This is aligned with (Hein & Andriushchenko, 2017) where the authors called it local Lipschitz and proved that the Lipschitz constant can be used to formally measure the robustness of machine learning models. The flatness preference of a robust model has been echoed by many empirical defense methods, such as hessian/curvature-based regularization (Moosavi-Dezfooli et al., 2019), gradient magnitude penalty (Wang & Zhang, 2019), smoothening with random noise (Liu et al., 2018), or entropy regularization (Jagatap et al., 2020). However, all those methods will incur (sometimes heavy) computational or memory overhead; and many can cause standard accuracy drops, e.g., hessian/curvature-based methods (Gupta et al., 2020).
+
+Stochastic weight averaging (SWA) (Izmailov et al., 2018) was proposed to enforce the weight smoothness, by simply averaging multiple checkpoints along the training trajectory. SWA is known to find much flatter solutions than SGD, is extremely easy to implement, improves standard generalization, and has almost no computational overhead. SWA has been successfully adopted in semi-supervised learning (Athiwaratkun et al., 2018), Bayesian inference (Maddox et al., 2019), and low-precision training (Yang et al., 2019a). In this paper, we introduce SWA to AT for the first time, in order to smooth the weights and find flatter minima that may improve the adversarially robust generalization. Note that we choose SWA mainly due to its simplicity for proofs-of-concept; while extensively comparing alternative “flatness” regularizations is beyond our current work’s scope.
+
+One additional bonus of adopting SWA in AT is the temporal ensemble effect of SWA. It has been widely observed (Tramer et al., 2018; Grefenstette et al., 2018; Wu et al., 2020b; Wang et al., 2021) \` that training a model with the attack transferred from another could reduce “trivial robustness” caused by locally nonlinear loss surfaces, and therefore constructed model ensembles for a stronger defense. SWA was interpreted as approximating the fast geometric ensembling (Garipov et al., 2018), by aggregating multiple checkpoint weights at different training time. Applying SWA to AT therefore may lead to stronger and more transferable attacks, and consequently stronger defense due to ensembling, with the convenience of a single model.
+
+Approach: How Following (Izmailov et al., 2018), applying SWA to AT is straightforward:
+
+$$
+\mathcal { W } _ { \mathrm { S W A } } ^ { \mathrm { T } } = \frac { \mathcal { W } _ { \mathrm { S W A } } ^ { \mathrm { T - 1 } } \times n + \mathcal { W } ^ { \mathrm { T } } } { n + 1 } , \quad \mathcal { W } ^ { \mathrm { T } } = \mathcal { W } ^ { \mathrm { T - 1 } } + \Delta \mathcal { W } ^ { \mathrm { T } }
+$$
+
+![](images/55d0381a4940d7033662ebcbccb6fb389196573d892facda083d81f020c0e80e.jpg)  
+Figure 3: Results of testing accuracy over epochs for ResNet-18 trained on CIFAR-10, CIFAR-100, STL-10, and Tiny ImageNet. Dash / Solid lines show the standard accuracy (SA) / robust accuracy (RA). Blue, Green, Black and Orange curves represent the performance of Baseline, KD, SWA and KD&SWA respectively.
+
+where $\mathrm { T }$ indexs the training epoch, $n$ the number of past checkpoints to be averaged, WSWA the averaged network weight, $\mathcal { W }$ the current network weight, and $\Delta { \boldsymbol { \nu } } _ { }$ the SGD update.
+
+# 3 EXPERIMENT AND ANALYSIS
+
+Datasets We consider five datasets in our experiments: CIFAR-10, CIFAR-100 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), STL-10 (Coates et al., 2011) and Tiny-ImageNet (Deng et al., 2009). In all experiments, we randomly split the original training set into one training and one validation sets with a 9:1 ratio. Due to the limited space, we place the SVHN results in Appendix A1.3. The ablation studies and the visualizations are mainly on CIFAR-10 and CIFAR-100.
+
+Attack Methods We consider three representative attacks: FGSM (Goodfellow et al., 2015), PGD (Madry et al., 2018), and TRADES (Zhang et al., 2019b). All of them are applied with $\ell _ { 2 } , \epsilon =$ 128/255) or $( \ell _ { \infty } , \epsilon = 8 / 2 5 5 )$ ) setting as in (Madry et al., 2018), to generate adversarial samples. We use FSGM-1/PGD-10/TRADES-10 for training and PGD-20 for testing as the default setting, following Madry et al. (2018); Chen et al. (2020a). In addition, we use Auto-Attack (Croce & Hein, 2020) and CW Attack (Carlini & Wagner, 2017) for a more rigorous evaluation. More details are provided in the Appendix A2.2.
+
+Training and Evaluation Details For all experiments, we by default use ResNet-18 (He et al., 2016), with the exception of VGG-16 (Simonyan & Zisserman, 2014) and Wide-ResNet (Zagoruyko & Komodakis, 2016) adopted in Table 3. For training, we adopt an SGD optimizer with a momentum of 0.9 and weight decay of $5 \times 1 0 ^ { - 4 }$ , for a total of 200 epochs, with a batch size of 128. The learning rate starts from 0.1 (0.01 for SVHN (Rice et al., 2020)), decay to one-tenth at epochs 50 and 150 respectively. For Tiny-ImageNet, we train for 100 epochs, and the learning rate decay at epochs 50 and 80 with other settings unchanged. The self-training KD regularization is applied throughout the entire training, and SWA is employed after the first learning rate decay (when the robust overfitting usually starts to occur). We evaluate two common metrics that are widely adopted (Zhang et al., 2019b; Chen et al., 2020a): Standard Testing Accuracy (SA), and Robust Testing Accuracy (RA), which are the classification accuracies on the original and the attacked test sets, respectively.
+
+# 3.1 TACKLING ROBUST OVERFITTING
+
+Superior Performance Across Datasets Table 1 demonstrates our proposal on STL-10, CIFAR10, CIFAR-100, and Tiny-ImageNet. We consider PGD-AT (Madry et al., 2018) as Baseline; and denote our two training techniques as $+ K \mathcal { D } _ { \mathrm { s t d \& a d v } }$ (KD with standard and robust self-teachers), and $+ S W A$ , respectively. To numerically show the gap of robust overfitting, we also report the best RA values when early stopping during training, the final RA in the last epoch, and as the difference between final minus best. For reference, we also report the corresponding SA for the same best-RA checkpoint (not the best SA value throughout training), the final epoch SA, and their difference.
+
+We first observe that the robust overfitting prevails in all Baseline cases, with RA differences between final and best early-stopping values as large as $9 . 3 4 \%$ (CIFAR-10). In comparison, SA stays stable (with negative gaps on STL-10 and CIFAR-10) or continues to improve along with more training epochs (with small positive gaps on CIFAR-100 and Tiny-ImageNet). Fortunately, the gaps were significantly reduced by $+ K \mathcal { D } _ { \mathrm { s t d \& a d v } }$ ; and further diminished to only $0 . 4 \%$ to $0 . 6 \%$ when SWA is also applied.
+
+Table 1: Performance showing the occurrence of robust overfitting across datasets and the effectiveness of our proposed remedies with ResNet-18. The difference between best and final robust accuracy indicates degradation in performance during training. We pick the checkpoint which has the best robust accuracy on the validation set. The best results and the smallest performance differences are marked in bold.   
+
+<table><tr><td>Dataset</td><td>Settings</td><td>Robust Accuracy Best Final</td><td></td><td>/ (RA) Diff.</td><td>Best Final</td><td>Standard Accuracy (SA)</td><td>Diff.</td></tr><tr><td rowspan="3">STL-10</td><td>Baseline</td><td>36.24</td><td>32.20</td><td>4.04</td><td>56.50</td><td>63.69</td><td>-7.19</td></tr><tr><td>Baseline + KDstd&amp;adv</td><td>36.39</td><td>34.83</td><td>1.56</td><td>62.44</td><td>64.81</td><td>-2.37</td></tr><tr><td>Baseline + KDstd&amp;radv + SWA</td><td>36.46</td><td>35.90</td><td>0.56</td><td>62.76</td><td>64.24</td><td>-1.48</td></tr><tr><td rowspan="3">CIFAR-10</td><td>Baseline</td><td>50.72</td><td>41.38</td><td>9.34</td><td>80.78</td><td>82.44</td><td>-1.66</td></tr><tr><td>Baseline + KDstd&amp;adv</td><td>50.89</td><td>48.26</td><td>2.63</td><td>83.67</td><td>85.25</td><td>-1.58</td></tr><tr><td> Baseline + KDstd&amp;adv + SWA</td><td>52.14</td><td>51.53</td><td>0.61</td><td>84.65</td><td>85.40</td><td>-0.75</td></tr><tr><td rowspan="3">CIFAR-100</td><td>Baseline</td><td>27.32</td><td>19.84</td><td>7.48</td><td>53.90</td><td>53.56</td><td>0.34</td></tr><tr><td>Baseline + KDstd&amp;adv</td><td>27.56</td><td>26.02</td><td>1.54</td><td>57.42</td><td>60.34</td><td>-2.92</td></tr><tr><td>Baseline + KDstd&amp;adv + SWA</td><td>28.28</td><td>27.69</td><td>0.59</td><td>60.58</td><td>60.85</td><td>-0.27</td></tr><tr><td rowspan="3">Tiny-ImageNet</td><td>Baseline</td><td>19.81</td><td>13.43</td><td>6.38</td><td>45.85</td><td>45.58</td><td>0.27</td></tr><tr><td>Baseline + KDstd&amp;adv</td><td>21.45</td><td>19.25</td><td>2.20</td><td>48.98</td><td>51.15</td><td>-2.17</td></tr><tr><td> Baseline + KDstd&amp;adv + SWA</td><td>21.84</td><td>21.45</td><td>0.39</td><td>50.57</td><td>51.38</td><td>-0.81</td></tr></table>
+
+Further, we observe our methods to push the best RA higher by $0 . 2 2 \% \sim 2 . 0 3 \%$ . For example, the best RA on Tiny-ImageNet rises from $1 9 . 8 1 \%$ to $2 1 . 8 \hat { 4 } \%$ . Meanwhile, since there is no longer robust overfitting early in training, the best RA checkpoints become to select late epochs (often close to the end). Consequently, the SA values of the selected best RA models are all substantially improved. For example on CIFAR-100, the standard accuracy of our methods (best RA checkpoint) surpasses the baseline’s best RA checkpoint by $6 . 6 8 \%$ , and by $7 . 2 9 \%$ for the final checkpoint.
+
+Figure 3 further plots the RA and SA curves during training, from which we can clearly observe the diminishing of robust overfitting, after applying $\kappa D _ { \mathrm { s t d \& a d v } }$ , SWA and a combination of two methods. The training curves robustly improve until the end, without compromising the best achievable RA results, and further leads to a much-improved trade-off between RA and SA by avoiding early stopping (e.g., selecting an early checkpoint for RA, when SA might still be half-baked).
+
+Across Perturbations and Robustified Methods Our success can extend beyond PGD-AT. Table 2 presents more results in different perturbations (i.e. $\ell _ { 2 } , \ell _ { \infty } )$ and diverse robustified methods (i.e. FSGM in (Wong et al., 2020), TRADES in (Zhang et al., 2019b)). Consistent observations can be made: almost eliminated robust overfitting gaps, and significant gains on RA (by $0 . 6 1 \% \sim 3 . 1 1 \%$ and SA (by $1 . 8 0 \% \sim 4 . 2 2 \%$ .
+
+We also compare with previous state-of-the-art results in (Rice et al., 2020) under the same setting. As shown in Table A7 (Appendix), our methods shrink the gap between the RA best checkpoint and the final epoch RA from $5 . 7 0 \%$ to $0 . 1 7 \%$ and simultaneously improve $4 . 5 0 \%$ by RA and $3 . 0 4 \%$ by SA. More results can be found in Appendix A1.
+
+Across Architectures and Improved Attacks Table 3 demonstrates the effectiveness of our methods across different architectures, including VGG-16, Wide-ResNet-34-4, and Wide-ResNet-34-10. Specifically, our methods reduce the drop of robust accuracy from $5 . 8 3 \%$ to $0 . 0 6 \%$ with VGG-16 on CIFAR-10 while achieving an extra robust accuracy improvement of $2 . 5 7 \%$ , $1 . 6 9 \%$ and $1 . 2 3 \%$ with VGG-16, Wide-ResNet-34-4 and Wide-ResNet-34-10 on CIFAR10, respectively. To further verify the improvements achieved by our methods, we conduct extra evaluations under improved attacks. As shown in Table 4, after applying the combination of KD and SWA, the overfitting problem is largely mitigated under both Auto-Attack (Croce & Hein, 2020) and CW attack (Carlini & Wagner, 2017). Take CIFAR- $1 0 \ell _ { \infty }$ adversary as an example, our approaches shrink the drop of robust accuracy from $7 . 0 4 \%$ to $- 0 . 0 9 \%$ under Auto-Attack, and $1 4 . { \bar { 9 6 \% } }$ to $0 . 7 9 \%$ under CW attack, when comparing the best model to the eventually converged model. These results indicate that our methods can generalize to different architectures and improved attacks.
+
+Excluding Obfuscated Gradients An often argued “counterfeit” of improved robustness is caused by less effectiveness of generated adversarial examples due to obfuscated gradients (Athalye et al., 2018). To exclude this possibility, we show that our methods maintain improved robustness under unseen transfer attacks. To start with, the left figure in Figure 4 shows the transfer testing performance on an unseen robust model (here we use a separately robustified ResNet-50 with PGD-10 on CIFAR-100), using attacks generated by the different epoch checkpoints of PGD-AT Baseline, Baseline $+ \mathcal { K } \mathcal { D } _ { \mathrm { s t d \& \mathrm { a d v } } }$ , and Baseline $+ \mathrm { \Delta } K \mathrm { \mathcal { D } } _ { \mathrm { s t d \& a d v } } + \mathrm { \Delta } \mathrm { S W A } .$ . A higher robust accuracy on the unseen robust model corresponds to a weaker attack. Apparently, our methods consistently yield stronger and more transferable attacks, while the attacking quality generated by the baseline quickly drops with deteriorated transferability. Similarly, the right figure of Figure 4 transfers the attack from
+
+Table 2: Controlled experiments on CIFAR-10. The difference between best and final robust accuracy indicates degradation in performance during training. We pick the checkpoint which has the best robust accuracy on the validation set. The best results and the smallest performance difference are marked in bold.   
+
+<table><tr><td rowspan="2">Adversary</td><td rowspan="2">Norm</td><td rowspan="2">Radius</td><td rowspan="2">Settings</td><td rowspan="2"></td><td colspan="2">Robust Accuracy ( (RA)</td><td colspan="3"> Standard Accuracy (SA)</td></tr><tr><td>Best Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td>PGD</td><td>l2</td><td>∈=</td><td>128 255</td><td>Baseline Our Methods</td><td>67.26 65.93 70.37 70.23</td><td>1.33 0.14</td><td>88.78 90.58</td><td>88.86 90.48</td><td>-0.08 0.10</td></tr><tr><td>FGSM</td><td>l2</td><td>∈二</td><td>128 255</td><td>Baseline Our Methods</td><td>66.48 64.12 69.53 69.49</td><td>2.36 0.04</td><td>88.77 91.11</td><td>89.40 91.15</td><td>-0.63 -0.04</td></tr><tr><td>FGSM</td><td>l8</td><td>e=</td><td>8 255</td><td>Baseline Our Methods</td><td>43.14 37.09 43.75 43.74</td><td>6.05 0.01</td><td>85.77 88.53</td><td>86.51 88.72</td><td>-0.74 -0.19</td></tr><tr><td>TRADES</td><td>l2</td><td>e=</td><td>128 255</td><td>Baseline Our Methods</td><td>69.07 64.81 70.30 69.60</td><td>4.26 0.70</td><td>85.43 89.65</td><td>85.61 89.55</td><td>-0.18 0.10</td></tr><tr><td>TRADES</td><td>l8</td><td>e=</td><td>8 255</td><td>Baseline Our Methods</td><td>51.07 47.32 52.92 52.28</td><td>3.75 0.64</td><td>79.67 82.95</td><td>81.81 83.31</td><td>-2.14 -0.36</td></tr></table>
+
+Table 3: Controlled experiments across different architecture on CIFAR-10/100 under $\ell _ { \infty }$ adversary. The difference between best and final robust accuracy indicates degradation in performance during training. We pick the checkpoint which has the best robust accuracy on the validation set. The best results and the smallest performance difference are marked in bold.   
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Dataset</td><td rowspan="2"> Settings</td><td colspan="3">Robust Accuracy ( (RA)</td><td colspan="3"> Standard Accuracy (SA)</td></tr><tr><td>Best</td><td>Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td rowspan="2">VGG-16</td><td>CIFAR-10</td><td>Baseline</td><td>46.42</td><td>40.59</td><td>5.83</td><td>75.29</td><td>79.54</td><td>-4.25</td></tr><tr><td rowspan="2"></td><td>Our Methods</td><td>48.99</td><td>48.93</td><td>0.06</td><td>79.00</td><td>79.69</td><td>-0.69</td></tr><tr><td>CIFAR-100</td><td>Baseline Our Methods</td><td>21.64 24.79</td><td>17.43</td><td>4.21</td><td>39.26</td><td>45.84</td><td>-6.58</td></tr><tr><td rowspan="2">WRN-34-4</td><td></td><td>Baseline</td><td></td><td>24.73</td><td>0.06</td><td>48.20</td><td>49.00</td><td>-0.80</td></tr><tr><td rowspan="2">CIFAR-10</td><td>Our Methods</td><td>52.59 54.28</td><td>43.06 53.90</td><td>9.53 0.38</td><td>81.53 85.17</td><td>83.28</td><td>-1.75</td></tr><tr><td rowspan="2"></td><td>Baseline</td><td>28.02</td><td>20.61</td><td>7.41</td><td>53.19</td><td>85.50 53.63</td><td>-0.33 -0.44</td></tr><tr><td rowspan="2">CIFAR-100</td><td>Our Methods</td><td>30.10</td><td>29.80</td><td>0.30</td><td>57.23</td><td>58.05</td><td>-0.82</td></tr><tr><td>CIFAR-10</td><td>Baseline</td><td>54.27</td><td>47.12</td><td>7.15</td><td>84.16</td><td></td><td></td></tr><tr><td rowspan="2">WRN-34-10 WRN-34-10</td><td></td><td>Our Methods</td><td>55.50</td><td>55.34</td><td>0.16</td><td>86.81</td><td>85.72 87.06</td><td>-1.56 -0.25</td></tr><tr><td rowspan="2">CIFAR-100</td><td>Baseline</td><td>29.95</td><td>24.02</td><td>5.93</td><td>56.56</td><td>56.42</td><td>0.14</td></tr><tr><td></td><td>Our Methods</td><td>31.93</td><td>31.51</td><td>0.42</td><td>60.86</td><td>61.78</td><td>-0.92</td></tr></table>
+
+Table 4: Evaluation under improved attacks on CIFAR-10/100 with ResNet-18. The difference between best and final robust accuracy indicates degradation in performance during training. We pick the checkpoint which has the best robust accuracy under PGD-20 attack on the validation set. The best results and the smallest performance difference are marked in bold.   
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Norm</td><td rowspan="2">Radius</td><td rowspan="2">Settings</td><td colspan="3">Auto-Attack</td><td colspan="3">CW Attack</td></tr><tr><td>Best</td><td>Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td rowspan="2">CIFAR-10</td><td rowspan="2">l2</td><td>128 e=</td><td>Baseline</td><td>67.18</td><td>64.29</td><td>2.89</td><td>73.80</td><td>53.77</td><td>20.03</td></tr><tr><td>255</td><td>Our Methods</td><td>68.87</td><td>68.90</td><td>-0.03</td><td>73.89</td><td>73.79</td><td>0.10</td></tr><tr><td rowspan="2">CIFAR-10</td><td rowspan="2">l8</td><td>8 e=</td><td>Baseline</td><td>47.00</td><td>39.96</td><td>7.04</td><td>75.48</td><td>60.52</td><td>14.96</td></tr><tr><td>255</td><td>Our Methods</td><td>49.35</td><td>49.44</td><td>-0.09</td><td>77.83</td><td>77.04</td><td>0.79</td></tr><tr><td rowspan="2">CIFAR-100</td><td rowspan="2">l2</td><td>128 ∈=</td><td>Baseline</td><td>37.16</td><td>33.43</td><td>3.73</td><td>48.43</td><td>37.73</td><td>10.70</td></tr><tr><td>255</td><td>Our Methods</td><td>40.56</td><td>40.61</td><td>-0.05</td><td>51.02</td><td>50.90</td><td>0.12</td></tr><tr><td rowspan="2">CIFAR-100</td><td rowspan="2">l8</td><td>8 E=</td><td>Baseline</td><td>22.73</td><td>18.11</td><td>4.62</td><td>45.89</td><td>37.76</td><td>8.13</td></tr><tr><td>255</td><td>Our Methods</td><td>25.42</td><td>25.35</td><td>0.07</td><td>49.46</td><td>49.07</td><td>0.39</td></tr></table>
+
+![](images/1938c5e9abe2c3f365eb73c25cfbc22dba4aed1edecbbd24674ec9da84484812.jpg)  
+Figure 4: (Left) Transfer attack performance on an unseen robust model, where attacks are generated by Baseline, KD, and KD&SWA’s different epoch checkpoints. (Right) Transfer attack performance on models from Baseline, KD, and KD&SWA, where attacks are generated by an unseen robust model. The unseen robust model is a ResNet-50 trained by PGD-10. All experiments are conducted on the CIFAR-100 dataset.
+
+Table 5: Ablation studies on CIFAR-10 with ResNet-18. Compared with Baseline (PGD-AT) methods, the performance improvements and degradations by adding each component are reported in red and blue numbers.   
+
+<table><tr><td rowspan="2">Settings</td><td colspan="3">Robust Accuracy (RA)</td><td colspan="3">Standard Accuracy (SA)</td></tr><tr><td>Best</td><td>Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td>Baseline</td><td>50.72</td><td>41.38</td><td>9.34</td><td>80.78</td><td>82.44</td><td>-1.66</td></tr><tr><td rowspan="3">KDstd KDadv SWA</td><td>48.72(↓ 2.00)</td><td>43.08(个 1.70)</td><td>5.64</td><td>83.40(个 2.62)</td><td>84.80(↑ 2.36)</td><td>-1.40</td></tr><tr><td>51.01 (个0.29)</td><td>47.72(个 6.34)</td><td>3.29</td><td>82.18(↑ 1.40)</td><td>83.51(个 1.07)</td><td>-1.33</td></tr><tr><td>51.65(个 0.93)</td><td>48.21(↑ 6.83)</td><td>3.44</td><td>83.42(个 2.64)</td><td>84.12(个 1.68)</td><td>-0.70</td></tr><tr><td>KDstd&amp;adv</td><td>50.89(↑ 0.17)</td><td>48.26(↑ 6.88)</td><td>2.63</td><td>83.67(个 2.89)</td><td>85.25(个 2.81)</td><td>-1.58</td></tr><tr><td> KDstd&amp;adv + SWA</td><td>52.14(个 1.42)</td><td>51.53(个 10.15)</td><td>0.61</td><td>84.65(个 3.87)</td><td>85.40(个 2.96)</td><td>-0.75</td></tr></table>
+
+an unseen robust model to the above three methods, while our methods consistently defend better.   
+Those empirical pieces of evidence suggest that our RA gains are not a result of gradient masking.
+
+# 3.2 ABLATION STUDY AND VISUALIZATION
+
+$\kappa D _ { \mathrm { a d v } }$ , $\kappa \mathcal { D } _ { \mathrm { s t d } }$ and SWA We study the effectiveness of each component in logit and weight smoothening. We also specifically decompose $\kappa D _ { \mathrm { s t d \& a d v } }$ into two ablation methods: $\kappa \mathcal { D } _ { \mathrm { s t d } }$ (by setting $\lambda _ { 2 } = 0$ in Eqn. (1)), and $\kappa D _ { \mathrm { a d v } }$ (by setting $\lambda _ { 1 } = 0$ ), respectively. Table 5 shows that $\kappa D _ { \mathrm { s t d } }$ , ${ \boldsymbol { \kappa } } { \boldsymbol { D } } _ { \mathrm { a d v } }$ and SWA all substantially contribute to suppressing the robust overfitting and enhancing the SA-RA trade-off. We notice that while $\kappa \mathcal { D } _ { \mathrm { s t d } }$ seems to (understandably) sacrifice the best RA a bit for improving TA, combining it with $\kappa D _ { \mathrm { a d v } }$ brings the RA compromise back and boosts them both.
+
+Table 6: Ablation of label smoothing versus $\kappa D$ on CIFAR-10.   
+
+<table><tr><td rowspan="2">Settings</td><td colspan="3">Robust Accuracy (RA)</td><td colspan="3">Standard Accuracy (SA)</td></tr><tr><td>Best</td><td>Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td>Label Smoothing</td><td>50.75</td><td>44.73</td><td>6.02</td><td>81.28</td><td>83.54</td><td>-2.26</td></tr><tr><td>Tf-KDreg</td><td>50.79</td><td>45.65</td><td>5.14</td><td>79.63</td><td>83.56</td><td>-3.93</td></tr><tr><td>KDstd&amp;adv-FGSM</td><td>50.81</td><td>48.32</td><td>2.49</td><td>84.79</td><td>85.59</td><td>-0.80</td></tr><tr><td>KDstd&amp;adv-PGD10</td><td>50.89</td><td>48.26</td><td>2.63</td><td>83.67</td><td>85.25</td><td>-1.58</td></tr><tr><td>KDstd&amp;adv-PGD100</td><td>50.92</td><td>48.47</td><td>2.45</td><td>83.31</td><td>84.64</td><td>-1.33</td></tr></table>
+
+Naive LS versus learned logit smoothening As $\kappa D$ could be viewed as a learned version of LS (Yuan et al., 2020), we next quantify the benefit of using $\kappa D _ { \mathrm { s t d \& a d v } }$ , compared to naive LS (Szegedy et al., 2016), and the teacher-free knowledge distillation regularization (Tf
+
+$\mathrm { K D } _ { \mathrm { r e g } } )$ ) in (Yuan et al., 2020), all incorporated with PGD-AT on CIFAR-10. Table 6 show that both naive LS and $\mathrm { T f - K D } _ { \mathrm { r e g } }$ also reduce robust overfitting to some extent, but far less competitive than $\kappa D _ { \mathrm { s t d \& a d v } }$ . Moreover, the robustness gains of naive LS and $\mathrm { T f - K D } _ { \mathrm { r e g } }$ no longer hold under transfer attacks, implying that they are susceptible to obfuscated gradients. Further visualization in Figure 5 demonstrates that our methods smooth the logits without compromising the class-wise discriminative information, while naive LS and Tf-KD might suffer from weaker gradients here.
+
+Quality of Self-Teachers An extra price for our learned logit smoothening is the pre-training of self-teachers, although this is already quite common in similar literature (Chen et al., 2020a;b). To further reduce this burden, we explore whether high-quality and more expensive pre-training is necessary for us, and fortunately find that is not the case. For example, Table 6 shows only marginal performance difference when the robust self-teacher is pre-trained using FGSM or PGD-10/100.
+
+![](images/f7651e188f85fac592057e60f8508a67441fa2545a84819507286cc419520801.jpg)  
+Figure 5: t-SNE results of different logits smoothing approaches on CIFAR-10. Dots and stars represent for clean and adversarial images respectively. Orange, Blue and Green represent classes A, B and C respectively. For each class, we visualize all testing images and their corresponding adversarial images from PGD-20.
+
+![](images/a05f5075fa91e62db719b009c439376db71444903655ac34d131a2589dc7ece4.jpg)  
+Figure 6: Comparison of loss landscapes of models trained by baseline (the first row) and our methods (the second row). Loss plots in each column are generated from the same original image randomly chosen from the CIFAR-100 test dataset. $z$ axis denotes the loss value. Following the settings in (Engstrom et al., 2018), we plot the loss landscape function: $z = \log ( x \cdot r _ { 1 } + y \cdot r _ { 2 } )$ , where $r _ { 1 } = \mathrm { s i g n } ( \nabla _ { i } f ( i ) )$ and $r _ { 2 } \sim$ Rademacher(0.5). Here $i$ denostes the original image, and $f ( \cdot )$ a trained model whose inputs are scaled to [0, 1].
+
+Visualizing Flatness and Local Linearity We expect SWA to find flatter minima for AT to improve its generalization, and we show it to indeed happen by visualizing the loss landscape w.r.t. both input and weight spaces. Figure 6 shows that our methods notably flatten the rugged landscape w.r.t. the input space, compared to the PGD-AT baseline, which aligns with the robust generalization claims in (Moosavi-Dezfooli et al., 2019; Wu et al., 2020a). Figure 7 follows (Izmailov et al., 2018) to perturb the trained model in the weight space and show how the robust testing loss changes over the perturbation radius. We perturb 10 different random directions at each different $\ell _ { 2 }$ distance. Our methods present better weight smoothness around the achieved local minima too, which suggests improved generalization (Dinh et al., 2017; Petzka et al., 2019).
+
+![](images/d8bba0b60785c739849c8c5ee73499c38b769b1cb5fdb3789cbdf5e343ac57d8.jpg)  
+Figure 7: Robust testing loss as a function of perturbed weight distance, starting from models trained by PGD-AT baseline and our methods on CIFAR-100.
+
+We additionally look at the local linearity measurement proposed in (Andriushchenko & Flammarion, 2020), which originally addresses catastrophic overfitting in fast AT. As shown in Figure A11, our methods also achieve consistently better local linearity.
+
+# 4 CONCLUSION
+
+This paper takes one more step towards addressing the recently discovered robust overfitting issue in AT. We present two empirical solutions to smooth the logits and weights respectively; both are motivated by successful practice in improving standard generalization, and we adapt them for AT. While Rice et al. (2020) found simpler regularizations unable to fix robustness overfitting, our learned smoothening regularization seems to largely mitigate that. Extensive experiments show our proposal to establish new state-of-the-art performance on AT. While promising progress has been made, the underlying cause of robust overfitting is not yet fully explained. Our future work will connect to more theoretical understandings of this issue (Wang et al., 2019; 2020b).
+
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+
+# A1 MORE EXPERIMENT RESULTS
+
+# A1.1 STATE-OF-THE-ART BENCHMARK ON CIFAR-100
+
+We implement our methods with exactly the same setting as (Rice et al., 2020) and compare it with the baseline result reported from the original paper. As shown in Table A7 and Figure A8, our methods achieve great improvements both on robust accuracy and standard accuracy ( $1 . 6 4 \%$ in RA and $3 . 7 8 \%$ in SA for $\ell _ { \infty }$ , $4 . 5 0 \%$ in RA and $3 . 0 4 \%$ in SA for $\ell _ { 2 }$ ), which establish a new state-of-theart bar.
+
+Table A7: Comparative Experiment on CIFAR-100, we follow the same setting and compare with the baseline result from (Rice et al., 2020). Best refers to the model with best robust accuracy during training and Final is an average of accuracy over last 5 epochs.   
+
+<table><tr><td rowspan="2">Adversary</td><td rowspan="2">Norm</td><td rowspan="2">Radius</td><td rowspan="2">Settings</td><td colspan="3">Robust Accuracy (RA)</td><td rowspan="2">Standard Accuracy</td><td colspan="2">(SA)</td></tr><tr><td>Best</td><td>Final</td><td>Diff. Best</td><td>Final</td><td>Diff.</td></tr><tr><td rowspan="2">PGD</td><td>l2</td><td>∈= 128</td><td>Baseline</td><td>43.20</td><td>37.50±0.09</td><td>5.70</td><td>62.50</td><td>60.10±0.22</td><td>2.40</td></tr><tr><td></td><td>255</td><td>Our Methods</td><td>47.70</td><td>47.53±0.03</td><td>0.17</td><td>65.54</td><td>65.56±0.01</td><td>-0.02</td></tr><tr><td rowspan="2">PGD</td><td>l</td><td>8</td><td>Baseline</td><td>28.10</td><td>21.40±0.39</td><td>6.70</td><td>52.70</td><td>54.10±0.23</td><td>-1.40</td></tr><tr><td></td><td>e=255</td><td>Our Methods</td><td>29.74</td><td>29.40±0.02</td><td>0.34</td><td>56.48</td><td>57.69±0.03</td><td>-1.21</td></tr></table>
+
+![](images/d7e6089ea2b5d13c2730b2f566c80e69f249ed99cd6e0a8a26d9f2a8e059de62.jpg)  
+Figure A8: Results of testing accuracy over epochs for ResNet-18 trained on CIFAR-100 with the same setting as Rice et al. (2020). Dash lines show the standard accuracy (SA); solid lines represent the robust accuracy (RA). Blue, Green and Orange curves represent the performance of Baseline, KD and KD&SWA respectively.
+
+# A1.2 T-SNE RESULT ON CIFAR-100
+
+We visualize the learned feature space with all training images and their corresponding adversarial images from PGD-10 on CIFAR-100. As shown in Figure A9, our learned features have a larger distance between classes while being more clustered within the same class. The more distinguishable feature embedding justifies the improvement of both robust and standard accuracy.
+
+# A1.3 SUPERIOR PERFORMANCE ON SVHN
+
+We conduct our experiments on SVHN with ResNet-18 (He et al., 2016) architecture and adopt an SGD optimizer with a momentum of 0.9 and a weight decay of $5 \times 1 0 ^ { - 4 }$ for 80 epochs in total with a batch size of 128. The learning rate starts from 0.01 and follows a cosine annealing schedule. The result can be found on Table A8 and Figure A10. As we can see, the robust accuracy of the best checkpoint for $\ell _ { \infty }$ is improved from $5 2 . 6 0 \%$ to $5 3 . 6 5 \%$ , and robust overfitting is alleviated by $6 . 3 0 \%$ . In the meantime, standard accuracy has also been improved by $2 . 4 7 \%$ . The superior performance on SVHN aligns with results on other datasets, which shows the effectiveness of our methods.
+
+![](images/a88d853639330843e71e1830d682c9f9949c2b9b439b14fd23f5f84c36964ca6.jpg)  
+Figure A9: t-SNE results of different models trained on CIFAR-100. Dots and stars represent for clean and adversarial images respectively. Red, Blue and Green represent classes A, B and C respectively. For each class, we visualize all training images and their corresponding adversarial images from PGD-10. The left figure is Baseline; the right figure is Our Methods.
+
+![](images/e21e8ec3480c4c96d61176fa2f88e892a3dcc14466178057e7510e60f63a3acd.jpg)  
+Figure A10: Results of testing accuracy over epochs for ResNet-18 trained on SVHN. Dash lines show the standard accuracy (SA); solid lines represent the robust accuracy (RA). Blue, Green, Black and Orange curves represent the performance of Baseline, KD, SWA and KD&SWA respectively.
+
+Table A8: Performance showing the occurrence of robust overfitting and effectiveness of our proposed remedies with ResNet-18 on SVHN. The difference between best and final robust accuracy indicates degradation in performance during training. We pick the checkpoint which has the best robust accuracy on validation dataset. The best results and the minimum performance difference are marked in bold.   
+
+<table><tr><td>Dataset</td><td>Settings</td><td>Robust Accuracy (RA) Best Final</td><td>Diff.</td><td></td><td>Standard Accuracy (SA) Best Final</td><td></td><td>Diff.</td></tr><tr><td rowspan="2">SVHN</td><td>Baseline</td><td>52.60</td><td>43.30</td><td>9.30</td><td>87.93</td><td>89.94</td><td>-2.01</td></tr><tr><td>Baseline + KDstd&amp;adv Baseline + KDstd&amp;adv + SWA</td><td>52.93 53.65</td><td>48.46 50.65</td><td>4.47 3.00</td><td>87.62 90.40</td><td>91.36 91.70</td><td>-3.74 -1.30</td></tr></table>
+
+# A1.4 LOCAL LINEARITY
+
+As proposed by (Andriushchenko & Flammarion, 2020), the catastrophic overfitting problem is mainly due to the local linearity reduction when adversarial training with FGSM(Rice et al., 2020). So we borrow this measurement in the robust overfitting scenario, which calculates the expectation of the cosine similarity of the gradient between the original input and randomly perturbed one with a uniform distribution, as shown in Eqn. 3. The result shown in figure A11 indicates that our methods help to slow the decline of local linearity and the maintenance of local linearity is also helpful for preventing robust overfitting.
+
+$$
+\mathbb { E } _ { ( \mathbf { x } , y ) \in \mathcal { D } , \eta \in \mathcal { U } ( [ - \epsilon , \epsilon ] ^ { d } ) } \bigg [ \cos \Big ( \nabla _ { \mathbf { x } } \mathcal { L } \big ( f ( \pmb { \theta } , \mathbf { x } ) , y \big ) , \nabla _ { \mathbf { x } } \mathcal { L } \big ( f ( \pmb { \theta } , \mathbf { x } + \pmb { \eta } ) , y \big ) \Big ) \bigg ]
+$$
+
+![](images/d44726726354895d815bb7c5a33ec203b58e82db82fb9aafd2d77f5eaef75e82.jpg)  
+Figure A11: The local linearity is calculated from all test images on CIFAR-100 with models from each period of the training process. Blue, Green and Orange curves represent the local linearity of Baseline, KD and KD&SWA respectively.
+
+# A1.5 ABLATION OF TRANSFER ATTACK
+
+With the purpose of fully comparing the effects of label smoothing and knowledge distillation, we introduce a transfer attack with an unseen non-robust model with the same architecture, follow the same setting as (Fu et al., 2020). A higher accuracy on the unseen model indicates a weaker attack generated by the corresponding setting while a higher accuracy from the unseen model means better robustness. As shown in Table A9, only knowledge distillation shows significant improvement with both accuracies, compared with baseline(PGD-AT) methods. The strength of generated adversarial images is improved by $4 . 3 2 \%$ and the robustness is improved by $2 . 9 1 \%$ for the best model. We also experiment with an unseen robust model and get consistent improvement. This improvement indicates that knowledge distillation introduces more discriminating information from teacher models, which is better than manually designed label smoothing methods.
+
+Table A9: Ablation of Transfer attack. The accuracy on unseen model is the accuracy of unseen model with adversarial images generated by source models from different settings and the accuracy from unseen model means the opposite. We generated adversarial images for all test images on CIFAR-10 with $\ell _ { \infty }$ PGD-20. Baseline represents the PGD-AT methods.   
+
+<table><tr><td rowspan="2">Settings</td><td colspan="2">Accuracy on unseen model</td><td colspan="2">Accuracy from unseen model</td></tr><tr><td>Best</td><td>Final</td><td>Best</td><td>Final</td></tr><tr><td>Baseline</td><td>69.87</td><td>80.43</td><td>79.72</td><td>81.77</td></tr><tr><td>Label smoothing</td><td>70.94</td><td>81.24</td><td>78.44</td><td>82.82</td></tr><tr><td>Tf-KDreg</td><td>73.29</td><td>82.21</td><td>79.47</td><td>82.71</td></tr><tr><td>KDstd&amp;adv-PGD10</td><td>65.55</td><td>72.63</td><td>82.63</td><td>84.01</td></tr></table>
+
+# A1.6 SWA VERSUS ISWA
+
+One possible extension of SWA is to replace $\mathcal { W } ^ { \mathrm { T - 1 } }$ with $\mathcal { W } _ { \mathrm { S W A } } ^ { \mathrm { T - 1 } }$ in Eqn. 2. We name this variant as iSWA and compare it with the original SWA in Table A10. Both weight smoothing techniques can mitigate robust overfitting, and iSWA performs slightly better on RA while sacrificing some SA.
+
+Table A10: Ablation of SWA on CIFAR-10.Best refers to the model selected with best robust accuracy on validation dataset and Final is the model at the end of training process.   
+
+<table><tr><td rowspan="2">Settings</td><td colspan="3">Robust Accuracy (RA)</td><td colspan="3"> Standard Accuracy (SA)</td></tr><tr><td>Best</td><td>Final</td><td>Diff.</td><td>Best</td><td>Final</td><td>Diff.</td></tr><tr><td>KDstd&amp;adv-PGD10 + SWA</td><td>52.14</td><td>51.53</td><td>0.61</td><td>84.65</td><td>85.40</td><td>-0.75</td></tr><tr><td>KDstd&amp;adv-PGD10 + iSWA</td><td>52.36</td><td>52.33</td><td>0.03</td><td>83.17</td><td>83.54</td><td>-0.37</td></tr></table>
+
+# A2 MORE METHODOLOGY AND IMPLEMENTATION DETAILS
+
+# A2.1 KNOWLEDGE DISTILLATION
+
+We state $\kappa D$ as follows:
+
+$$
+\begin{array} { c } { \displaystyle \mathcal { K D } ( { \pmb y } , \hat { \pmb y } ) = - \mathcal { H } ( t ( { \pmb y } ) , t ( \hat { \pmb y } ) ) } \\ { = - \displaystyle \sum _ { j } t ( { \pmb y } ) _ { j } \log t ( \hat { \pmb y } ) _ { j } } \end{array}
+$$
+
+where t(y)i $\begin{array} { r } { t ( \pmb { y } ) _ { i } = \frac { ( \pmb { y } _ { i } ) ^ { 1 / \operatorname { T } } } { \sum _ { j } ( \pmb { y } _ { j } ) ^ { 1 / \operatorname { T } } } } \end{array}$ = (yi)1/TPj (yj )1/T , T = 2 in our case, following the standard setting in (Hinton et al., 2015;   
+Li & Hoiem, 2017).
+
+# A2.2 ADVERSARIAL TRAINING
+
+Adversarial training incorporates generated adversarial examples into the training process and significantly improves the robustness of networks. In our paper, we implemented three different adversarial training schemes: FGSM, PGD and TRADES, which can be described as the optimization problem below: Eqn.4 for FGSM and PGD, Eqn.5 for TRADES.
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathbb { E } _ { ( \mathbf { x } , y ) \in \mathcal { D } } \operatorname* { m a x } _ { \pmb { \delta } \in \mathrm { B } _ { \epsilon } ( \mathbf { x } ) } \mathcal { L } \big ( f ( \pmb { \theta } , \mathbf { x } + \pmb { \delta } ) , y \big )
+$$
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathbb { E } _ { ( \mathbf { x } , \pmb { y } ) \in \mathcal { D } } \bigg [ \mathcal { L } \big ( f ( \pmb { \theta } , \mathbf { x } ) , \pmb { y } \big ) + \beta \cdot \operatorname* { m a x } _ { \pmb { \delta } \in \mathrm { B } _ { \epsilon } ( \mathbf { x } ) } \mathcal { K L } \big ( f ( \pmb { \theta } , \mathbf { x } + \pmb { \delta } ) , f ( \pmb { \theta } , \mathbf { x } ) \big ) \bigg ]
+$$
+
+As for the maximization process, FGSM perturbs the input with a single step in the direction of the sign of the gradient and PGD is the iterative form of FGSM with random restarts, which works as follows.
+
+$$
+\delta ^ { t + 1 } = \mathrm { p r o j } _ { \mathcal { P } } \Big ( \delta ^ { t } + \alpha \cdot \mathrm { s g n } \big ( \nabla _ { \mathbf { x } } \mathcal { L } ( f ( \pmb { \theta } , \mathbf { x } + \delta ^ { t } ) , y ) \big ) \Big )
+$$
+
+$$
+\delta ^ { t + 1 } = \mathrm { p r o j } _ { \mathcal { P } } \Big ( \delta ^ { t } + \alpha \cdot \mathrm { s g n } \big ( \nabla _ { \mathbf { x } } \mathcal { K } \mathcal { L } ( f ( \pmb { \theta } , \mathbf { x } + \delta ) , f ( \pmb { \theta } , \mathbf { x } ) \big ) \Big )
+$$
+
+TRADES (Eqn.7) replaces the cross entropy loss in PGD with the Kull-back–Leibler divergence of network output for the clean input and the adversarial input. Where f is the network with parameters $\theta , ( \mathbf { x } , y )$ is the data. $\alpha$ is the step size and $\delta ^ { t }$ is the adversarial perturbation after t times iterations. The perturbation is constrained in an $\ell _ { p }$ norm ball, i.e. $\| \delta \| _ { p } \leq \epsilon$ , which is realized by projection. we consider both $\ell _ { \infty }$ and $\ell _ { 2 }$ in our paper. For $\ell _ { \infty }$ adversary, we use $\begin{array} { r } { \epsilon = \frac { 8 } { 2 5 5 } } \end{array}$ and $\textstyle \alpha = { \frac { 2 } { 2 5 5 } }$ for PGD and TRADES with 10 steps in training and 20 steps in testing, while using $\begin{array} { r } { \alpha = \frac { 7 } { 2 5 5 } } \end{array}$ for FGSM during training. As for $\ell _ { 2 }$ adversary, we use $\begin{array} { r } { \epsilon = \frac { 1 2 8 } { 2 5 5 } } \end{array}$ and $\textstyle \alpha = { \frac { 1 5 } { 2 5 5 } }$ with the same steps as $\ell _ { \infty }$ adversary in all three attack methods.
+
+For a comprehensive evaluation, we consider two improved attacks, i.e., Auto-Attack (Croce & Hein, 2020) and CW Attack (Carlini $\&$ Wagner, 2017). We use the official implementation and default settings for Auto-Attack ( $\ell _ { \infty }$ with $\epsilon = \frac { 8 } { 2 5 5 }$ and $\ell _ { 2 }$ with $\begin{array} { r } { \epsilon = \frac { 1 2 8 } { 2 5 5 } . } \end{array}$ ) and the implementation from AdverTorch (Ding et al., 2019) for CW attack with the same setting as Rony et al. (2019), specifically, 1 search step on C with an initial constant of 0.1, with 100 iterations for each search step and 0.01 learning rate. Detailed links are provided below:
+
+• The Official Repository: https://github.com/fra31/auto-attack • The Leaderboard: https://robustbench.github.io/